Integrated Math II Honors

Unit 1 10 topics

Number Systems

The Number System, and Introduction to Complex Numbers.

1.1 The Number System

Every natural number is whole, every whole number is an integer, and every integer is rational. Rational and irrational numbers sit side by side and together make up the reals.

1.1.1
The Real Number System
Natural numbers are $1, 2, 3, \dots$ ; whole numbers add $0$ ; integers add the negatives. Rational numbers are fractions $\tfrac{p}{q}$ of integers ( $q \ne 0$ ), with decimals that end or repeat. Irrational numbers have decimals that never end or repeat, and rationals and irrationals together make up the real numbers.
e.g. $0.75 = \tfrac34$ and $-5$ are rational; $\sqrt2$ and $\pi$ are irrational
$\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
$\displaystyle \mathbb{R} = \mathbb{Q} \cup \text{irrationals}$
1.1.2
Writing Repeating Decimals as Fractions
Set $x$ equal to the decimal. Multiply by $10^n$ , where $n$ is the number of repeating digits, and subtract $x$ so the repeating tail cancels. If the repeat starts later, first multiply to move the decimal point to where the repeat begins.
e.g. $x = 0.1\overline{6}$ : $100x - 10x = 16.\overline{6} - 1.\overline{6} = 15$ , so $x = \tfrac{15}{90} = \tfrac16$
$\displaystyle x = 0.\overline{36}$
$\displaystyle 100x = 36.\overline{36}$
$\displaystyle 100x - x = 36$
$\displaystyle \Rightarrow\; x = \frac{36}{99} = \frac{4}{11}$
1.1.3
Sums and Products of Rational and Irrational Numbers
Rational plus rational and rational times rational are always rational. Rational plus irrational is irrational, and a nonzero rational times an irrational is irrational. Two irrationals can give either kind.
e.g. $3 + \sqrt2$ is irrational; $\sqrt2 \cdot \sqrt2 = 2$ is rational; $\sqrt2 \cdot \sqrt3 = \sqrt6$ is irrational

1.2 Introduction to Complex Numbers

The number 3 + 2i sits 3 units along the real axis and 2 units up the imaginary axis. Each multiplication by i turns a quarter turn, so the powers of i cycle i, −1, −i, 1.

1.2.1
Imaginary Numbers
The imaginary unit $i$ is defined by $i^2 = -1$ . For $b > 0$ , $\sqrt{-b} = i\sqrt{b}$ . Rewrite with $i$ before multiplying square roots of negatives.
e.g. $\sqrt{-18} = 3i\sqrt2$ ; $\sqrt{-4} \cdot \sqrt{-9} = (2i)(3i) = 6i^2 = -6$
$\displaystyle i = \sqrt{-1}$
$\displaystyle i^2 = -1$
$\displaystyle \sqrt{-b} = i\sqrt{b} \quad (b > 0)$
1.2.2
Quadratic Equations with Purely Imaginary Solutions
Isolate $x^2$ . If it equals a negative number, take $\pm$ the square root to get two imaginary solutions.
e.g. $2x^2 + 18 = 0 \Rightarrow x^2 = -9 \Rightarrow x = \pm 3i$
$\displaystyle x^2 = -k \quad (k > 0)$
$\displaystyle \Rightarrow\; x = \pm i\sqrt{k}$
1.2.3
Complex Numbers
A complex number has the form $a + bi$ with $a$ and $b$ real. $a$ is the real part and $b$ is the imaginary part. Every real number is complex (with $b = 0$ ), and two complex numbers are equal only when both parts match.
e.g. $-3 + 4i$ : real part $-3$ , imaginary part $4$
$\displaystyle z = a + bi$
$\displaystyle \operatorname{Re}(z) = a,\; \operatorname{Im}(z) = b$
1.2.4
Adding and Subtracting Complex Numbers
Combine real parts with real parts and imaginary parts with imaginary parts. When subtracting, distribute the minus sign to both parts.
e.g. $(5 - 2i) - (3 - 6i) = 2 + 4i$
$\displaystyle (a + bi) + (c + di)$
$\displaystyle = (a + c) + (b + d)i$
$\displaystyle (a + bi) - (c + di)$
$\displaystyle = (a - c) + (b - d)i$
1.2.5
Multiplying Complex Numbers
Multiply like binomials, replace $i^2$ with $-1$ , then combine like terms. A number times its conjugate is real.
e.g. $(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i^2 = 14 - 5i$
$\displaystyle (a + bi)(c + di)$
$\displaystyle = (ac - bd) + (ad + bc)i$
$\displaystyle (a + bi)(a - bi) = a^2 + b^2$
1.2.6
Solving Quadratic Equations With Complex Roots
When the discriminant $b^2 - 4ac$ is negative, the quadratic formula gives two complex roots. They are conjugates, $p \pm qi$ .
e.g. $x^2 - 4x + 13 = 0$ : $x = \tfrac{4 \pm \sqrt{-36}}{2} = 2 \pm 3i$
$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$\displaystyle b^2 - 4ac < 0 \Rightarrow \text{complex roots}$
1.2.7
The Cyclic Property of the Imaginary Unit
Powers of $i$ repeat every 4 steps: $i, -1, -i, 1$ . To find $i^n$ , divide $n$ by 4 and use the remainder $r$ (a remainder of 0 gives 1).
e.g. $i^{43}$ : $43 = 4 \cdot 10 + 3$ , so $i^{43} = i^3 = -i$
$\displaystyle i^1 = i,\; i^2 = -1$
$\displaystyle i^3 = -i,\; i^4 = 1$
$\displaystyle i^n = i^r,\; r = \text{remainder of } n \div 4$

Unit 2 31 topics

Polynomials

Polynomials, Quadratic Equations, Factoring Polynomials, and Polynomial Equations.

2.3 Polynomials

A square with side a + b splits into four pieces. Its area is not a² + b²; the two rectangles make up the difference.

2.3.1
The Least Common Multiple of Two Monomials
Take the LCM of the coefficients. Then take each variable to the highest power it has in either monomial.
e.g. LCM of $6x^2y$ and $8xy^3$ is $24x^2y^3$
$\displaystyle \text{LCM}(x^m, x^n) = x^{\max(m, n)}$
2.3.2
The Least Common Multiple of Two Polynomials
Factor each polynomial completely. The LCM uses every factor that appears, each raised to the highest power it has in either polynomial.
e.g. $x^2 - 9 = (x - 3)(x + 3)$ and $x^2 + 6x + 9 = (x + 3)^2$ : LCM $= (x - 3)(x + 3)^2$
2.3.3
Describing Relationships Using Polynomial Identities
A polynomial identity is an equation that is true for every value of its variables. Prove one by expanding or factoring one side until it matches the other. Identities explain number patterns.
e.g. $(n + 1)^2 - n^2 = 2n + 1$ , so consecutive squares always differ by an odd number: $5^2 - 4^2 = 9$
$\displaystyle (a + b)^2 = a^2 + 2ab + b^2$
$\displaystyle (a - b)^2 = a^2 - 2ab + b^2$
$\displaystyle a^2 - b^2 = (a + b)(a - b)$

2.4 Quadratic Equations

D = b² − 4ac. The solutions are where the parabola meets the x-axis, so D says whether it crosses twice, touches once, or misses.

2.4.1
Introduction to Quadratic Equations
Degree 2. Written in standard form, it has at most two real solutions.
$\displaystyle ax^2 + bx + c = 0 \quad (a \ne 0)$
2.4.2
Solving Perfect Square Quadratic Equations
Write it as a square equal to a number, then take $\pm$ square roots.
e.g. $(x - 2)^2 = 9 \Rightarrow x = 5$ or $-1$
$\displaystyle (x + p)^2 = k \;\Rightarrow\; x = -p \pm \sqrt{k}$
2.4.3
Perfect Square Quadratic Equations with One or No Solutions
$k = 0$ gives one solution. $k < 0$ gives none, since no real number squares to a negative.
$\displaystyle (x + p)^2 = 0 \;\Rightarrow\; x = -p$
$\displaystyle (x + p)^2 = -4 \;\Rightarrow\; \text{no real solution}$
2.4.4
The Zero Product Rule for Solving Quadratic Equations
If a product is zero, one of the factors is zero. One side must be 0 first.
e.g. $(x - 3)(x + 5) = 0 \Rightarrow x = 3$ or $-5$
$\displaystyle ab = 0 \;\Rightarrow\; a = 0 \text{ or } b = 0$
2.4.5
Solving Quadratic Equations Using a Difference of Squares
Factor the difference of squares and use the zero product rule.
e.g. $4x^2 - 25 = 0 \Rightarrow x = \pm\tfrac52$
$\displaystyle x^2 - k^2 = 0$
$\displaystyle \Rightarrow\; (x - k)(x + k) = 0$
$\displaystyle \Rightarrow\; x = \pm k$
2.4.6
Solving Quadratic Equations with No Constant Term
Factor out $x$ . Zero is always a solution; dividing by $x$ would lose it.
$\displaystyle ax^2 + bx = 0$
$\displaystyle \Rightarrow\; x(ax + b) = 0$
$\displaystyle \Rightarrow\; x = 0 \text{ or } x = -\tfrac{b}{a}$
2.4.7
Solving Quadratic Equations by Factoring
Get 0 on one side, factor, then set each factor to zero.
e.g. $x = 3$ or $-2$
$\displaystyle x^2 - x - 6 = 0$
$\displaystyle \Rightarrow\; (x - 3)(x + 2) = 0$
2.4.8
Solving Quadratic Equations with Leading Coefficients by Factoring
Factor $ax^2 + bx + c$ with the $ac$ method, then set each factor to zero.
e.g. $x = -\tfrac13$ or $2$
$\displaystyle 3x^2 - 5x - 2 = 0$
$\displaystyle \Rightarrow\; (3x + 1)(x - 2) = 0$
2.4.9
Completing the Square
Add the square of half the $x$ -coefficient.
e.g. $x^2 + 6x + 9 = (x + 3)^2$
$\displaystyle x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2$
2.4.10
Completing the Square With Odd Linear Terms
The same rule; half of an odd number is a fraction.
$\displaystyle x^2 + 5x + \tfrac{25}{4} = \left(x + \tfrac{5}{2}\right)^2$
2.4.11
Completing the Square With Leading Coefficients
Factor $a$ out of the $x$ -terms, complete the square inside, and adjust the constant.
$\displaystyle 2x^2 + 12x + 5 = 2(x + 3)^2 - 13$
2.4.12
Solving Quadratic Equations by Completing the Square
Move $c$ across, complete the square on both sides, take $\pm$ roots.
e.g. $x^2 + 6x = 7 \Rightarrow (x + 3)^2 = 16 \Rightarrow x = 1$ or $-7$
2.4.13
Solving Quadratic Equations With Leading Coefficients by Completing the Square
Divide every term by $a$ first, then complete the square.
e.g. $2x^2 - 8x - 10 = 0 \Rightarrow x^2 - 4x = 5 \Rightarrow (x - 2)^2 = 9 \Rightarrow x = 5$ or $-1$
2.4.14
The Quadratic Formula
Works for every quadratic in standard form.
e.g. $x^2 - 4x + 1 = 0 \Rightarrow x = 2 \pm \sqrt3$
$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
2.4.15
The Discriminant of a Quadratic Equation
$b^2 - 4ac$ tells how many real solutions there are without solving.
$\displaystyle D = b^2 - 4ac$
$\displaystyle D > 0:\ \text{two solutions}$
$\displaystyle D = 0:\ \text{one solution}$
$\displaystyle D < 0:\ \text{no real solution}$
2.4.16
Determining Unknown Parameters in Quadratic Equations With One Real Solution
Set the discriminant equal to zero and solve for the parameter.
e.g. $x^2 + kx + 9 = 0$ has one solution when $k^2 = 36$ , $k = \pm 6$
$\displaystyle b^2 - 4ac = 0$
2.4.17
Modeling With Quadratic Equations
Write the area, motion, or number relationship as a quadratic, solve, and reject answers that make no sense, such as negative lengths.
e.g. Length 3 more than width, area 40 → $w(w + 3) = 40 \Rightarrow w = 5$
2.4.18
Equating Polynomial Coefficients
Two polynomials equal for every $x$ have equal coefficients term by term.
e.g. $(x + k)^2 = x^2 + 8x + m \Rightarrow k = 4,\ m = 16$
$\displaystyle ax^2 + bx + c \equiv px^2 + qx + r$
$\displaystyle \Rightarrow\; a = p,\ b = q,\ c = r$
2.4.19
Vieta's Formulas
The sum and product of the roots come straight from the coefficients.
e.g. $x^2 - 7x + 10$ : roots sum to 7 and multiply to 10 (they are 2 and 5)
$\displaystyle r_1 + r_2 = -\frac{b}{a}$
$\displaystyle r_1r_2 = \frac{c}{a}$

2.5 Factoring Polynomials

Factoring runs the box backward: the corners hold x² and the constant, the two middle cells add to the middle term, and the edges are the factors.

2.5.1
Further Factoring of Polynomials Using GCFs
Factor out the GCF first, including common variable powers (and $-1$ if the leading term is negative). The GCF can be a binomial. Then factor what remains.
e.g. $3x^3 - 12x^2 - 36x = 3x(x^2 - 4x - 12) = 3x(x - 6)(x + 2)$
2.5.2
Factoring Higher-Order Polynomials as a Difference of Squares
Write both terms as squares and use $a^2 - b^2 = (a + b)(a - b)$ . Check whether a factor is again a difference of squares. A sum of squares $a^2 + b^2$ does not factor over the reals.
e.g. $x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$
$\displaystyle a^2 - b^2 = (a + b)(a - b)$
$\displaystyle x^4 - y^4$
$\displaystyle = (x^2 + y^2)(x + y)(x - y)$
2.5.3
Factoring Cubic Expressions by Grouping
Split the four terms into two pairs. Factor the GCF out of each pair, then factor out the common binomial.
e.g. $x^3 + 3x^2 - 4x - 12 = x^2(x + 3) - 4(x + 3) = (x + 3)(x + 2)(x - 2)$
$\displaystyle x^3 + ax^2 + bx + ab$
$\displaystyle = x^2(x + a) + b(x + a)$
$\displaystyle = (x + a)(x^2 + b)$
2.5.4
Factoring Sums and Differences of Cubes
Write both terms as cubes and use the pattern. Signs follow SOAP: Same, Opposite, Always Positive. The trinomial factor does not factor further over the reals.
e.g. $8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)$
$\displaystyle a^3 + b^3$
$\displaystyle = (a + b)(a^2 - ab + b^2)$
$\displaystyle a^3 - b^3$
$\displaystyle = (a - b)(a^2 + ab + b^2)$
2.5.5
Factoring Biquadratic Expressions
A biquadratic has the form $ax^4 + bx^2 + c$ . Substitute $u = x^2$ , factor the quadratic in $u$ , then put $x^2$ back and factor further if possible.
e.g. $x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4) = (x - 1)(x + 1)(x - 2)(x + 2)$
$\displaystyle ax^4 + bx^2 + c$
$\displaystyle u = x^2:\; au^2 + bu + c$

2.6 Polynomial Equations

2.6.1
Determining the Roots of Polynomials
The roots are the $x$ -values that make the polynomial equal 0. Factor completely and set each factor equal to 0. A factor $(x - r)^k$ gives the root $r$ with multiplicity $k$ .
e.g. $p(x) = (x - 2)^2(x + 5)$ : roots $2$ (multiplicity 2) and $-5$
$\displaystyle (x - r)(x - s) = 0$
$\displaystyle \Rightarrow\; x = r \;\text{or}\; x = s$
2.6.2
Solving Polynomial Equations Using the GCF
Move every term to one side so the equation equals 0. Factor out the GCF, then set each factor equal to 0. Never divide both sides by $x$ , since that loses the root $x = 0$ .
e.g. $2x^3 = 18x \Rightarrow 2x(x - 3)(x + 3) = 0 \Rightarrow x = 0, \pm 3$
2.6.3
Solving Cubic Equations by Grouping
Set the cubic equal to 0, factor by grouping, then set each factor equal to 0.
e.g. $x^3 - 2x^2 - 9x + 18 = 0 \Rightarrow (x - 2)(x - 3)(x + 3) = 0 \Rightarrow x = 2, \pm 3$
$\displaystyle x^2(x + a) + b(x + a) = 0$
$\displaystyle \Rightarrow\; (x + a)(x^2 + b) = 0$
2.6.4
Solving Biquadratic Equations
Let $u = x^2$ and solve the quadratic in $u$ . Then solve $x^2 = u$ for each value of $u$ . A negative $u$ gives imaginary solutions, so there can be up to 4 solutions.
e.g. $x^4 - 5x^2 - 36 = 0 \Rightarrow (x^2 - 9)(x^2 + 4) = 0 \Rightarrow x = \pm 3, \pm 2i$
$\displaystyle ax^4 + bx^2 + c = 0$
$\displaystyle u = x^2:\; au^2 + bu + c = 0$
$\displaystyle x = \pm\sqrt{u}$

Unit 3 18 topics

Quadratic Functions

Quadratic Functions, and Modeling With Quadratic Functions.

3.7 Quadratic Functions

The axis of symmetry runs through the vertex, halfway between the roots. Each form of the equation shows one of these features at a glance.

3.7.1
Graphing Elementary Quadratic Functions
$y = x^2$ is a U-shaped parabola with its vertex at the origin. In $y = ax^2$ , a larger $\lvert a \rvert$ makes it narrower.
$\displaystyle y = ax^2$
3.7.2
Vertical Reflections of Quadratic Functions
A negative $a$ opens the parabola downward, so the vertex becomes a maximum.
$\displaystyle y = -x^2$
3.7.3
Graphs of General Quadratic Functions
$a$ sets direction and width; $c$ is the $y$ -intercept.
$\displaystyle y = ax^2 + bx + c$
3.7.4
Roots of Quadratic Functions
Where the parabola meets the $x$ -axis: solve $ax^2 + bx + c = 0$ .
3.7.5
The Discriminant of a Quadratic Function
Counts the $x$ -intercepts: two, one (vertex on the axis), or none.
$\displaystyle D = b^2 - 4ac$
3.7.6
The Axis of Symmetry of a Parabola
A vertical line through the vertex. The parabola mirrors across it.
e.g. $y = x^2 - 6x + 5$ : axis $x = 3$
$\displaystyle x = -\frac{b}{2a}$
3.7.7
The Average of the Roots Formula
The axis sits exactly halfway between the roots.
e.g. Roots 1 and 5 → vertex at $x = 3$
$\displaystyle x_{\text{vertex}} = \frac{r_1 + r_2}{2}$
3.7.8
The Vertex Form of a Parabola
The vertex is at $(h, k)$ . Watch the sign inside: $(x - h)$ .
e.g. $y = 2(x + 1)^2 - 3$ : vertex $(-1, -3)$
$\displaystyle y = a(x - h)^2 + k$
3.7.9
Writing the Equation of a Parabola in Vertex Form
Complete the square, or find $h = -\tfrac{b}{2a}$ and then $k = f(h)$ .
e.g. $y = x^2 - 6x + 5 = (x - 3)^2 - 4$
3.7.10
Domain and Range of Quadratic Functions
The domain is all real numbers. The range starts at the vertex: $[k, \infty)$ if $a > 0$ , $(-\infty, k]$ if $a < 0$ .
3.7.11
Solving Systems of Nonlinear Equations
Substitute one equation into the other, set it to zero, and solve the quadratic.
e.g. Solutions $(2, 4)$ and $(-1, 1)$
$\displaystyle y = x^2,\ y = x + 2$
$\displaystyle \Rightarrow\; x^2 - x - 2 = 0$
3.7.12
Finding Intersections of Lines and Quadratic Functions
Set the two equal. The discriminant of the result gives 2 crossings, 1 (the line just touches), or 0.
$\displaystyle ax^2 + bx + c = mx + d$

3.8 Modeling With Quadratic Functions

A thrown object’s height is a downward parabola in time. The vertex gives the peak; the positive root gives the landing.

3.8.1
Modeling Downwards Vertical Motion
A dropped object falls $16t^2$ feet in $t$ seconds.
e.g. Dropped from 144 ft → lands when $16t^2 = 144$ , $t = 3$ s
$\displaystyle h(t) = -16t^2 + h_0$
$\displaystyle h(t) = -4.9t^2 + h_0 \;\text{(meters)}$
3.8.2
Modeling Upwards Vertical Motion
Launched upward at speed $v_0$ from height $h_0$ .
$\displaystyle h(t) = -16t^2 + v_0t + h_0$
3.8.3
Vertical Motion
Maximum height is at the vertex. It lands when $h(t) = 0$ ; keep the positive root.
e.g. $h = -16t^2 + 64t$ peaks at $t = 2$ s, 64 ft
$\displaystyle t_{\text{peak}} = -\frac{b}{2a} = \frac{v_0}{32}$
3.8.4
Revenue, Cost, and Profit Functions
Revenue is price times quantity. Profit is revenue minus cost.
$\displaystyle R = p\,q$
$\displaystyle P = R - C$
3.8.5
Constructing Revenue, Cost, and Profit Functions
When price depends on quantity, revenue is quadratic.
$\displaystyle p = 50 - 2q$
$\displaystyle \Rightarrow\; R = q(50 - 2q) = 50q - 2q^2$
3.8.6
Maximizing Profit and Break-Even Points
Maximum profit is at the vertex of $P$ . Break-even is where $R = C$ , which means $P = 0$ .
e.g. $P = -2q^2 + 40q - 150$ : best at $q = 10$ , profit 50
$\displaystyle q_{\max} = -\frac{b}{2a}$

Unit 4 30 topics

Functions

Functions, and Graph Transformations of Functions.

4.9 Functions

A function sends each input to exactly one output. On a graph, a vertical line that hits twice means some input has two outputs.

4.9.1
The Arithmetic of Functions
Add, subtract, multiply, or divide the outputs at the same input.
$\displaystyle (f + g)(x) = f(x) + g(x)$
$\displaystyle (fg)(x) = f(x)\,g(x)$
$\displaystyle \left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)},\ g(x) \ne 0$
4.9.2
Function Composition
Feed one function’s output into another. Work from the inside out.
e.g. $f(x) = x^2$ , $g(x) = x + 1$ : $f(g(2)) = 9$ , $g(f(2)) = 5$
$\displaystyle (f \circ g)(x) = f(g(x))$
4.9.3
Describing Function Composition
Read $f(g(x))$ as “do $g$ , then $f$ .” Order matters: usually $f(g(x)) \ne g(f(x))$ .
4.9.4
Local Extrema of Functions
A local maximum is a point higher than all nearby points, and a local minimum is lower than all nearby points. They sit at the tops of hills and bottoms of valleys, where the graph switches between increasing and decreasing. The extreme value is the $y$ -coordinate.
e.g. $f(x) = x^3 - 3x$ : local max $2$ at $x = -1$ , local min $-2$ at $x = 1$
4.9.5
One-To-One Functions
A function is one-to-one if different inputs always give different outputs. Horizontal line test: no horizontal line crosses the graph more than once.
e.g. $f(x) = 2x + 1$ is one-to-one; $g(x) = x^2$ is not, since $g(-2) = g(2) = 4$
$\displaystyle f(a) = f(b) \Rightarrow a = b$
4.9.6
Introduction to Inverse Functions
$f^{-1}$ undoes $f$ . To find it, swap $x$ and $y$ and solve for $y$ . Its graph is the reflection of $f$ in $y = x$ .
$\displaystyle f(f^{-1}(x)) = x$
$\displaystyle f(x) = 2x + 3 \;\Rightarrow\; f^{-1}(x) = \frac{x - 3}{2}$
4.9.7
Calculating the Inverse of a Function
The inverse $f^{-1}$ undoes $f$ . Replace $f(x)$ with $y$ , swap $x$ and $y$ , then solve for $y$ . Note $f^{-1}(x)$ does not mean $\tfrac{1}{f(x)}$ .
e.g. $f(x) = 3x - 6$ : $x = 3y - 6 \Rightarrow f^{-1}(x) = \tfrac{x}{3} + 2$
$\displaystyle f(a) = b \iff f^{-1}(b) = a$
$\displaystyle f(f^{-1}(x)) = f^{-1}(f(x)) = x$
4.9.8
Inverses of Quadratic Functions
A quadratic is not one-to-one, so first restrict its domain to one side of the vertex. Then swap $x$ and $y$ , solve with square roots, and keep the sign that matches the restricted domain.
e.g. $f(x) = (x - 2)^2 + 1$ , $x \ge 2$ : $f^{-1}(x) = 2 + \sqrt{x - 1}$ , $x \ge 1$
$\displaystyle f(x) = a(x - h)^2 + k,\; x \ge h$
$\displaystyle f^{-1}(x) = h + \sqrt{\frac{x - k}{a}}$
4.9.9
Graphs of Inverse Functions
The graph of $f^{-1}$ is the graph of $f$ reflected across the line $y = x$ . Each point $(a, b)$ on $f$ becomes $(b, a)$ on $f^{-1}$ .
e.g. $f(x) = x^3$ passes through $(2, 8)$ , so $f^{-1}(x) = \sqrt[3]{x}$ passes through $(8, 2)$
$\displaystyle (a, b) \to (b, a)$
4.9.10
Domain and Range of Inverse Functions
Inputs and outputs trade places. The domain of $f^{-1}$ is the range of $f$ , and the range of $f^{-1}$ is the domain of $f$ .
e.g. $f(x) = \sqrt{x - 3}$ : domain $x \ge 3$ , range $y \ge 0$ . $f^{-1}(x) = x^2 + 3$ : domain $x \ge 0$ , range $y \ge 3$
$\displaystyle \text{domain of } f^{-1} = \text{range of } f$
$\displaystyle \text{range of } f^{-1} = \text{domain of } f$
4.9.11
Invertible Functions
A function has an inverse function exactly when it is one-to-one, so it passes the horizontal line test. A function that is not one-to-one can be made invertible by restricting its domain.
e.g. $f(x) = x^3$ is invertible; $g(x) = \lvert x \rvert$ is not, but it is on $x \ge 0$
4.9.12
Plotting X as a Function of Y
In $x = g(y)$ , $y$ is the input. Pick $y$ -values, compute $x$ , and plot the points $(x, y)$ . The graph is the graph of $y = g(x)$ reflected across $y = x$ .
e.g. $x = y^2$ : $y = -2, 0, 2$ give $(4, -2), (0, 0), (4, 2)$ , a parabola opening right
$\displaystyle x = g(y)$
4.9.13
Periodic Functions
A periodic function repeats its values at regular intervals. The period $p$ is the smallest positive shift that maps the graph onto itself.
e.g. Period 4 and $f(1) = 7$ : then $f(9) = 7$ and $f(-3) = 7$
$\displaystyle f(x + p) = f(x) \;\text{for all } x$
4.9.14
Even and Odd Functions
Even: $f(-x) = f(x)$ , symmetric about the $y$ -axis. Odd: $f(-x) = -f(x)$ , symmetric about the origin. Most functions are neither.
e.g. $f(x) = x^3 - x$ : $f(-x) = -x^3 + x = -f(x)$ , so odd. $x^2 + 1$ is even.
$\displaystyle f(-x) = f(x) \quad \text{even}$
$\displaystyle f(-x) = -f(x) \quad \text{odd}$
4.9.15
Products of Even and Odd Functions
These multiply like signs, with even as $+$ and odd as $-$ . Even times even and odd times odd are even; even times odd is odd. Quotients follow the same rules.
e.g. $x \cdot x^3 = x^4$ (odd times odd, even); $x^2 \cdot x^3 = x^5$ (even times odd, odd)
$\displaystyle \text{even} \cdot \text{even} = \text{even}$
$\displaystyle \text{odd} \cdot \text{odd} = \text{even}$
$\displaystyle \text{even} \cdot \text{odd} = \text{odd}$
4.9.16
Unbounded Behavior of Functions Near a Point
A function is unbounded near $x = a$ if its values grow without limit (toward $\infty$ or $-\infty$ ) as $x$ gets close to $a$ . The graph has a vertical asymptote at $x = a$ , often where a denominator is 0.
e.g. $f(x) = \tfrac{1}{x - 2}$ : as $x$ approaches 2 from the right, $f(x) \to \infty$ ; from the left, $f(x) \to -\infty$
$\displaystyle x \to a \;\Rightarrow\; f(x) \to \pm\infty$
4.9.17
The Average Rate of Change of a Function
The change in output divided by the change in input between two points. It is the slope of the line through those two points on the graph.
e.g. $f(x) = x^2$ from $x = 1$ to $x = 4$ : $\tfrac{16 - 1}{4 - 1} = 5$
$\displaystyle \frac{f(b) - f(a)}{b - a}$

4.10 Graph Transformations of Functions

Replacing x with x − 2 and adding 1 moves every point of y = x² right 2 and up 1. Changes inside the function act on x and run opposite to how they look.

4.10.1
Vertical Translations of Functions
$f(x) + k$ shifts the graph up $k$ units if $k > 0$ and down if $k < 0$ . Only the $y$ -coordinates change.
e.g. $y = x^2 - 3$ is $y = x^2$ shifted down 3; vertex $(0, -3)$
$\displaystyle y = f(x) + k$
$\displaystyle (x, y) \to (x, y + k)$
4.10.2
Horizontal Translations of Functions
$f(x - h)$ shifts the graph right $h$ units if $h > 0$ and left if $h < 0$ . The shift is opposite the sign inside, so $f(x + 2)$ moves left 2.
e.g. $y = \sqrt{x + 2}$ is $y = \sqrt{x}$ shifted left 2; it starts at $(-2, 0)$
$\displaystyle y = f(x - h)$
$\displaystyle (x, y) \to (x + h, y)$
4.10.3
Vertical Stretches of Functions
For $a > 0$ , $a f(x)$ multiplies every $y$ -coordinate by $a$ . It is a stretch if $a > 1$ and a compression if $0 < a < 1$ . Points on the $x$ -axis stay fixed.
e.g. $y = 3x^2$ : the point $(2, 4)$ on $y = x^2$ moves to $(2, 12)$
$\displaystyle y = a f(x)$
$\displaystyle (x, y) \to (x, ay)$
4.10.4
Horizontal Stretches of Functions
For $b > 0$ , $f(bx)$ divides every $x$ -coordinate by $b$ . It is a compression if $b > 1$ and a stretch if $0 < b < 1$ . Points on the $y$ -axis stay fixed.
e.g. $y = f(2x)$ : the point $(6, 1)$ on $f$ moves to $(3, 1)$
$\displaystyle y = f(bx)$
$\displaystyle (x, y) \to \left(\frac{x}{b}, y\right)$
4.10.5
Vertical Reflections of Functions
$-f(x)$ reflects the graph across the $x$ -axis. Every $y$ -coordinate changes sign.
e.g. $y = -\sqrt{x}$ passes through $(4, -2)$ instead of $(4, 2)$
$\displaystyle y = -f(x)$
$\displaystyle (x, y) \to (x, -y)$
4.10.6
Horizontal Reflections of Functions
$f(-x)$ reflects the graph across the $y$ -axis. Every $x$ -coordinate changes sign.
e.g. $y = \sqrt{-x}$ passes through $(-4, 2)$ instead of $(4, 2)$ ; domain $x \le 0$
$\displaystyle y = f(-x)$
$\displaystyle (x, y) \to (-x, y)$
4.10.7
Combining Graph Transformations: Two Operations
Handle outputs and inputs separately. For outputs, follow order of operations: stretch or reflect first, then shift. A horizontal change and a vertical change can be done in either order.
e.g. $y = 2f(x) + 3$ : the point $(1, 4)$ on $f$ moves to $(1, 11)$
$\displaystyle y = a f(x) + k$
$\displaystyle (x, y) \to (x, ay + k)$
4.10.8
Combining Graph Transformations: Three or More Operations
Write the function as $a f(b(x - h)) + k$ , factoring $b$ out of the inside first. Horizontally, stretch by $\tfrac{1}{b}$ , then shift by $h$ . Vertically, stretch or reflect by $a$ , then shift by $k$ .
e.g. $y = -2f(3x - 6) + 1 = -2f(3(x - 2)) + 1$ : $(6, 5) \to (4, -9)$
$\displaystyle y = a f(b(x - h)) + k$
$\displaystyle (x, y) \to \left(\frac{x}{b} + h,\; ay + k\right)$
4.10.9
Constructing Functions Using Transformations
Start with the parent function and apply each step in the order given. Horizontal changes go inside on $x$ (shifts with the opposite sign); vertical changes go outside. Each later step acts on everything built so far.
e.g. Shift $y = x^2$ right 3, stretch vertically by 2, then shift up 1: $y = 2(x - 3)^2 + 1$
4.10.10
Combining Reflections With Other Graph Transformations
A reflection is a stretch by $-1$ . For outputs, reflect before shifting. For inputs, factor out the negative: $f(-x + h) = f(-(x - h))$ means reflect across the $y$ -axis, then shift right $h$ .
e.g. $y = \sqrt{-x + 3} = \sqrt{-(x - 3)}$ : reflect $\sqrt{x}$ across the $y$ -axis, shift right 3; starts at $(3, 0)$ , domain $x \le 3$
$\displaystyle y = -f(x) + k$
$\displaystyle y = f(-(x - h))$
4.10.11
Finding Points on Transformed Curves
Map a known point $(x, y)$ on $f$ . Set the inside expression equal to the old $x$ and solve for the new $x$ . Run the old $y$ through the outside operations to get the new $y$ .
e.g. $(2, 3)$ on $f$ , $g(x) = 4f(x + 1) - 5$ : $x + 1 = 2 \Rightarrow x = 1$ , $4(3) - 5 = 7$ , so $(1, 7)$
$\displaystyle y = a f(b(x - h)) + k$
$\displaystyle (x, y) \to \left(\frac{x}{b} + h,\; ay + k\right)$
4.10.12
The Domain and Range of Transformed Functions
Horizontal changes affect only the domain: divide by $b$ , then add $h$ . Vertical changes affect only the range: multiply by $a$ , then add $k$ . A negative multiplier flips the interval, so reorder its endpoints.
e.g. $f$ : domain $[0, 4]$ , range $[-1, 2]$ . $g(x) = -3f(x - 1) + 2$ : domain $[1, 5]$ , range $[-4, 5]$
$\displaystyle y = a f(b(x - h)) + k$
4.10.13
Absolute Value Graph Transformations
$y = a\lvert x - h \rvert + k$ is a V shape with vertex $(h, k)$ . It opens up if $a > 0$ and down if $a < 0$ . The right arm has slope $a$ and the left arm has slope $-a$ .
e.g. $y = -2\lvert x + 1 \rvert + 3$ : vertex $(-1, 3)$ , opens down, slopes $-2$ (right) and $2$ (left)
$\displaystyle y = a\lvert x - h \rvert + k$
$\displaystyle \text{vertex } (h, k)$

Unit 5 8 topics

Absolute Value Functions

Absolute Value Functions.

5.11 Absolute Value Functions

Every absolute value graph is the V of y = |x| moved, flipped, or stretched. Find the vertex first.

5.11.1
Absolute Value Graphs
$y = \lvert x \rvert$ is a V with its corner at the origin. Shifted, the corner (vertex) is at $(h, k)$ .
$\displaystyle y = a\lvert x - h \rvert + k$
5.11.2
Vertical Reflections of Absolute Value Graphs
A negative sign in front flips the V upside down.
$\displaystyle y = -\lvert x \rvert$
5.11.3
Stretches of Absolute Value Graphs
$\lvert a \rvert > 1$ narrows the V; $0 < \lvert a \rvert < 1$ widens it. The arms have slopes $a$ and $-a$ .
5.11.4
Combining Transformations of Absolute Value Graphs
Plot the vertex $(h, k)$ , then step out with slopes $\pm a$ . Note that $\lvert x + 3 \rvert$ shifts left 3.
e.g. $y = -2\lvert x + 3 \rvert + 4$ : vertex $(-3, 4)$ , opens down
5.11.5
Domain and Range of Absolute Value Functions
The domain is all real numbers. The range starts at $k$ : $[k, \infty)$ if it opens up, $(-\infty, k]$ if down.
5.11.6
Roots of Absolute Value Functions
Set $y = 0$ and solve the absolute value equation. There can be zero, one, or two roots.
e.g. $\lvert x - 2 \rvert - 3 = 0 \Rightarrow x = 5$ or $-1$
5.11.7
Equations Connecting Absolute Value and Linear Functions
When the other side contains $x$ , split into two cases and keep only answers where that side is $\ge 0$ .
5.11.8
Absolute Value Equations With Extraneous Solutions
Check every answer in the original equation. Some solutions of the split equations fail.
e.g. $\lvert x + 1 \rvert = 2x$ → $x = 1$ works; $x = -\tfrac13$ fails

Unit 6 35 topics

Exponentials & Logarithms

Introduction to Logarithms, The Laws of Logarithms, Graphs of Exponential Functions, and more.

6.12 Introduction to Logarithms

log₂ 8 = 3 and 2³ = 8 say the same thing. The base stays the base, the log’s value becomes the exponent, and the number inside the log is the result.

6.12.1
Converting From Exponential to Logarithmic Form
A logarithm is an exponent. $\log_b x$ is the power you raise $b$ to in order to get $x$ . The base must satisfy $b > 0$ , $b \ne 1$ , and the input must satisfy $x > 0$ .
e.g. $2^5 = 32 \Rightarrow \log_2 32 = 5$
$\displaystyle b^y = x \iff \log_b x = y$
6.12.2
Converting From Logarithmic to Exponential Form
The base of the log becomes the base of the power, and the value of the log becomes the exponent.
e.g. $\log_3 81 = 4 \Rightarrow 3^4 = 81$
$\displaystyle \log_b x = y \iff b^y = x$
6.12.3
Evaluating Logarithms
Rewrite the input as a power of the base, then read off the exponent.
e.g. $\log_2 \tfrac18 = \log_2 2^{-3} = -3$
$\displaystyle \log_b b^k = k$
$\displaystyle \log_b 1 = 0,\; \log_b b = 1$
6.12.4
The Natural Logarithm
$\ln x$ is the logarithm with base $e \approx 2.718$ . It follows every rule of other logs.
e.g. $\ln e^7 = 7$
$\displaystyle \ln x = \log_e x$
$\displaystyle \ln e = 1,\; \ln 1 = 0$
$\displaystyle \ln e^x = x,\; e^{\ln x} = x$
6.12.5
The Common Logarithm
$\log x$ with no base written means base 10.
e.g. $\log 0.001 = \log 10^{-3} = -3$
$\displaystyle \log x = \log_{10} x$
$\displaystyle \log 10^k = k$
6.12.6
Simplifying Logarithmic Expressions
A log and a power with the same base undo each other.
e.g. $5^{\log_5 9} = 9$ and $\log_3 3^{2x} = 2x$
$\displaystyle \log_b b^x = x$
$\displaystyle b^{\log_b x} = x \quad (x > 0)$

6.13 The Laws of Logarithms

6.13.1
The Product Rule for Logarithms
The log of a product is the sum of the logs. Both inputs must be positive.
e.g. $\log_2 8 + \log_2 4 = \log_2 32 = 5$
$\displaystyle \log_b (MN) = \log_b M + \log_b N$
6.13.2
The Quotient Rule for Logarithms
The log of a quotient is the log of the top minus the log of the bottom.
e.g. $\log_3 54 - \log_3 2 = \log_3 27 = 3$
$\displaystyle \log_b \frac{M}{N} = \log_b M - \log_b N$
6.13.3
The Power Rule for Logarithms
An exponent on the input moves to the front as a multiplier.
e.g. $\log_2 8^5 = 5 \log_2 8 = 15$
$\displaystyle \log_b M^p = p \log_b M$
6.13.4
Combining the Laws of Logarithms
Use the product, quotient, and power rules together to expand or condense. When condensing, move coefficients up as exponents first.
e.g. $2\log 5 + \log 4 = \log(25 \cdot 4) = \log 100 = 2$
$\displaystyle \log_b \frac{x^2 y}{z}$
$\displaystyle = 2\log_b x + \log_b y - \log_b z$
6.13.5
The Change of Base Formula for Logarithms
Rewrites a log in any base you like, such as base 10 or $e$ for a calculator.
e.g. $\log_4 8 = \tfrac{\log_2 8}{\log_2 4} = \tfrac32$
$\displaystyle \log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}$

6.14 Graphs of Exponential Functions

Adding 3 outside the exponential lifts the whole graph, including its asymptote. The y-intercept moves from (0, 1) to (0, 4).

6.14.1
Vertical Translations of Exponential Growth Functions
For $b > 1$ , adding $k$ shifts the graph up $k$ units (down if $k < 0$ ). The horizontal asymptote moves from $y = 0$ to $y = k$ , and the range becomes $y > k$ .
e.g. $y = 2^x - 3$ : asymptote $y = -3$ , $y$ -intercept $1 - 3 = -2$
$\displaystyle y = b^x + k$
$\displaystyle \text{asymptote: } y = k$
6.14.2
Vertical Translations of Exponential Decay Functions
For $0 < b < 1$ , adding $k$ shifts the graph up $k$ units (down if $k < 0$ ). The graph falls toward the asymptote $y = k$ as $x$ increases.
e.g. $y = \left(\tfrac12\right)^x + 4$ : asymptote $y = 4$ , $y$ -intercept $5$
$\displaystyle y = b^x + k$
$\displaystyle \text{asymptote: } y = k$
6.14.3
Interpreting Graphs of Exponential Functions
The $y$ -intercept is the starting value $a$ . The ratio of outputs one $x$ -unit apart is the factor $b$ : growth if $b > 1$ , decay if $0 < b < 1$ . The asymptote is the level the quantity approaches.
e.g. Graph through $(0, 3)$ and $(1, 6)$ : $a = 3$ , $b = 2$ , so $y = 3 \cdot 2^x$
$\displaystyle y = a \cdot b^x$
6.14.4
Vertical Reflections of Exponential Functions
A negative sign in front reflects the graph across the $x$ -axis. The asymptote stays $y = 0$ , but the graph lies below it, so the range is $y < 0$ . Note $-2^x$ means $-(2^x)$ .
e.g. $y = -2^x$ passes through $(0, -1)$ and $(1, -2)$
$\displaystyle y = -b^x$
6.14.5
Combining Graph Transformations of Exponential Functions
Shift right $h$ and up $k$ . A negative $a$ reflects across the $x$ -axis, and $\lvert a \rvert$ stretches or shrinks vertically. Apply the stretch and reflection before the vertical shift.
e.g. $y = -3 \cdot 2^{x-1} + 5$ : reflect, stretch by 3, right 1, up 5; asymptote $y = 5$ , passes $(1, 2)$
$\displaystyle y = a \cdot b^{x - h} + k$
6.14.6
Properties of Transformed Exponential Functions
For $y = a \cdot b^{x-h} + k$ , the domain is all real numbers and the horizontal asymptote is $y = k$ . The sign of $a$ decides which side of the asymptote the graph is on.
e.g. $y = -2 \cdot 3^x + 1$ : range $y < 1$ , $y$ -intercept $-1$
$\displaystyle \text{Domain: } (-\infty, \infty)$
$\displaystyle \text{Asymptote: } y = k$
$\displaystyle \text{Range: } y > k \;\; (a > 0)$
$\displaystyle \text{Range: } y < k \;\; (a < 0)$

6.15 Graphs of Logarithmic Functions

y = 2ˣ and y = log₂ x are inverses, so each is the other reflected across y = x. The exponential never touches y = 0 and the log never touches x = 0.

6.15.1
Graphing Logarithmic Functions
$y = \log_b x$ is the reflection of $y = b^x$ across the line $y = x$ . It has domain $x > 0$ , vertical asymptote $x = 0$ , and passes through $(1, 0)$ and $(b, 1)$ .
e.g. $y = \log_2 x$ passes $(\tfrac12, -1)$ , $(1, 0)$ , $(2, 1)$ , $(4, 2)$
$\displaystyle (\tfrac1b, -1),\; (1, 0),\; (b, 1)$
6.15.2
Combining Graph Transformations of Logarithmic Functions
Shift right $h$ and up $k$ . A negative $a$ reflects across the $x$ -axis, and $\lvert a \rvert$ stretches or shrinks vertically. The vertical asymptote moves to $x = h$ .
e.g. $y = \log_2(x + 3) - 1$ : left 3, down 1; asymptote $x = -3$ , passes $(-2, -1)$
$\displaystyle y = a \log_b (x - h) + k$
6.15.3
Properties of Transformed Logarithmic Functions
For $y = a \log_b(x - h) + k$ , the input must be positive, so the domain is $x > h$ . The range is all real numbers. Find the $x$ -intercept by setting $y = 0$ .
e.g. $y = \ln(x - 2) + 1$ : domain $x > 2$ , asymptote $x = 2$ , $x$ -intercept $2 + e^{-1}$
$\displaystyle \text{Domain: } x > h$
$\displaystyle \text{Range: } (-\infty, \infty)$
$\displaystyle \text{Asymptote: } x = h$
6.15.4
Inverses of Exponential and Logarithmic Functions
$b^x$ and $\log_b x$ undo each other. To find an inverse, swap $x$ and $y$ , then solve for $y$ .
e.g. $f(x) = 3^x + 2$ : $x = 3^y + 2 \Rightarrow f^{-1}(x) = \log_3(x - 2)$
$\displaystyle f(x) = b^x \iff f^{-1}(x) = \log_b x$

6.16 Exponential Equations

6.16.1
Solving Exponential Equations Using Logarithms
Isolate the power, then take the log of both sides or rewrite in log form.
e.g. $3^x = 20 \Rightarrow x = \tfrac{\ln 20}{\ln 3} \approx 2.73$
$\displaystyle b^x = c$
$\displaystyle \Rightarrow\; x = \log_b c = \frac{\ln c}{\ln b}$
6.16.2
Solving Equations Containing the Exponential Function
Isolate $e^{(\ldots)}$ , then take $\ln$ of both sides. There is no solution if $e^{(\ldots)}$ equals zero or a negative number.
e.g. $4e^{2x} = 36 \Rightarrow e^{2x} = 9 \Rightarrow x = \tfrac{\ln 9}{2} = \ln 3$
$\displaystyle e^x = c \;\Rightarrow\; x = \ln c \;\; (c > 0)$
6.16.3
Solving Exponential Equations With Different Bases
Write both sides as powers of a common base, then set the exponents equal.
e.g. $4^x = 8^{x-1} \Rightarrow 2^{2x} = 2^{3x-3} \Rightarrow x = 3$
$\displaystyle b^M = b^N \;\Rightarrow\; M = N$
6.16.4
Solving Exponential Equations With Different Bases Using Logarithms
When no common base exists, take $\ln$ of both sides, bring exponents down with the power rule, and collect the $x$ terms.
e.g. $2^x = 3^{x-1} \Rightarrow x\ln 2 = (x-1)\ln 3 \Rightarrow x = \tfrac{\ln 3}{\ln 3 - \ln 2} \approx 2.71$
$\displaystyle a^{f(x)} = b^{g(x)}$
$\displaystyle \Rightarrow\; f(x)\ln a = g(x)\ln b$
6.16.5
Solving Exponential Equations Using the Zero-Product Property
Factor, then set each factor to zero. For forms like $b^{2x} + \ldots$ , substitute $u = b^x$ . Since $b^x > 0$ always, a factor like $e^x$ is never zero and $b^x = -2$ has no solution.
e.g. $4^x - 6 \cdot 2^x + 8 = 0$ : $u = 2^x$ , $(u - 2)(u - 4) = 0$ , so $x = 1$ or $x = 2$
$\displaystyle e^x(x - a) = 0 \;\Rightarrow\; x = a$

6.17 Logarithmic Equations

6.17.1
Solving Logarithmic Equations
Isolate the log, rewrite in exponential form, and solve. Check that every log input is positive.
e.g. $\log_3(2x - 1) = 2 \Rightarrow 2x - 1 = 9 \Rightarrow x = 5$
$\displaystyle \log_b x = c \;\Rightarrow\; x = b^c$
6.17.2
Solving Logarithmic Equations Containing the Natural Logarithm
Isolate $\ln$ , then rewrite with base $e$ .
e.g. $2\ln(x + 1) = 6 \Rightarrow \ln(x + 1) = 3 \Rightarrow x = e^3 - 1 \approx 19.09$
$\displaystyle \ln x = c \;\Rightarrow\; x = e^c$
6.17.3
Solving Logarithmic Equations Using the Laws of Logarithms
Combine the logs into a single log, convert to exponential form, and solve. Reject any answer that makes an original log input zero or negative.
e.g. $\log_2 x + \log_2(x - 2) = 3 \Rightarrow x(x - 2) = 8 \Rightarrow x = 4$ ( $x = -2$ rejected)
$\displaystyle \log_b M + \log_b N = c$
$\displaystyle \Rightarrow\; MN = b^c$
6.17.4
Solving Logarithmic Equations by Combining the Laws of Logarithms
Use the power rule to move coefficients up first, then the product and quotient rules to reach one log. Check answers in the original equation.
e.g. $2\log x - \log 4 = 2 \Rightarrow \log \tfrac{x^2}{4} = 2 \Rightarrow x^2 = 400 \Rightarrow x = 20$ ( $x = -20$ rejected)
$\displaystyle a\log_b M - \log_b N = \log_b \frac{M^a}{N}$
6.17.5
Solving Logarithmic Equations With Logarithms on Both Sides
With a single log of the same base on each side, set the inputs equal. Check that both original inputs are positive.
e.g. $\ln(3x - 4) = \ln(x + 6) \Rightarrow 3x - 4 = x + 6 \Rightarrow x = 5$
$\displaystyle \log_b M = \log_b N \;\Rightarrow\; M = N$
6.17.6
Solving Logarithmic Equations Using the Zero-Product Property
Move everything to one side and factor, often by treating $\log_b x$ as a single variable. Set each factor to zero and solve each. The log itself may be zero or negative; only its input must be positive.
e.g. $(\log x)^2 - 3\log x = 0 \Rightarrow \log x(\log x - 3) = 0 \Rightarrow x = 1$ or $x = 1000$

6.18 Modeling with Exponential Functions

6.18.1
Modeling With Compound Interest
$P$ is the principal, $r$ the annual rate as a decimal, $n$ the number of compounding periods per year, and $t$ the time in years.
e.g. $500 at 4 percent compounded quarterly for 3 years: $500(1.01)^{12} \approx 563.41$
$\displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}$
6.18.2
Continuously Compounded Interest
Interest compounded continuously uses base $e$ . The rate $r$ is a decimal and $t$ is in years.
e.g. $2,000 at 5 percent for 10 years: $2000e^{0.5} \approx 3297.44$
$\displaystyle A = Pe^{rt}$
6.18.3
Converting Between Exponents
Rewrite an exponential with a different base or time unit using exponent rules. Use $b = e^{\ln b}$ to switch to base $e$ , and $e^k = b$ to switch from base $e$ . To change the time unit, split the exponent.
e.g. $500(1.08)^t = 500e^{0.077t}$ , since $\ln 1.08 \approx 0.077$
$\displaystyle b^t = e^{t \ln b}$
$\displaystyle b^t = \left(b^{1/m}\right)^{mt}$

Unit 7 17 topics

Sequences & Series

Geometric Sequences, Arithmetic Series, and The Binomial Theorem.

7.19 Geometric Sequences

Arithmetic sequences grow by equal steps, like linear functions. Geometric sequences grow by equal factors, like exponential functions.

7.19.1
Introduction to Geometric Sequences
Multiply by the same common ratio $r$ each time.
e.g. 3, 6, 12, 24: $r = 2$
$\displaystyle r = \frac{a_{n+1}}{a_n}$
7.19.2
The Recursive Formula for a Geometric Sequence
Give the first term and multiply by $r$ for the next.
$\displaystyle a_1 = \text{first term},\quad a_n = r\,a_{n-1}$
7.19.3
The Nth Term of a Geometric Sequence
Start at $a_1$ and multiply by $r$ a total of $n - 1$ times.
e.g. 2, 6, 18, … → $a_6 = 2 \cdot 3^5 = 486$
$\displaystyle a_n = a_1r^{n-1}$
7.19.4
Translating Between Explicit and Recursive Formulas for Geometric Sequences
Both use the same $a_1$ and $r$ .
$\displaystyle a_n = r\,a_{n-1}$
$\displaystyle \leftrightarrow\; a_n = a_1r^{n-1}$
7.19.5
Finding the Common Ratio of a Geometric Sequence Given Two Terms
Divide the later term by the earlier one, then take the root for the number of steps between them.
e.g. $a_2 = 6$ , $a_5 = 162$ → $r^3 = 27$ , $r = 3$
$\displaystyle r^{m-k} = \frac{a_m}{a_k}$
7.19.6
Determining Indexes of Terms in Geometric Sequences
Set $a_1r^{n-1}$ equal to the value and match the powers of $r$ .
e.g. 5, 10, 20, …: $640 = 5 \cdot 2^7$ → $n = 8$

7.20 Arithmetic Series

Write the series forward and backward and add: every pair has the same total. Half of n pairs gives the sum.

7.20.1
Sigma Notation
$\sum$ means add. Plug each integer from the lower limit to the upper limit into the expression and add the results. There are (upper minus lower plus 1) terms.
e.g. $\sum_{k=1}^{4} (2k + 1) = 3 + 5 + 7 + 9 = 24$
$\displaystyle \sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$
7.20.2
Properties of Finite Series
A sum can be split across addition, and a constant factor can be pulled out. Summing a constant $n$ times gives $n$ times the constant.
e.g. $\sum_{i=1}^{10} (3i + 2) = 3 \cdot 55 + 2 \cdot 10 = 185$
$\displaystyle \sum (a_i + b_i) = \sum a_i + \sum b_i$
$\displaystyle \sum c\,a_i = c \sum a_i$
$\displaystyle \sum_{i=1}^{n} c = cn$
$\displaystyle \sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
7.20.3
Expressing an Arithmetic Series in Sigma Notation
Write the general term $a_1 + (k - 1)d$ , simplify it, and sum from $k = 1$ to the number of terms.
e.g. $4 + 7 + 10 + \cdots + 31 = \sum_{k=1}^{10} (3k + 1)$
$\displaystyle \sum_{k=1}^{n} \big(a_1 + (k - 1)d\big)$
7.20.4
Finding the Sum of an Arithmetic Series
Average the first and last terms, then multiply by the number of terms. Use the second form when the last term is unknown.
e.g. $2 + 5 + 8 + \cdots + 29$ : $n = 10$ , $S = \tfrac{10(2 + 29)}{2} = 155$
$\displaystyle S_n = \frac{n(a_1 + a_n)}{2}$
$\displaystyle S_n = \frac{n}{2}\big(2a_1 + (n - 1)d\big)$
7.20.5
Finding the First Term of an Arithmetic Series
Substitute the known values into a sum formula and solve for $a_1$ .
e.g. $S_8 = 100$ , $d = 3$ : $100 = \tfrac82(2a_1 + 21) \Rightarrow 25 = 2a_1 + 21 \Rightarrow a_1 = 2$
$\displaystyle a_1 = \frac{2S_n}{n} - a_n$
7.20.6
Calculating the Number of Terms in an Arithmetic Series
If the last term is known, use the formula below. If only the sum is known, set up $S_n$ , solve the quadratic in $n$ , and keep the positive whole-number answer.
e.g. $5 + 9 + 13 + \cdots + 81$ : $n = \tfrac{81 - 5}{4} + 1 = 20$
$\displaystyle n = \frac{a_n - a_1}{d} + 1$
7.20.7
Modeling With Arithmetic Series
Use when a quantity changes by the same amount each step and you need the total. Identify $a_1$ , $d$ , and $n$ , then apply the sum formula.
e.g. 15 rows, 20 seats in row 1, 2 more per row: last row $20 + 14 \cdot 2 = 48$ , total $\tfrac{15(20 + 48)}{2} = 510$

7.21 The Binomial Theorem

Each number in Pascal’s triangle is the sum of the two above it. Row 4 gives the coefficients 1, 4, 6, 4, 1 of (a + b)⁴.

7.21.1
Pascal's Triangle and the Binomial Coefficients
Each entry is the sum of the two entries above it. Row $n$ (counting the top as row 0) lists $\binom{n}{0}$ through $\binom{n}{n}$ .
e.g. Row 4 is $1, 4, 6, 4, 1$ , so $\binom42 = 6$
$\displaystyle \binom{n}{r} = \frac{n!}{r!\,(n - r)!}$
$\displaystyle \binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$
7.21.2
Expanding a Binomial Using Binomial Coefficients
Powers of $a$ fall from $n$ to 0 while powers of $b$ rise from 0 to $n$ . Coefficients come from row $n$ of Pascal's triangle. Keep any negative sign with $b$ .
e.g. $(x - 2)^3 = x^3 - 6x^2 + 12x - 8$
$\displaystyle (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
7.21.3
The Special Case of the Binomial Theorem
With $a = 1$ , the expansion of $(1 + x)^n$ uses the binomial coefficients directly. Setting $x = 1$ shows that row $n$ of Pascal's triangle sums to $2^n$ .
e.g. $(1 + x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$
$\displaystyle (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k$
$\displaystyle = 1 + nx + \binom{n}{2}x^2 + \cdots$
7.21.4
Approximating Values Using the Binomial Theorem
Write the number as $(1 + x)^n$ with $x$ small and keep the first few terms. Later terms are tiny.
e.g. $1.02^5 \approx 1 + 5(0.02) + 10(0.0004) = 1.104$
$\displaystyle (1 + x)^n \approx 1 + nx + \binom{n}{2}x^2$

Unit 8 7 topics

Rationals & Radicals

Rational Expressions, and Radical Equations.

8.22 Rational Expressions

8.22.1
Simplifying Rational Expressions Using Polynomial Factorization
Factor the numerator and denominator fully, then cancel common factors, never single terms. Values that make the original denominator zero stay excluded.
e.g. $\tfrac{x^2 - 9}{x^2 + x - 6} = \tfrac{(x - 3)(x + 3)}{(x + 3)(x - 2)} = \tfrac{x - 3}{x - 2}$ , $x \ne -3, 2$
$\displaystyle \frac{ac}{bc} = \frac{a}{b} \quad (b, c \ne 0)$
8.22.2
Adding and Subtracting Rational Expressions
Factor the denominators, find the least common denominator, rewrite each fraction over it, and combine numerators. When subtracting, distribute the minus sign to the whole numerator.
e.g. $\tfrac{x}{x^2 - 1} - \tfrac{1}{x - 1} = \tfrac{x - (x + 1)}{(x - 1)(x + 1)} = \tfrac{-1}{x^2 - 1}$
$\displaystyle \frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$
8.22.3
Adding Rational Expressions With No Common Factors in the Denominator
When the denominators share no factor, the common denominator is their product.
e.g. $\tfrac{2}{x} + \tfrac{3}{x + 1} = \tfrac{2(x + 1) + 3x}{x(x + 1)} = \tfrac{5x + 2}{x(x + 1)}$
$\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
8.22.4
Multiplying Rational Expressions
Factor everything, cancel common factors between any numerator and any denominator, then multiply what remains.
e.g. $\tfrac{x^2 - 4}{3x} \cdot \tfrac{6}{x + 2} = \tfrac{2(x - 2)}{x}$
$\displaystyle \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
8.22.5
Dividing Rational Expressions
Multiply by the reciprocal of the divisor, then simplify. Also exclude values that make the divisor's numerator zero.
e.g. $\tfrac{x + 3}{x} \div \tfrac{x^2 - 9}{x^2} = \tfrac{x + 3}{x} \cdot \tfrac{x^2}{(x - 3)(x + 3)} = \tfrac{x}{x - 3}$
$\displaystyle \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

8.23 Radical Equations

8.23.1
Solving Radical Equations
Isolate the radical, raise both sides to its power, solve, then check every answer. Squaring can create false solutions.
e.g. Gives $x = 1$ or $6$ ; only $6$ checks
$\displaystyle \sqrt{x + 3} = x - 3$
$\displaystyle \Rightarrow\; x + 3 = x^2 - 6x + 9$
8.23.2
Solving Advanced Radical Equations
Isolate one radical and square both sides. If a radical remains, isolate it and square again. Check every answer in the original equation, since squaring can create extraneous solutions.
e.g. $\sqrt{x + 5} = 1 + \sqrt{x} \Rightarrow x + 5 = 1 + 2\sqrt{x} + x \Rightarrow \sqrt{x} = 2 \Rightarrow x = 4$
$\displaystyle \sqrt{A} = B \;\Rightarrow\; A = B^2$
$\displaystyle \text{valid only if } B \ge 0$

Unit 9 53 topics

Geometry

Similarity, Circles, Polygons, and more.

9.24 Similarity

Multiplying every side by k = 2 keeps the angles the same. Area grows by k², so the bigger triangle has 4 times the area.

9.24.1
Similarity and Similar Polygons
Two polygons are similar ( $\sim$ ) when all corresponding angles are congruent and all corresponding sides are in the same ratio. That ratio is the scale factor $k$ .
e.g. A $2 \times 3$ rectangle and a $4 \times 6$ rectangle are similar with $k = 2$ .
$\displaystyle ABCD \sim EFGH$
$\displaystyle \frac{EF}{AB} = \frac{FG}{BC} = \cdots = k$
9.24.2
Side Lengths and Angle Measures of Similar Polygons
Corresponding angles have equal measures. To find a missing side, set up a proportion using sides in the same order as the similarity statement.
e.g. $\triangle ABC \sim \triangle DEF$ , $AB = 4$ , $DE = 6$ , $BC = 5$ : $\tfrac46 = \tfrac{5}{EF} \Rightarrow EF = 7.5$
$\displaystyle \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$
9.24.3
Areas of Similar Polygons
If the scale factor between similar polygons is $k$ , the ratio of their areas is $k^2$ .
e.g. Scale factor 3 and smaller area 5 cm²: larger area $= 5 \cdot 9 = 45$ cm²
$\displaystyle \frac{A_2}{A_1} = \left(\frac{s_2}{s_1}\right)^2 = k^2$
9.24.4
Working With Areas of Similar Polygons
To go from an area ratio back to a side ratio, take the square root.
e.g. Areas 16 and 25 give side ratio $4 : 5$ . A side of 8 on the smaller figure matches $10$ on the larger.
$\displaystyle k = \sqrt{\frac{A_2}{A_1}}$
9.24.5
Similarity Transformations
A similarity transformation is a dilation combined with any rigid motions (translations, rotations, reflections). Two figures are similar exactly when one maps onto the other by such a transformation.
e.g. Dilate $(1, 3)$ by 2, then shift right 1: $(1, 3) \to (2, 6) \to (3, 6)$
$\displaystyle \text{dilation about origin:}$
$\displaystyle (x, y) \to (kx, ky)$
9.24.6
The AA Similarity Criterion
If two angles of one triangle are congruent to two angles of another, the triangles are similar.
e.g. Two triangles each have angles $40^\circ$ and $60^\circ$ , so they are similar (both have a third angle of $80^\circ$ ).
$\displaystyle \angle A \cong \angle D,\; \angle B \cong \angle E$
$\displaystyle \Rightarrow\; \triangle ABC \sim \triangle DEF$
9.24.7
The SSS Similarity Criterion
If all three pairs of corresponding sides are in the same ratio, the triangles are similar.
e.g. Sides 3, 4, 5 and 6, 8, 10: every ratio is 2, so the triangles are similar.
$\displaystyle \frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$
9.24.8
The SAS Similarity Criterion
If two pairs of sides are in the same ratio and the angles between them (the included angles) are congruent, the triangles are similar.
e.g. $AB = 4$ , $AC = 6$ , $DE = 6$ , $DF = 9$ , $\angle A = \angle D = 50^\circ$ : both ratios are 1.5, so similar.
$\displaystyle \frac{DE}{AB} = \frac{DF}{AC},\; \angle A \cong \angle D$
9.24.9
Combining Similarity Criteria for Triangles
Pick the criterion that fits the given facts: AA for angles, SSS for three sides, SAS for two sides and the included angle. Look for hidden facts such as a shared angle, vertical angles, or parallel lines.
e.g. $D$ on $\overline{AB}$ , $E$ on $\overline{AC}$ , $AD = 2$ , $AB = 6$ , $AE = 3$ , $AC = 9$ : ratios both $\tfrac13$ with shared $\angle A$ , so $\triangle ADE \sim \triangle ABC$ by SAS.
9.24.10
Proving Similarity Statements
Back every condition of a criterion with a reason: given, reflexive property, vertical angles, or angles from parallel lines. After proving similarity, corresponding sides are proportional.
e.g. $\overline{AB} \parallel \overline{DE}$ with $\overline{AE}$ and $\overline{BD}$ crossing at $C$ : vertical angles at $C$ and alternate interior angles $\angle A \cong \angle E$ give $\triangle ABC \sim \triangle EDC$ by AA.
$\displaystyle \angle A \cong \angle A \quad \text{reflexive}$
$\displaystyle \angle ADE \cong \angle B \quad \text{corr. angles}$
$\displaystyle \triangle ADE \sim \triangle ABC \quad \text{AA}$
9.24.11
The Midpoint Theorem
The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.
e.g. $D$ , $E$ are midpoints of $\overline{AB}$ , $\overline{AC}$ and $BC = 14$ , so $DE = 7$ .
$\displaystyle DE \parallel BC$
$\displaystyle DE = \tfrac12 BC$
9.24.12
Proving the Midpoint Theorem
Show the small triangle is similar to the big one by SAS. Then the side ratio gives the length, and congruent corresponding angles give the parallel lines.
e.g. $\angle ADE \cong \angle B$ are corresponding angles, so $DE \parallel BC$ by the converse of the corresponding angles theorem.
$\displaystyle \tfrac{AD}{AB} = \tfrac{AE}{AC} = \tfrac12 \quad \text{midpoints}$
$\displaystyle \angle A \cong \angle A \quad \text{reflexive}$
$\displaystyle \triangle ADE \sim \triangle ABC \quad \text{SAS}$
$\displaystyle \Rightarrow\; DE = \tfrac12 BC,\; DE \parallel BC$
9.24.13
The Triangle Proportionality Theorem
A line parallel to one side of a triangle divides the other two sides proportionally. The converse is also true.
e.g. $AD = 4$ , $DB = 6$ , $AE = 6$ : $\tfrac46 = \tfrac{6}{EC} \Rightarrow EC = 9$
$\displaystyle DE \parallel BC$
$\displaystyle \Rightarrow\; \frac{AD}{DB} = \frac{AE}{EC}$

9.25 Circles

An inscribed angle is half the central angle that stands on the same arc. A diameter cuts off a 180° arc, so any angle standing on it is 90°.

9.25.1
Inscribed Angles
An inscribed angle has its vertex on the circle and chords as sides. It measures half its intercepted arc. A central angle equals its arc.
e.g. Intercepted arc $110^\circ$ : inscribed angle $55^\circ$
$\displaystyle m\angle ACB = \tfrac12\, m\overset{\frown}{AB}$
9.25.2
Problem-Solving With Inscribed Angles
Inscribed angles that intercept the same arc are congruent. An inscribed angle is half the central angle on the same arc. Arcs around a circle total $360^\circ$ .
e.g. $\angle ACB$ and $\angle ADB$ intercept the same arc: $3x + 5 = 5x - 15 \Rightarrow x = 10$ , so each angle is $35^\circ$ .
$\displaystyle m\angle \text{inscribed} = \tfrac12\, m\angle \text{central}$
9.25.3
Thales' Theorem
An angle inscribed in a semicircle is a right angle. Conversely, the hypotenuse of a right triangle inscribed in a circle is a diameter.
e.g. Inscribed right triangle with legs 6 and 8: hypotenuse $= 10$ is a diameter, so the radius is 5.
$\displaystyle AB \text{ diameter} \Rightarrow m\angle ACB = 90^\circ$
9.25.4
Angles in Inscribed Right Triangles
If one side of an inscribed triangle is a diameter, the angle opposite it is $90^\circ$ and the other two angles add to $90^\circ$ .
e.g. $\overline{AB}$ is a diameter and $m\angle CAB = 35^\circ$ : $m\angle ABC = 55^\circ$
$\displaystyle m\angle A + m\angle B = 90^\circ$
9.25.5
Further Angles in Inscribed Right Triangles
Draw the radius $\overline{OC}$ to the right-angle vertex. It splits the triangle into two isosceles triangles, since all radii are equal. A central angle is twice the inscribed angle on the same arc.
e.g. $m\angle CAB = 25^\circ$ : $m\angle OCA = 25^\circ$ , $m\angle AOC = 130^\circ$ , $m\angle BOC = 50^\circ$ , $m\angle ABC = 65^\circ$
$\displaystyle OA = OB = OC = r$
$\displaystyle m\angle BOC = 2\, m\angle BAC$
9.25.6
Inscribed Quadrilaterals
A quadrilateral inscribed in a circle has opposite angles that are supplementary. The converse is also true.
e.g. $m\angle A = 70^\circ \Rightarrow m\angle C = 110^\circ$
$\displaystyle m\angle A + m\angle C = 180^\circ$
$\displaystyle m\angle B + m\angle D = 180^\circ$
9.25.7
Tangent Lines to Circles
A tangent line touches the circle at exactly one point and is perpendicular to the radius at that point. This makes a right triangle with the center and any outside point on the tangent.
e.g. Radius 5, center to outside point 13: tangent length $= \sqrt{169 - 25} = 12$
$\displaystyle \overline{OP} \perp \text{tangent at } P$
$\displaystyle r^2 + t^2 = d^2$
9.25.8
Tangent-Tangent Lines to Circles
Two tangent segments from the same outside point are congruent. The angle between them is half the difference of the far and near arcs, which equals $180^\circ$ minus the central angle.
e.g. Near arc $120^\circ$ , far arc $240^\circ$ : $m\angle P = \tfrac12(240 - 120) = 60^\circ$
$\displaystyle PA = PB$
$\displaystyle m\angle P = \tfrac12(\text{far} - \text{near})$
$\displaystyle = 180^\circ - m\angle AOB$
9.25.9
Secant-Secant Angles to Circles
Two secants meeting outside the circle form an angle equal to half the difference of the intercepted arcs. Two chords crossing inside form an angle equal to half the sum of the intercepted arcs.
e.g. Outside, arcs $100^\circ$ and $30^\circ$ : $35^\circ$ . Inside, arcs $80^\circ$ and $40^\circ$ : $60^\circ$ .
$\displaystyle \text{outside: } \tfrac12(\text{far} - \text{near})$
$\displaystyle \text{inside: } \tfrac12(\text{arc}_1 + \text{arc}_2)$
9.25.10
Secant-Tangent Angles to Circles
A secant and a tangent meeting outside the circle form an angle equal to half the difference of the intercepted arcs.
e.g. Arcs $150^\circ$ and $50^\circ$ : $m\angle P = \tfrac12(100) = 50^\circ$
$\displaystyle m\angle P = \tfrac12(\text{far} - \text{near})$
9.25.11
The Tangent-Chord Theorem
The angle between a tangent and a chord at the point of tangency is half the intercepted arc. It equals any inscribed angle on that same arc.
e.g. Tangent and chord intercept a $140^\circ$ arc: the angle is $70^\circ$ .
$\displaystyle m\angle = \tfrac12\, m\overset{\frown}{AB}$
9.25.12
Circle Similarity
All circles are similar. The scale factor is the ratio of the radii, so circumferences scale by $k$ and areas by $k^2$ .
e.g. Radii 3 and 6: $k = 2$ , circumference ratio 2, area ratio 4
$\displaystyle k = \frac{r_2}{r_1}$
$\displaystyle \frac{C_2}{C_1} = k,\quad \frac{A_2}{A_1} = k^2$
9.25.13
Proving Circle Similarity
Translate one circle so its center lands on the other's center. Then dilate about that center by the ratio of the radii. A similarity transformation maps one onto the other.
e.g. Center $(1, 2)$ , $r = 2$ and center $(4, 0)$ , $r = 6$ : translate by $\langle 3, -2 \rangle$ , then dilate by 3 about $(4, 0)$ .
$\displaystyle \text{translate } O_1 \text{ to } O_2$
$\displaystyle \text{dilate by } k = \tfrac{r_2}{r_1} \text{ about } O_2$
$\displaystyle \Rightarrow\; \text{the circles are similar}$

9.26 Polygons

Arrowheads mark parallel sides, tick marks mark equal lengths, and matching arcs mark equal angles. The diagonals cut each other in half at M.

9.26.1
Proving Triangle Theorems
The angles of a triangle add to $180^\circ$ . An exterior angle equals the sum of the two remote interior angles. Proof of the sum: draw a line through one vertex parallel to the opposite side.
e.g. Remote interior angles $50^\circ$ and $70^\circ$ : exterior angle $120^\circ$
$\displaystyle \text{draw } \ell \parallel BC \text{ through } A$
$\displaystyle \angle 1 \cong \angle B,\ \angle 3 \cong \angle C \quad \text{alt. int.}$
$\displaystyle \angle 1 + \angle A + \angle 3 = 180^\circ \quad \text{line}$
9.26.2
Proofs Involving Isosceles Triangles
Base angles theorem: if two sides are congruent, the angles opposite them are congruent, and conversely. Proof: bisect the vertex angle with $\overline{AD}$ , which is shared by both halves (reflexive).
e.g. Vertex angle $40^\circ$ : each base angle is $\tfrac12(180 - 40) = 70^\circ$
$\displaystyle AB \cong AC \quad \text{given}$
$\displaystyle \angle BAD \cong \angle CAD \quad \text{bisector}$
$\displaystyle \triangle ABD \cong \triangle ACD \quad \text{SAS}$
$\displaystyle \angle B \cong \angle C \quad \text{CPCTC}$
9.26.3
Proofs Involving Medians and Centroids
A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid $G$ , which divides each median $2 : 1$ from the vertex. Proof below: medians $\overline{BE}$ and $\overline{CF}$ meet at $G$ .
e.g. A median of length 18: vertex to centroid 12, centroid to midpoint 6
$\displaystyle FE \parallel BC,\ FE = \tfrac12 BC \quad \text{midpt.}$
$\displaystyle \triangle GFE \sim \triangle GCB \quad \text{AA}$
$\displaystyle \Rightarrow\; BG = 2\,GE,\; CG = 2\,GF$
9.26.4
Diagonals of Squares
A diagonal cuts a square into two isosceles right triangles.
e.g. Side 7 → diagonal $7\sqrt2$
$\displaystyle d = s\sqrt{2}$
$\displaystyle A = \tfrac{1}{2}d^2$
9.26.5
Properties of Parallelograms
Opposite sides are parallel and congruent. Opposite angles are congruent and consecutive angles are supplementary. The diagonals bisect each other.
e.g. Diagonals meet at $E$ with $AE = 2x + 1$ , $EC = 3x - 4$ : $x = 5$ , so $AC = 22$
$\displaystyle AB = CD,\; BC = DA$
$\displaystyle m\angle A = m\angle C$
$\displaystyle m\angle A + m\angle B = 180^\circ$
9.26.6
Proving Properties of Parallelograms
Draw diagonal $\overline{AC}$ , which is shared by both triangles (reflexive). Each pair of parallel sides gives alternate interior angles, so the triangles are congruent by ASA.
e.g. CPCTC also gives $\angle B \cong \angle D$ , so opposite angles are congruent.
$\displaystyle \angle BAC \cong \angle DCA \quad \text{alt. int.}$
$\displaystyle \angle BCA \cong \angle DAC \quad \text{alt. int.}$
$\displaystyle \triangle ABC \cong \triangle CDA \quad \text{ASA}$
$\displaystyle \Rightarrow\; AB \cong CD,\ BC \cong DA$
9.26.7
Proving Criteria for Parallelograms and Rectangles
A quadrilateral is a parallelogram if both pairs of opposite sides are congruent, one pair is both congruent and parallel, both pairs of opposite angles are congruent, or the diagonals bisect each other. A parallelogram with congruent diagonals or one right angle is a rectangle.
e.g. The diagonals of $ABCD$ bisect each other and are both 10 long, so $ABCD$ is a rectangle.
$\displaystyle AE \cong EC,\ BE \cong ED \quad \text{given}$
$\displaystyle \triangle AEB \cong \triangle CED \quad \text{SAS}$
$\displaystyle \angle BAE \cong \angle DCE \quad \text{CPCTC}$
$\displaystyle \Rightarrow\; AB \parallel CD \quad \text{alt. int.}$

9.27 Circles

A circle is every point a fixed distance r from its center. The distance formula squared gives the equation, so the center is (2, 1) and the radius is 3.

9.27.1
Circles in the Coordinate Plane
A circle is every point at distance $r$ from the center. Use the distance formula to find a radius or test whether a point is on the circle.
e.g. Center $(1, 2)$ through $(4, 6)$ : $r = \sqrt{9 + 16} = 5$
$\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
9.27.2
Equations of Circles Centered at the Origin
A circle with center $(0, 0)$ and radius $r$ .
e.g. Through $(3, 4)$ : $r^2 = 9 + 16 = 25$ , so $x^2 + y^2 = 25$
$\displaystyle x^2 + y^2 = r^2$
9.27.3
Equations of Circles
Standard form for center $(h, k)$ and radius $r$ . Watch the signs: $(x + 3)$ means $h = -3$ .
e.g. Center $(2, -3)$ , $r = 4$ : $(x - 2)^2 + (y + 3)^2 = 16$
$\displaystyle (x - h)^2 + (y - k)^2 = r^2$
9.27.4
Determining Circle Properties by Completing the Square
Group the $x$ terms and the $y$ terms, move the constant, and complete the square on each. Add the same amounts to both sides.
e.g. $x^2 + y^2 - 6x + 4y - 3 = 0 \Rightarrow (x - 3)^2 + (y + 2)^2 = 16$ : center $(3, -2)$ , $r = 4$
$\displaystyle x^2 + Dx = \left(x + \tfrac{D}{2}\right)^2 - \tfrac{D^2}{4}$
9.27.5
Calculating Circle Intercepts
For $x$ -intercepts set $y = 0$ . For $y$ -intercepts set $x = 0$ . A circle can have 0, 1, or 2 of each.
e.g. $(x - 3)^2 + y^2 = 25$ : $x = 8, -2$ and $y = \pm 4$
$\displaystyle x\text{-intercepts: } y = 0$
$\displaystyle y\text{-intercepts: } x = 0$
9.27.6
Intersections of Circles with Lines
Substitute the line into the circle and solve the quadratic. Two solutions means the line crosses twice, one means it is tangent, none means it misses.
e.g. $x^2 + y^2 = 25$ , $y = x + 1$ : $x^2 + x - 12 = 0$ , so $(3, 4)$ and $(-4, -3)$
9.27.7
Equations of Tangent Lines to Circles
The tangent at point $P$ is perpendicular to the radius to $P$ . Its slope is the negative reciprocal of the radius slope. A horizontal radius gives a vertical tangent, and vice versa.
e.g. $x^2 + y^2 = 25$ at $(3, 4)$ : radius slope $\tfrac43$ , tangent $y - 4 = -\tfrac34(x - 3)$
$\displaystyle m_{\text{tan}} = -\frac{1}{m_{\text{radius}}}$
$\displaystyle y - y_1 = m(x - x_1)$
9.27.8
Perpendicular Bisectors of Diameters in the Coordinate Plane
The midpoint of a diameter is the center. The perpendicular bisector of a diameter passes through the center with the negative reciprocal slope, and it contains the perpendicular diameter.
e.g. Diameter from $(1, 2)$ to $(5, 6)$ : center $(3, 4)$ , slope 1, bisector $y = -x + 7$
$\displaystyle M = \left(\tfrac{x_1 + x_2}{2}, \tfrac{y_1 + y_2}{2}\right)$
$\displaystyle m_\perp = -\tfrac{1}{m}$
9.27.9
Perpendicular Bisectors of Chords
The perpendicular bisector of any chord passes through the center. A radius perpendicular to a chord bisects it, which forms a right triangle with the radius.
e.g. Radius 5, chord 8: distance from center $= \sqrt{25 - 16} = 3$
$\displaystyle \left(\tfrac{c}{2}\right)^2 + d^2 = r^2$
9.27.10
Perpendicular Bisectors of Chords in the Coordinate Plane
Find the perpendicular bisectors of two chords. They meet at the center. Then the radius is the distance from the center to any point on the circle.
e.g. Points $(0, 0)$ , $(6, 0)$ , $(0, 4)$ : bisectors $x = 3$ and $y = 2$ , center $(3, 2)$ , $r = \sqrt{13}$
9.27.11
Thales' Theorem in the Coordinate Plane
If $\overline{AB}$ is a diameter and $C$ is another point on the circle, then $\overline{CA} \perp \overline{CB}$ , so their slopes multiply to $-1$ . Conversely, the hypotenuse of an inscribed right triangle is a diameter, and its midpoint is the center.
e.g. $x^2 + y^2 = 25$ , $A(-5, 0)$ , $B(5, 0)$ , $C(3, 4)$ : slopes $\tfrac12$ and $-2$ , product $-1$
$\displaystyle m_{CA} \cdot m_{CB} = -1$

9.28 Parabolas

Every point on a parabola is the same distance from the focus as from the directrix. For x² = 4y, p = 1, so the focus is 1 above the vertex and the directrix is 1 below.

9.28.1
Upward and Downward Opening Parabolas
A parabola with a vertical axis has vertex $(h, k)$ . It opens up if $p > 0$ and down if $p < 0$ .
e.g. $x^2 = -12y$ : $4p = -12$ , $p = -3$ , opens down
$\displaystyle (x - h)^2 = 4p(y - k)$
$\displaystyle p > 0 \text{ up},\quad p < 0 \text{ down}$
9.28.2
Left and Right Opening Parabolas
When $y$ is the squared variable, the axis is horizontal. It opens right if $p > 0$ and left if $p < 0$ .
e.g. $y^2 = -8x$ : $4p = -8$ , $p = -2$ , opens left
$\displaystyle (y - k)^2 = 4p(x - h)$
$\displaystyle p > 0 \text{ right},\quad p < 0 \text{ left}$
9.28.3
The Vertex of a Parabola
The vertex is the turning point, $(h, k)$ in either form. It lies on the axis of symmetry, halfway between the focus and the directrix.
e.g. $(y + 1)^2 = 12(x - 4)$ : vertex $(4, -1)$ , opens right
$\displaystyle (x - h)^2 = 4p(y - k) \to (h, k)$
$\displaystyle (y - k)^2 = 4p(x - h) \to (h, k)$
9.28.4
Calculating the Vertex of a Parabola by Completing the Square
Put the squared variable's terms on one side, complete the square, and factor the other side into the form $4p(\ldots)$ . If the squared term has a coefficient, divide by it first.
e.g. $y = x^2 - 6x + 5 \Rightarrow (x - 3)^2 = y + 4$ : vertex $(3, -4)$
$\displaystyle x^2 + bx = \left(x + \tfrac{b}{2}\right)^2 - \tfrac{b^2}{4}$
9.28.5
The Focus-Directrix Property of a Parabola
Every point on a parabola is the same distance from the focus (a point) and the directrix (a line). The vertex is midway between them, $\lvert p \rvert$ from each.
e.g. Focus $(0, 2)$ , directrix $y = -2$ : $x^2 = 8y$ . Point $(4, 2)$ is 4 from both.
$\displaystyle PF = PD$
$\displaystyle \sqrt{x^2 + (y - p)^2} = \lvert y + p \rvert$
$\displaystyle \Rightarrow\; x^2 = 4py$
9.28.6
Calculating the Focus of a Parabola
Find the vertex and $p$ , then move $p$ units from the vertex along the axis, toward the inside of the curve.
e.g. $(x - 1)^2 = 8(y + 2)$ : $p = 2$ , vertex $(1, -2)$ , focus $(1, 0)$
$\displaystyle (x - h)^2 = 4p(y - k) \to (h, k + p)$
$\displaystyle (y - k)^2 = 4p(x - h) \to (h + p, k)$
9.28.7
Calculating the Directrix of a Parabola
The directrix is the line $p$ units from the vertex on the opposite side from the focus.
e.g. $(y - 3)^2 = -12(x - 1)$ : $p = -3$ , vertex $(1, 3)$ , directrix $x = 4$
$\displaystyle (x - h)^2 = 4p(y - k) \to y = k - p$
$\displaystyle (y - k)^2 = 4p(x - h) \to x = h - p$
9.28.8
Calculating Intercepts of Parabolas
For $x$ -intercepts set $y = 0$ . For $y$ -intercepts set $x = 0$ . A parabola can have 0, 1, or 2 of each.
e.g. $(x - 2)^2 = 4(y + 1)$ : $y = 0$ gives $x = 0, 4$ . $x = 0$ gives $y = 0$ .
$\displaystyle x\text{-intercepts: } y = 0$
$\displaystyle y\text{-intercepts: } x = 0$
9.28.9
Intersections of Parabolas With Lines
Substitute the line into the parabola and solve the quadratic. Two solutions means two crossing points, one means the line touches once, none means no intersection.
e.g. $x^2 = 4y$ , $y = x + 3$ : $x^2 - 4x - 12 = 0$ , so $(6, 9)$ and $(-2, 1)$

Unit 10 26 topics

Solid Geometry

Introduction to Solid Geometry, Rectangular Solids and Pyramids, and Non-Polyhedrons.

10.29 Introduction to Solid Geometry

Six squares in a cross fold up into a cube. Counting vertices, edges, and faces on any polyhedron like this one gives V − E + F = 2.

10.29.1
Identifying Three-Dimensional Shapes
A prism has two congruent, parallel bases joined by rectangles or parallelograms. A pyramid has one polygon base and triangular faces that meet at a point (the apex). Cylinders, cones, and spheres have curved surfaces, so they are not polyhedrons.
e.g. Two triangular bases and three rectangular faces: a triangular prism
10.29.2
Faces, Vertices, and Edges of Polyhedrons
A polyhedron is a solid whose faces are all flat polygons. Edges are where two faces meet, and vertices are the corners where edges meet. For a base with $n$ sides:
e.g. Hexagonal prism ( $n = 6$ ): $F = 8$ , $V = 12$ , $E = 18$
$\displaystyle \text{Prism: } F = n + 2,\; V = 2n,\; E = 3n$
$\displaystyle \text{Pyramid: } F = n + 1,\; V = n + 1,\; E = 2n$
10.29.3
Nets of Polyhedrons
A net is a flat pattern that folds up into a solid. Every face appears exactly once, and faces that share an edge in the solid must connect correctly when folded.
e.g. A square pyramid's net is a square with a triangle attached to each of its four sides.
10.29.4
Finding Surface Areas Using Nets
Surface area is the total area of all the faces. Unfold the solid into a net, find the area of each face, and add them.
e.g. $2 \times 3 \times 4$ box: faces $6, 6, 8, 8, 12, 12$ , so $SA = 52$
10.29.5
The Distance Formula in Three Dimensions
Like the 2D distance formula, with a third squared difference for $z$ .
e.g. $(1, 2, 3)$ to $(3, 5, 9)$ : $\sqrt{4 + 9 + 36} = \sqrt{49} = 7$
$\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
10.29.6
Euler's Formula for Polyhedra
For any convex polyhedron, vertices minus edges plus faces equals 2. Use it to find a missing count.
e.g. $V = 12$ , $E = 30$ : $F = 2 - 12 + 30 = 20$
$\displaystyle V - E + F = 2$
10.29.7
The Five Platonic Solids
A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces meeting at every vertex. There are exactly five: tetrahedron (4 triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), and icosahedron (20 triangles).
e.g. Octahedron: $E = \tfrac{8 \times 3}{2} = 12$ , $V = 2 - 8 + 12 = 6$
$\displaystyle E = \frac{F \times (\text{sides per face})}{2}$
10.29.8
Surface Areas and Volumes of Similar Solids
If two similar solids have scale factor $k$ (ratio of corresponding lengths), their surface areas are in ratio $k^2$ and their volumes in ratio $k^3$ .
e.g. Lengths $2 : 3$ , so volumes $8 : 27$ . Small volume 16 → large volume $16 \cdot \tfrac{27}{8} = 54$
$\displaystyle \frac{A_1}{A_2} = k^2$
$\displaystyle \frac{V_1}{V_2} = k^3$

10.30 Rectangular Solids and Pyramids

The face diagonal across the bottom and the height form a right triangle whose hypotenuse is the space diagonal. Using the Pythagorean theorem twice gives d = √(l² + w² + h²).

10.30.1
Volumes of Cubes
Cube the edge length. To find the edge from the volume, take the cube root.
e.g. $s = 4$ : $V = 64$ . $V = 125$ : $s = \sqrt[3]{125} = 5$
$\displaystyle V = s^3$
10.30.2
Surface Areas of Cubes
A cube has 6 congruent square faces.
e.g. $s = 3$ : $SA = 6 \cdot 9 = 54$
$\displaystyle SA = 6s^2$
10.30.3
Face Diagonals of Cubes
A face diagonal crosses one square face from corner to corner. It is the hypotenuse of a 45-45-90 triangle with legs $s$ .
e.g. $s = 5$ : $d = 5\sqrt2 \approx 7.07$
$\displaystyle d = s\sqrt{2}$
10.30.4
Diagonals of Cubes
The space diagonal runs through the inside of the cube between opposite vertices. It is the hypotenuse of a right triangle with legs $s$ and $s\sqrt2$ .
e.g. $s = 4$ : $D = 4\sqrt3 \approx 6.93$
$\displaystyle D = s\sqrt{3}$
10.30.5
Volumes of Rectangular Solids
Multiply length, width, and height.
e.g. $3 \times 4 \times 5$ : $V = 60$
$\displaystyle V = lwh$
10.30.6
Surface Areas of Rectangular Solids
Three pairs of congruent rectangular faces.
e.g. $3 \times 4 \times 5$ : $2(12 + 15 + 20) = 94$
$\displaystyle SA = 2(lw + lh + wh)$
10.30.7
Diagonals of Rectangular Solids
The space diagonal connects opposite corners through the interior. It extends the Pythagorean theorem to three dimensions.
e.g. $3 \times 4 \times 12$ : $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$
$\displaystyle d = \sqrt{l^2 + w^2 + h^2}$
10.30.8
Volumes of Pyramids
One third of base area times height. The height $h$ is perpendicular from the apex to the base, not the slant height.
e.g. Square base side 6, $h = 5$ : $V = \tfrac13 \cdot 36 \cdot 5 = 60$
$\displaystyle V = \frac{1}{3}Bh$
10.30.9
Surface Areas of Pyramids
Add the base area to the areas of the triangular faces. For a regular pyramid, the lateral area is half the base perimeter times the slant height $\ell$ .
e.g. Square base side 6, $\ell = 5$ : $36 + \tfrac12 \cdot 24 \cdot 5 = 36 + 60 = 96$
$\displaystyle SA = B + L$
$\displaystyle L = \frac{1}{2}P\ell \quad \text{(regular)}$

10.31 Non-Polyhedrons

A cone holds exactly one third of the cylinder with the same base and height. The cone’s slant height ℓ, not its height h, goes in the surface area.

10.31.1
Volumes of Cylinders
Base area times height, like a prism with a circular base.
e.g. $r = 3$ , $h = 5$ : $V = 45\pi \approx 141.4$
$\displaystyle V = \pi r^2 h$
10.31.2
Surface Areas of Cylinders
Two circular bases plus the lateral surface, which unrolls into a rectangle $2\pi r$ wide and $h$ tall.
e.g. $r = 3$ , $h = 5$ : $18\pi + 30\pi = 48\pi \approx 150.8$
$\displaystyle SA = 2\pi r^2 + 2\pi r h$
10.31.3
Volumes of Right Cones
One third of a cylinder with the same base and height. Use the perpendicular height, not the slant height.
e.g. $r = 3$ , $h = 4$ : $V = \tfrac13 \cdot 9 \cdot 4\pi = 12\pi \approx 37.7$
$\displaystyle V = \frac{1}{3}\pi r^2 h$
10.31.4
Slant Heights of Right Cones
The radius, height, and slant height form a right triangle with the slant height $\ell$ as the hypotenuse.
e.g. $r = 5$ , $h = 12$ : $\ell = \sqrt{169} = 13$
$\displaystyle \ell = \sqrt{r^2 + h^2}$
10.31.5
Surface Areas of Right Cones
Circular base plus the lateral area $\pi r \ell$ , where $\ell$ is the slant height.
e.g. $r = 3$ , $\ell = 5$ : $9\pi + 15\pi = 24\pi \approx 75.4$
$\displaystyle SA = \pi r^2 + \pi r \ell$
10.31.6
Volumes of Spheres
A hemisphere has half the volume of the sphere.
e.g. $r = 3$ : $V = \tfrac43 \cdot 27\pi = 36\pi \approx 113.1$
$\displaystyle V = \frac{4}{3}\pi r^3$
10.31.7
Surface Areas of Spheres
Four times the area of a great circle. A hemisphere's curved surface is $2\pi r^2$ ; add $\pi r^2$ for its flat base.
e.g. $r = 5$ : $SA = 100\pi \approx 314.2$
$\displaystyle SA = 4\pi r^2$
10.31.8
Conical Frustums
A frustum is the part of a cone left after cutting off the top with a plane parallel to the base. It has radii $R$ (bottom) and $r$ (top) and height $h$ . You can also find its volume as the big cone minus the small cone.
e.g. $R = 4$ , $r = 2$ , $h = 3$ : $V = \pi(16 + 8 + 4) = 28\pi$
$\displaystyle V = \frac{1}{3}\pi h\,(R^2 + Rr + r^2)$
$\displaystyle \ell = \sqrt{h^2 + (R - r)^2}$
$\displaystyle L = \pi (R + r)\ell$
10.31.9
Volumes of Revolution
Spinning a flat region around an axis sweeps out a solid. A rectangle spun around a side makes a cylinder, a right triangle spun around a leg makes a cone, and a semicircle spun around its diameter makes a sphere. Identify $r$ and $h$ , then use the solid's volume formula.
e.g. Right triangle with legs 3 and 6, spun around the leg of 6: cone with $r = 3$ , $h = 6$ , $V = \tfrac13 \cdot 9 \cdot 6\pi = 18\pi$

Unit 11 57 topics

Trigonometry

Introduction to Trigonometry, Radians, The Unit Circle, and more.

11.32 Introduction to Trigonometry

Each trig ratio compares two sides of a right triangle, named from the angle θ. The hypotenuse is always the longest side, across from the right angle.

11.32.1
Angles and Sides in Right Triangles
The hypotenuse is opposite the right angle and is the longest side. For an acute angle $\theta$ , the opposite leg is across from $\theta$ , and the adjacent leg is the leg that forms $\theta$ . The two acute angles add to $90^\circ$ .
e.g. $\angle C = 90^\circ$ , $\angle A = 35^\circ$ : $\angle B = 55^\circ$ . Side $a$ is opposite $A$ ; side $b$ is adjacent to $A$
$\displaystyle \angle A + \angle B = 90^\circ$
$\displaystyle a^2 + b^2 = c^2$
11.32.2
The Trigonometric Ratios
SOH-CAH-TOA. Each ratio compares two sides of a right triangle relative to an acute angle $\theta$ .
e.g. Legs 3 and 4, hypotenuse 5, $\theta$ opposite 3: $\sin\theta = \tfrac35$ , $\cos\theta = \tfrac45$ , $\tan\theta = \tfrac34$
$\displaystyle \sin\theta = \frac{\text{opp}}{\text{hyp}}$
$\displaystyle \cos\theta = \frac{\text{adj}}{\text{hyp}}$
$\displaystyle \tan\theta = \frac{\text{opp}}{\text{adj}}$
11.32.3
Calculating Trigonometric Ratios Using the Pythagorean Theorem
Given one ratio, label two sides of a right triangle, find the third side with the Pythagorean theorem, then read off any other ratio.
e.g. $\sin\theta = \tfrac{5}{13}$ : adj $= \sqrt{169 - 25} = 12$ , so $\cos\theta = \tfrac{12}{13}$ , $\tan\theta = \tfrac{5}{12}$
$\displaystyle \text{opp}^2 + \text{adj}^2 = \text{hyp}^2$
11.32.4
Calculating Side Lengths of Right Triangles Using Trigonometry
Pick the ratio that uses the known angle, the known side, and the unknown side. Solve for the unknown. Set the calculator to degree mode.
e.g. $\theta = 40^\circ$ , adj $= 8$ : opp $= 8\tan 40^\circ \approx 6.71$
$\displaystyle \text{opp} = \text{hyp} \cdot \sin\theta$
$\displaystyle \text{adj} = \text{hyp} \cdot \cos\theta$
$\displaystyle \text{opp} = \text{adj} \cdot \tan\theta$
11.32.5
Calculating Angles in Right Triangles Using Trigonometry
Find a ratio from two known sides, then use the inverse function. $\sin^{-1}$ means inverse sine, not $\tfrac{1}{\sin}$ .
e.g. opp $= 5$ , adj $= 12$ : $\theta = \tan^{-1}\!\left(\tfrac{5}{12}\right) \approx 22.6^\circ$
$\displaystyle \theta = \sin^{-1}\!\left(\frac{\text{opp}}{\text{hyp}}\right)$
$\displaystyle \theta = \cos^{-1}\!\left(\frac{\text{adj}}{\text{hyp}}\right)$
$\displaystyle \theta = \tan^{-1}\!\left(\frac{\text{opp}}{\text{adj}}\right)$
11.32.6
Modeling With Trigonometry
Sketch the right triangle and label what you know. The angle of elevation is measured up from the horizontal, and the angle of depression is measured down from the horizontal. The two are equal when they describe the same line of sight.
e.g. Standing 20 ft from a tree, angle of elevation $50^\circ$ : height $= 20\tan 50^\circ \approx 23.8$ ft
11.32.7
The Reciprocal Trigonometric Ratios
Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent. Flip the fraction.
e.g. Legs 3 and 4, hypotenuse 5, $\theta$ opposite 3: $\csc\theta = \tfrac53$ , $\sec\theta = \tfrac54$ , $\cot\theta = \tfrac43$
$\displaystyle \csc\theta = \frac{1}{\sin\theta} = \frac{\text{hyp}}{\text{opp}}$
$\displaystyle \sec\theta = \frac{1}{\cos\theta} = \frac{\text{hyp}}{\text{adj}}$
$\displaystyle \cot\theta = \frac{1}{\tan\theta} = \frac{\text{adj}}{\text{opp}}$
11.32.8
Trigonometric Ratios in Similar Right Triangles
Right triangles with the same acute angle are similar, so their sides are proportional. A trig ratio depends only on the angle, not on the size of the triangle.
e.g. 3-4-5 and 6-8-10 triangles: $\sin\theta = \tfrac35 = \tfrac{6}{10}$
11.32.9
Trigonometric Functions of Complementary Angles
The two acute angles of a right triangle are complementary, and the side opposite one is adjacent to the other. So each function equals its cofunction of the complement.
e.g. $\sin 25^\circ = \cos 65^\circ$ . If $\sin x = \cos 2x$ : $x + 2x = 90^\circ$ , so $x = 30^\circ$
$\displaystyle \sin\theta = \cos(90^\circ - \theta)$
$\displaystyle \tan\theta = \cot(90^\circ - \theta)$
$\displaystyle \sec\theta = \csc(90^\circ - \theta)$
11.32.10
Special Trigonometric Ratios
Exact values come from the 30-60-90 triangle (sides $1 : \sqrt3 : 2$ ) and the 45-45-90 triangle (sides $1 : 1 : \sqrt2$ ).
e.g. 30-60-90 with hypotenuse 10: short leg 5, long leg $5\sqrt3$
$\displaystyle \sin 30^\circ = \cos 60^\circ = \frac{1}{2}$
$\displaystyle \sin 45^\circ = \cos 45^\circ = \frac{\sqrt2}{2}$
$\displaystyle \sin 60^\circ = \cos 30^\circ = \frac{\sqrt3}{2}$
$\displaystyle \tan 30^\circ = \frac{\sqrt3}{3},\; \tan 45^\circ = 1$
$\displaystyle \tan 60^\circ = \sqrt3$
11.32.11
Calculating Areas of Right Triangles Using Trigonometry
The area is half the product of the legs. If a leg is missing, find it with trig first.
e.g. Hypotenuse 10, angle $30^\circ$ : legs 5 and $5\sqrt3$ , so $A = \tfrac{25\sqrt3}{2} \approx 21.65$
$\displaystyle A = \frac{1}{2} \cdot \text{leg}_1 \cdot \text{leg}_2$
11.32.12
Solving Multiple Right Triangles Using Trigonometry
Start with the triangle that has enough information. Use the side the triangles share to carry the result into the next triangle.
e.g. $\overline{AD} \perp \overline{BC}$ , $AD = 6$ , $\angle B = 45^\circ$ , $\angle C = 30^\circ$ : $BD = 6$ , $DC = \tfrac{6}{\tan 30^\circ} = 6\sqrt3$ , so $BC = 6 + 6\sqrt3$

11.33 Radians

One radian is the angle whose arc is exactly as long as the radius. About 6.28 radii fit around the circle, so 2π radians make a full turn.

11.33.1
Introduction to Radians
One radian is the central angle whose arc equals the radius. A full turn is $2\pi$ radians, so $180^\circ = \pi$ radians.
e.g. $135^\circ = \tfrac{135\pi}{180} = \tfrac{3\pi}{4}$ ; $\tfrac{\pi}{6} = 30^\circ$
$\displaystyle 180^\circ = \pi \text{ rad}$
$\displaystyle \text{rad} = \text{deg} \cdot \frac{\pi}{180}$
$\displaystyle \text{deg} = \text{rad} \cdot \frac{180}{\pi}$
11.33.2
Calculating Arc Length Using Radians
Arc length is radius times the central angle, with $\theta$ in radians. Convert degrees first.
e.g. $r = 6$ , $\theta = \tfrac{2\pi}{3}$ : $s = 4\pi \approx 12.57$
$\displaystyle s = r\theta$
11.33.3
Calculating Areas of Sectors Using Radians
A sector is the slice of a circle between two radii. Use $\theta$ in radians.
e.g. $r = 6$ , $\theta = \tfrac{\pi}{3}$ : $A = \tfrac12 \cdot 36 \cdot \tfrac{\pi}{3} = 6\pi \approx 18.85$
$\displaystyle A = \frac{1}{2}r^2\theta$
11.33.4
Trigonometric Ratios With Radians
The ratios are the same; only the angle units change. Convert to degrees or memorize the special angles, and set the calculator to radian mode.
e.g. $\sin\tfrac{\pi}{6} = \tfrac12$ , $\cos\tfrac{\pi}{4} = \tfrac{\sqrt2}{2}$ , $\tan\tfrac{\pi}{3} = \sqrt3$
$\displaystyle \frac{\pi}{6} = 30^\circ,\; \frac{\pi}{4} = 45^\circ,\; \frac{\pi}{3} = 60^\circ$

11.34 The Unit Circle

On the unit circle, the point at angle θ is (cos θ, sin θ), so the legs of the triangle are cos θ and sin θ. The quadrant labels show which ratios are positive there.

11.34.1
Angles in the Coordinate Plane
An angle in standard position has its vertex at the origin and its initial side on the positive $x$ -axis. A positive angle rotates counterclockwise to its terminal side. The quadrant of the terminal side names the angle's quadrant.
e.g. $200^\circ$ ends in Quadrant III
$\displaystyle \text{QI: } 0^\circ < \theta < 90^\circ$
$\displaystyle \text{QII: } 90^\circ < \theta < 180^\circ$
$\displaystyle \text{QIII: } 180^\circ < \theta < 270^\circ$
$\displaystyle \text{QIV: } 270^\circ < \theta < 360^\circ$
11.34.2
Negative Angles in the Coordinate Plane
A negative angle rotates clockwise from the positive $x$ -axis. Go clockwise through IV, III, II, I.
e.g. $-120^\circ$ goes $30^\circ$ past the negative $y$ -axis, so it ends in Quadrant III
$\displaystyle -90^\circ < \theta < 0^\circ \;\to\; \text{QIV}$
$\displaystyle -180^\circ < \theta < -90^\circ \;\to\; \text{QIII}$
11.34.3
Coterminal Angles
Coterminal angles share the same terminal side. Add or subtract full turns to find them.
e.g. $60^\circ$ is coterminal with $420^\circ$ and $-300^\circ$
$\displaystyle \theta + 360^\circ k \quad (k \text{ an integer})$
$\displaystyle \theta + 2\pi k \quad \text{in radians}$
11.34.4
Calculating Reference Angles
The reference angle is the acute angle between the terminal side and the $x$ -axis. First find a coterminal angle from $0^\circ$ to $360^\circ$ , then use its quadrant.
e.g. $210^\circ$ is in QIII, so its reference angle is $210^\circ - 180^\circ = 30^\circ$
$\displaystyle \text{QI: } \theta$
$\displaystyle \text{QII: } 180^\circ - \theta$
$\displaystyle \text{QIII: } \theta - 180^\circ$
$\displaystyle \text{QIV: } 360^\circ - \theta$
11.34.5
Properties of the Unit Circle in the First Quadrant
The unit circle has radius 1 and center at the origin. The terminal side of angle $\theta$ meets it at the point $(\cos\theta, \sin\theta)$ .
e.g. At $30^\circ$ the point is $\left(\tfrac{\sqrt3}{2}, \tfrac12\right)$ , and $\tfrac34 + \tfrac14 = 1$
$\displaystyle x^2 + y^2 = 1$
$\displaystyle (x, y) = (\cos\theta, \sin\theta)$
11.34.6
Extending the Trigonometric Ratios Using the Unit Circle
For any angle, use a point $(x, y)$ on the terminal side at distance $r$ from the origin. On the unit circle $r = 1$ . Signs by quadrant: all positive in I, only sin in II, only tan in III, only cos in IV.
e.g. Through $(-3, 4)$ : $r = 5$ , $\sin\theta = \tfrac45$ , $\cos\theta = -\tfrac35$ , $\tan\theta = -\tfrac43$
$\displaystyle \sin\theta = \frac{y}{r},\; \cos\theta = \frac{x}{r}$
$\displaystyle \tan\theta = \frac{y}{x},\; r = \sqrt{x^2 + y^2}$
11.34.7
Extending the Trigonometric Ratios Using Angles in Radians
A full turn is $2\pi$ radians, so $\pi$ radians is $180^\circ$ . Find the reference angle in radians, take the ratio of that angle, then attach the quadrant's sign.
e.g. $\cos\tfrac{4\pi}{3}$ : reference angle $\tfrac{\pi}{3}$ in QIII, so $\cos\tfrac{4\pi}{3} = -\tfrac12$
$\displaystyle \text{rad} = \text{deg} \cdot \frac{\pi}{180}$
$\displaystyle \text{QII: } \pi - \theta,\; \text{QIII: } \theta - \pi$
$\displaystyle \text{QIV: } 2\pi - \theta$
11.34.8
Extending the Trigonometric Ratios to Negative Angles
Use a positive coterminal angle, or use the symmetry rules. Cosine is even. Sine and tangent are odd.
e.g. $\sin(-150^\circ) = -\sin 150^\circ = -\tfrac12$
$\displaystyle \sin(-\theta) = -\sin\theta$
$\displaystyle \cos(-\theta) = \cos\theta$
$\displaystyle \tan(-\theta) = -\tan\theta$
11.34.9
Extending the Trigonometric Ratios to Large Angles
Subtract $360^\circ$ (or $2\pi$ ) until the angle is between $0^\circ$ and $360^\circ$ . Coterminal angles have the same trig ratios.
e.g. $\sin 750^\circ = \sin(750^\circ - 720^\circ) = \sin 30^\circ = \tfrac12$
$\displaystyle \sin(\theta + 360^\circ k) = \sin\theta$
$\displaystyle \cos(\theta + 360^\circ k) = \cos\theta$
11.34.10
Using the Pythagorean Identity in the First Quadrant
Since $(\cos\theta, \sin\theta)$ is on the unit circle, the squares add to 1. Given one ratio, solve for the other. In Quadrant I every ratio is positive.
e.g. $\sin\theta = \tfrac35$ in QI: $\cos\theta = \sqrt{1 - \tfrac{9}{25}} = \tfrac45$ , $\tan\theta = \tfrac34$
$\displaystyle \sin^2\theta + \cos^2\theta = 1$
$\displaystyle \cos\theta = \sqrt{1 - \sin^2\theta}$
11.34.11
Extending the Pythagorean Identity to All Quadrants
The identity holds for every angle. Solving gives $\pm$ , so choose the sign from the quadrant.
e.g. $\sin\theta = -\tfrac{5}{13}$ in QIII: $\cos\theta = -\sqrt{1 - \tfrac{25}{169}} = -\tfrac{12}{13}$
$\displaystyle \sin^2\theta + \cos^2\theta = 1$
$\displaystyle \cos\theta = \pm\sqrt{1 - \sin^2\theta}$
$\displaystyle \sin\theta = \pm\sqrt{1 - \cos^2\theta}$

11.35 Special Trigonometric Ratios

These exact values come from the 30-60-90 and 45-45-90 triangles. At 45° sine and cosine are equal, so tangent is 1.

11.35.1
Finding Trigonometric Ratios of Quadrantal Angles
Quadrantal angles land on an axis. Read $\cos\theta$ and $\sin\theta$ from the point. A ratio is undefined when its denominator is 0: tan and sec at $90^\circ$ and $270^\circ$ , cot and csc at $0^\circ$ and $180^\circ$ .
e.g. $\cos 180^\circ = -1$ , $\sin 270^\circ = -1$ , $\tan 90^\circ$ is undefined
$\displaystyle 0^\circ \to (1, 0),\; 90^\circ \to (0, 1)$
$\displaystyle 180^\circ \to (-1, 0)$
$\displaystyle 270^\circ \to (0, -1)$
11.35.2
Trigonometric Ratios of Quadrantal Angles Outside the Standard Range
Add or subtract full turns to reach $0^\circ$ , $90^\circ$ , $180^\circ$ , or $270^\circ$ , then read the point.
e.g. $\sin(-450^\circ) = \sin 270^\circ = -1$ ; $\cos 5\pi = \cos\pi = -1$
11.35.3
Finding Trigonometric Ratios of Special Angles Using the Unit Circle
Memorize the first-quadrant points. For other angles, use the reference angle's values and the quadrant's signs.
e.g. $\cos 135^\circ = -\tfrac{\sqrt2}{2}$ (reference $45^\circ$ , QII); $\sin 300^\circ = -\tfrac{\sqrt3}{2}$
$\displaystyle 30^\circ\;(\tfrac{\pi}{6}):\; \left(\tfrac{\sqrt3}{2}, \tfrac12\right)$
$\displaystyle 45^\circ\;(\tfrac{\pi}{4}):\; \left(\tfrac{\sqrt2}{2}, \tfrac{\sqrt2}{2}\right)$
$\displaystyle 60^\circ\;(\tfrac{\pi}{3}):\; \left(\tfrac12, \tfrac{\sqrt3}{2}\right)$
11.35.4
Evaluating Trigonometric Expressions
Replace each ratio with its exact value, then simplify. $\sin^2\theta$ means $(\sin\theta)^2$ .
e.g. $2\sin 30^\circ + \cos^2 45^\circ = 2\left(\tfrac12\right) + \tfrac12 = \tfrac32$
11.35.5
Further Extensions of the Special Trigonometric Ratios
Reduce negative or large angles to a coterminal angle from $0^\circ$ to $360^\circ$ , then use the reference angle. The reciprocal ratios flip sin, cos, and tan.
e.g. $\sec(-405^\circ) = \sec 315^\circ = \dfrac{1}{\sqrt2/2} = \sqrt2$
$\displaystyle \csc\theta = \frac{1}{\sin\theta},\; \sec\theta = \frac{1}{\cos\theta}$
$\displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta}$

11.36 Graphing Trigonometric Functions

Sine starts at 0 and cosine starts at 1, but they are the same wave shifted by π/2. Both repeat every 2π and stay between −1 and 1.

11.36.1
Graphing Sine and Cosine
Both waves have period $2\pi$ and range $[-1, 1]$ . Plot the five key points over one period and repeat. Sine starts at the midline going up. Cosine starts at its maximum.
e.g. $\sin\tfrac{5\pi}{2} = \sin\tfrac{\pi}{2} = 1$ because the graph repeats every $2\pi$
$\displaystyle x:\; 0,\; \tfrac{\pi}{2},\; \pi,\; \tfrac{3\pi}{2},\; 2\pi$
$\displaystyle \sin x:\; 0,\; 1,\; 0,\; -1,\; 0$
$\displaystyle \cos x:\; 1,\; 0,\; -1,\; 0,\; 1$
11.36.2
Graphing Tangent and Cotangent
Both have period $\pi$ and a vertical asymptote wherever the denominator is 0. Tangent rises left to right between asymptotes. Cotangent falls.
e.g. $\tan$ passes through $(-\tfrac{\pi}{4}, -1)$ , $(0, 0)$ , $(\tfrac{\pi}{4}, 1)$
$\displaystyle \tan x = \frac{\sin x}{\cos x},\; \cot x = \frac{\cos x}{\sin x}$
$\displaystyle \tan:\; \text{asymptotes } x = \tfrac{\pi}{2} + k\pi$
$\displaystyle \cot:\; \text{asymptotes } x = k\pi$
11.36.3
Graphing Secant and Cosecant
Sketch cos (for sec) or sin (for csc) first. Draw asymptotes where that curve is 0. Draw U-shaped branches from each maximum and minimum, opening away from the $x$ -axis. Period $2\pi$ .
e.g. $\csc\tfrac{\pi}{2} = 1$ : a branch opens up from $(\tfrac{\pi}{2}, 1)$ between asymptotes $x = 0$ and $x = \pi$
$\displaystyle \sec x = \frac{1}{\cos x},\; \csc x = \frac{1}{\sin x}$

11.37 Properties of Trigonometric Functions

Tangent is sine divided by cosine, so it blows up wherever cosine is 0. Its branches repeat every π, half the period of sine.

11.37.1
Describing Properties of the Sine Function
Sine is odd, so its graph is symmetric about the origin. It increases on $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$ . Here $k$ is any integer.
e.g. Zeros of $\sin x$ on $[0, 2\pi]$ : $x = 0, \pi, 2\pi$
$\displaystyle \text{Domain } \mathbb{R},\; \text{range } [-1, 1]$
$\displaystyle \text{Period } 2\pi,\; \text{zeros } x = k\pi$
$\displaystyle \text{Max } 1 \text{ at } x = \tfrac{\pi}{2} + 2k\pi$
$\displaystyle \text{Min } -1 \text{ at } x = \tfrac{3\pi}{2} + 2k\pi$
11.37.2
Describing Properties of the Cosine Function
Cosine is even, so its graph is symmetric about the $y$ -axis. It decreases on $(0, \pi)$ . Here $k$ is any integer.
e.g. Zeros of $\cos x$ on $[0, 2\pi]$ : $x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}$
$\displaystyle \text{Domain } \mathbb{R},\; \text{range } [-1, 1]$
$\displaystyle \text{Period } 2\pi,\; \text{zeros } x = \tfrac{\pi}{2} + k\pi$
$\displaystyle \text{Max } 1 \text{ at } x = 2k\pi$
$\displaystyle \text{Min } -1 \text{ at } x = \pi + 2k\pi$
11.37.3
Describing Properties of the Tangent Function
Tangent is odd and increases between each pair of asymptotes. It has no maximum or minimum. Here $k$ is any integer.
e.g. On $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$ the only zero is $x = 0$
$\displaystyle \text{Domain } x \ne \tfrac{\pi}{2} + k\pi$
$\displaystyle \text{Range } \mathbb{R},\; \text{period } \pi$
$\displaystyle \text{Zeros } x = k\pi$
11.37.4
Describing Properties of the Secant Function
Secant is even and has no zeros. Its asymptotes are at the zeros of cosine. Here $k$ is any integer.
e.g. $\sec 0 = 1$ and $\sec\pi = -1$
$\displaystyle \text{Domain } x \ne \tfrac{\pi}{2} + k\pi$
$\displaystyle \text{Range } (-\infty, -1] \cup [1, \infty)$
$\displaystyle \text{Period } 2\pi$
$\displaystyle \text{Local min } 1 \text{ at } x = 2k\pi$
$\displaystyle \text{Local max } -1 \text{ at } x = \pi + 2k\pi$
11.37.5
Describing Properties of the Cosecant Function
Cosecant is odd and has no zeros. Its asymptotes are at the zeros of sine. Here $k$ is any integer.
e.g. $\csc\tfrac{\pi}{2} = 1$ and $\csc\tfrac{3\pi}{2} = -1$
$\displaystyle \text{Domain } x \ne k\pi$
$\displaystyle \text{Range } (-\infty, -1] \cup [1, \infty)$
$\displaystyle \text{Period } 2\pi$
$\displaystyle \text{Local min } 1 \text{ at } x = \tfrac{\pi}{2} + 2k\pi$
$\displaystyle \text{Local max } -1 \text{ at } x = \tfrac{3\pi}{2} + 2k\pi$
11.37.6
Describing Properties of the Cotangent Function
Cotangent is odd and decreases between each pair of asymptotes. It has no maximum or minimum. Here $k$ is any integer.
e.g. On $(0, \pi)$ the only zero is $x = \tfrac{\pi}{2}$
$\displaystyle \text{Domain } x \ne k\pi$
$\displaystyle \text{Range } \mathbb{R},\; \text{period } \pi$
$\displaystyle \text{Zeros } x = \tfrac{\pi}{2} + k\pi$

11.38 Graph Transformations of Trigonometric Functions

A sinusoid is the sine wave stretched, shifted, and lifted. Read the midline first; the amplitude is how far the wave rises above it, and the period is the length of one full cycle.

11.38.1
Vertical Translations of Trigonometric Functions
Adding $D$ shifts the whole graph up $D$ units, or down if $D < 0$ . For sine and cosine the midline moves to $y = D$ .
e.g. $y = \cos x - 2$ : midline $y = -2$ , range $[-3, -1]$
$\displaystyle y = f(x) + D$
11.38.2
Vertical Stretches of Trigonometric Functions
Multiplying by $A$ multiplies every $y$ -value by $A$ . For sine and cosine the amplitude is $\lvert A \rvert$ . Tangent, cotangent, secant, and cosecant have no amplitude, but their points still scale.
e.g. $y = 3\sin x$ : amplitude 3, range $[-3, 3]$ ; $y = 2\tan x$ passes through $(\tfrac{\pi}{4}, 2)$
$\displaystyle y = A f(x)$
$\displaystyle \text{amplitude} = \lvert A \rvert$
11.38.3
Horizontal Translations of Trigonometric Functions
Replacing $x$ with $x - C$ shifts the graph right $C$ units, or left if $C < 0$ . This shift is the phase shift.
e.g. $y = \cos(x + \tfrac{\pi}{3})$ : shift left $\tfrac{\pi}{3}$ , so a maximum is at $x = -\tfrac{\pi}{3}$
$\displaystyle y = f(x - C)$
11.38.4
Horizontal Stretches of Trigonometric Functions
Replacing $x$ with $Bx$ ( $B > 0$ ) changes the period. $B > 1$ squeezes the graph and $0 < B < 1$ stretches it.
e.g. $y = \sin 2x$ has period $\pi$ ; $y = \cos\tfrac{x}{2}$ has period $4\pi$
$\displaystyle \sin, \cos, \sec, \csc:\; P = \frac{2\pi}{B}$
$\displaystyle \tan, \cot:\; P = \frac{\pi}{B}$
11.38.5
Combining Graph Transformations of Sine and Cosine
Write the function in this form with $B > 0$ , factoring $B$ out of the parentheses first. Then read off all four features.
e.g. $y = 2\sin(3x - \pi) + 1 = 2\sin(3(x - \tfrac{\pi}{3})) + 1$ : amplitude 2, period $\tfrac{2\pi}{3}$ , right $\tfrac{\pi}{3}$ , midline $y = 1$
$\displaystyle y = A\sin(B(x - C)) + D$
$\displaystyle \text{amplitude } \lvert A \rvert,\; \text{period } \frac{2\pi}{B}$
$\displaystyle \text{phase shift } C,\; \text{midline } y = D$
11.38.6
Graph Transformations of Tangent and Cotangent
The same transformations apply, but the period is $\tfrac{\pi}{B}$ . $A$ is a vertical stretch, not an amplitude. Shifts move the asymptotes too.
e.g. $y = \tan 2x$ : period $\tfrac{\pi}{2}$ , asymptotes $x = \tfrac{\pi}{4} + \tfrac{k\pi}{2}$
$\displaystyle y = A\tan(B(x - C)) + D$
$\displaystyle \text{period} = \frac{\pi}{B}$
11.38.7
Combining Graph Transformations of Tangent and Cotangent
Find the asymptotes by setting the inside equal to where the parent is undefined. Tangent's center point moves to $(C, D)$ , halfway between two asymptotes.
e.g. $y = 2\tan(x - \tfrac{\pi}{4}) + 1$ : period $\pi$ , center $(\tfrac{\pi}{4}, 1)$ , asymptotes $x = \tfrac{3\pi}{4} + k\pi$
$\displaystyle \tan:\; B(x - C) = \tfrac{\pi}{2} + k\pi$
$\displaystyle \cot:\; B(x - C) = k\pi$
11.38.8
Combining Graph Transformations of Secant and Cosecant
Graph the matching cosine or sine with the same $A$ , $B$ , $C$ , $D$ . Asymptotes go where that guide curve crosses its midline. Branches start at its maximums and minimums.
e.g. $y = 2\sec 2x + 1$ : period $\pi$ , asymptotes $x = \tfrac{\pi}{4} + \tfrac{k\pi}{2}$ , range $(-\infty, -1] \cup [3, \infty)$
$\displaystyle y = A\sec(B(x - C)) + D$
$\displaystyle \text{period} = \frac{2\pi}{B}$
11.38.9
Graphing Reflections of Trigonometric Functions
A negative $A$ flips the graph over the $x$ -axis (over the midline if there is a vertical shift). A negative sign on $x$ flips it over the $y$ -axis, and the even and odd rules simplify that.
e.g. $y = -2\cos x$ starts at a minimum $(0, -2)$ instead of a maximum
$\displaystyle y = -f(x):\; \text{flip over } x\text{-axis}$
$\displaystyle y = f(-x):\; \text{flip over } y\text{-axis}$
$\displaystyle \sin(-x) = -\sin x,\; \cos(-x) = \cos x$
11.38.10
Graphing Reflections of Trigonometric Functions: Three or More Transformations
Rewrite in the form $y = A f(B(x - C)) + D$ . Move each key point of the parent graph with the rule below. A negative $A$ reflects.
e.g. $y = -3\sin(2x - \tfrac{\pi}{2}) + 2 = -3\sin(2(x - \tfrac{\pi}{4})) + 2$ : reflected, amplitude 3, period $\pi$ , right $\tfrac{\pi}{4}$ , range $[-1, 5]$
$\displaystyle (x, y) \to \left(\frac{x}{B} + C,\; Ay + D\right)$

11.39 Properties of Transformed Trigonometric Functions

11.39.1
Properties of Transformed Sine and Cosine Functions
For $y = A\sin(B(x - C)) + D$ or the cosine version, the domain is all real numbers. The graph swings $\lvert A \rvert$ above and below the midline $y = D$ .
e.g. $y = 4\cos(\tfrac{\pi}{2}x) - 1$ : period $\tfrac{2\pi}{\pi/2} = 4$ , range $[-5, 3]$
$\displaystyle \max = D + \lvert A \rvert,\; \min = D - \lvert A \rvert$
$\displaystyle \text{range } [D - \lvert A \rvert,\; D + \lvert A \rvert]$
$\displaystyle \text{period} = \frac{2\pi}{B}$
11.39.2
Finding Zeros and Extrema of Transformed Sine and Cosine Functions
Let $u = B(x - C)$ . For zeros, set $y = 0$ and solve for $u$ , then for $x$ . Extrema occur where the parent hits 1 or $-1$ . If $A < 0$ , the maximums and minimums swap.
e.g. $y = 2\sin 2x$ : zeros $x = \tfrac{k\pi}{2}$ ; max 2 at $x = \tfrac{\pi}{4} + k\pi$ ; min $-2$ at $x = \tfrac{3\pi}{4} + k\pi$
$\displaystyle \sin u = 1 \text{ at } u = \tfrac{\pi}{2} + 2k\pi$
$\displaystyle \sin u = -1 \text{ at } u = \tfrac{3\pi}{2} + 2k\pi$
$\displaystyle \cos u = 1 \text{ at } u = 2k\pi$
$\displaystyle \cos u = -1 \text{ at } u = \pi + 2k\pi$
11.39.3
Properties of Transformed Tangent and Cotangent Functions
The range is all real numbers and there are no extrema. The domain leaves out the asymptotes. With $D = 0$ , tangent's zeros are at $B(x - C) = k\pi$ and cotangent's at $B(x - C) = \tfrac{\pi}{2} + k\pi$ .
e.g. $y = 3\cot\tfrac{x}{2}$ : period $2\pi$ , asymptotes $x = 2k\pi$ , zeros $x = \pi + 2k\pi$
$\displaystyle \text{period} = \frac{\pi}{B}$
$\displaystyle \tan \text{ asymptotes: } B(x - C) = \tfrac{\pi}{2} + k\pi$
$\displaystyle \cot \text{ asymptotes: } B(x - C) = k\pi$
11.39.4
Properties of Transformed Secant and Cosecant Functions
The range is shown below. There are zeros only if the vertical shift moves a branch across the $x$ -axis. Asymptotes sit where the guide cosine or sine is 0: secant at $B(x - C) = \tfrac{\pi}{2} + k\pi$ , cosecant at $B(x - C) = k\pi$ .
e.g. $y = 3\csc 2x + 1$ : period $\pi$ , asymptotes $x = \tfrac{k\pi}{2}$ , range $(-\infty, -2] \cup [4, \infty)$
$\displaystyle \text{period} = \frac{2\pi}{B}$
$\displaystyle (-\infty, D - \lvert A \rvert] \cup [D + \lvert A \rvert, \infty)$
11.39.5
Interpreting Trigonometric Models
In a real-world model, $D$ is the average level, $\lvert A \rvert$ is how far it swings above and below that average, the period is the time for one full cycle, and $C$ tells when a cycle starts.
e.g. $T(t) = 15\cos\left(\tfrac{\pi}{6}(t - 7)\right) + 60$ ( $t$ in months): average $60^\circ$ F, swing 15 degrees, period 12 months, high of $75^\circ$ F at $t = 7$
11.39.6
Modeling With Trigonometric Functions
Get $A$ and $D$ from the max and min, and $B$ from the period. Use cosine if the data starts at a maximum, $-\cos$ if it starts at a minimum, and sine if it starts at the midline going up.
e.g. Ferris wheel from 5 ft to 45 ft, one turn per 8 min, rider starts at the bottom: $h(t) = -20\cos\left(\tfrac{\pi}{4}t\right) + 25$
$\displaystyle A = \frac{\max - \min}{2}$
$\displaystyle D = \frac{\max + \min}{2}$
$\displaystyle B = \frac{2\pi}{\text{period}}$

Unit 12 43 topics

Probability & Statistics

Introduction to Probability, Compound Events in Probability, Conditional Probability, and more.

12.40 Introduction to Probability

A ∪ B is everything in either circle, and A ∩ B is the shaded overlap. Adding P(A) and P(B) counts the overlap twice, so subtract it once.

12.40.1
Sets
A set is a collection of distinct objects called elements. $x \in A$ means $x$ is an element of $A$ . The empty set $\varnothing$ has no elements, and $A \subseteq B$ means every element of $A$ is also in $B$ .
e.g. Even numbers from 1 to 10: $\{2, 4, 6, 8, 10\}$ , a set with 5 elements
$\displaystyle A = \{1, 2, 3\}$
$\displaystyle 2 \in A,\; 5 \notin A$
12.40.2
Experimental Probability
The fraction of trials in which the event actually happened. More trials usually bring it closer to the theoretical probability.
e.g. 27 heads in 50 flips: $P(\text{heads}) \approx \tfrac{27}{50} = 0.54$
$\displaystyle P(A) \approx \frac{\text{successes}}{\text{trials}}$
12.40.3
Using Experimental Probability to Make Predictions
Multiply the experimental probability by the number of future trials to predict how often the event will happen.
e.g. Made 18 of 30 free throws. In 50 more shots, expect $\tfrac{18}{30} \times 50 = 30$ makes
$\displaystyle \text{expected count} = P(A) \times n$
12.40.4
Sample Spaces and Events in Probability
The sample space $S$ is the set of all possible outcomes. An event is any subset of $S$ .
e.g. Flip two coins: $S = \{HH, HT, TH, TT\}$ , and "at least one head" $= \{HH, HT, TH\}$
$\displaystyle S = \{1, 2, 3, 4, 5, 6\}$
$\displaystyle \text{even} = \{2, 4, 6\}$
12.40.5
The Likelihood of an Event
Probability runs from 0 (impossible) to 1 (certain). Near 0 is unlikely, $\tfrac12$ is as likely as not, and near 1 is likely.
e.g. One die: $P(7) = 0$ is impossible, and $P(\text{less than } 7) = 1$ is certain
$\displaystyle 0 \le P(A) \le 1$
12.40.6
Single Events in Probability
When all outcomes are equally likely, divide the number of favorable outcomes by the total number of outcomes.
e.g. One die: $P(\text{prime}) = \tfrac{3}{6} = \tfrac12$ (2, 3, 5)
$\displaystyle P(A) = \frac{n(A)}{n(S)}$
12.40.7
Using Probability to Make Predictions
Expected number of occurrences = probability × number of trials. It is a long-run average, not a guarantee.
e.g. Roll a die 120 times: expect $120 \cdot \tfrac16 = 20$ sixes
$\displaystyle \text{expected count} = n \cdot P(A)$
12.40.8
The Complement of an Event
The complement $A'$ (also written $A^c$ ) is every outcome in $S$ that is not in $A$ . Its probability is 1 minus $P(A)$ .
e.g. $P(\text{rain}) = 0.3 \Rightarrow P(\text{no rain}) = 0.7$
$\displaystyle P(A') = 1 - P(A)$
12.40.9
Venn Diagrams in Probability
The rectangle is the sample space and each circle is an event. Fill each region with a count or probability; all regions together total $n(S)$ or 1.
e.g. 30 students: 12 soccer only, 8 both, 6 basketball only, 4 neither. $P(\text{soccer}) = \tfrac{12 + 8}{30} = \tfrac23$
12.40.10
Geometric Probability
When a point is chosen at random in a region, the probability is the favorable length, area, or volume divided by the total.
e.g. A 4 by 4 square target on a 10 by 10 board: $P = \tfrac{16}{100} = 0.16$
$\displaystyle P = \frac{\text{favorable area}}{\text{total area}}$

12.41 Compound Events in Probability

12.41.1
The Union of Sets
$A \cup B$ ("A or B") is everything in $A$ , in $B$ , or in both. Each element is listed once.
e.g. $\{1, 2, 3\} \cup \{3, 4\} = \{1, 2, 3, 4\}$
$\displaystyle A \cup B = \{x : x \in A \text{ or } x \in B\}$
12.41.2
The Intersection of Sets
$A \cap B$ ("A and B") is the elements in both $A$ and $B$ . If they share nothing, $A \cap B = \varnothing$ and the sets are disjoint.
e.g. $\{1, 2, 3\} \cap \{2, 3, 4\} = \{2, 3\}$
$\displaystyle A \cap B = \{x : x \in A \text{ and } x \in B\}$
12.41.3
Compound Events in Probability From Experimental Data
For "A and B," count the trials where both happened. For "A or B," count the trials where either happened, counting the overlap once. Divide by the total number of trials.
e.g. 50 people: 20 own a dog, 15 a cat, 5 both. $P(\text{dog or cat}) = \tfrac{20 + 15 - 5}{50} = 0.6$
$\displaystyle P(A \cap B) \approx \frac{n(A \cap B)}{\text{total}}$
12.41.4
Computing Probabilities for Compound Events Using Venn Diagrams
Add the regions that make up the event. $A \cup B$ is every region inside either circle; $A \cap B$ is only the overlap.
e.g. A only 0.3, both 0.1, B only 0.2, neither 0.4: $P(A \cup B) = 0.3 + 0.1 + 0.2 = 0.6$ , $P(A \cap B) = 0.1$
12.41.5
Computing Probabilities of Events Containing Complements Using Venn Diagrams
$A'$ is every region outside circle $A$ , including the region outside both circles. Work region by region.
e.g. A only 0.3, both 0.1, B only 0.2, neither 0.4: $P(A \cap B') = 0.3$ , $P(A') = 0.2 + 0.4 = 0.6$
$\displaystyle A \cap B' = A \text{ only}$
$\displaystyle (A \cup B)' = \text{neither}$
12.41.6
Computing Probabilities for Three Events Using Venn Diagrams
Three circles make 8 regions. Fill from the center out: the triple overlap first, then each pair overlap minus the center, then each "only" region.
e.g. 40 students: 20 Spanish, 15 French, 10 art; 5 Spanish and French, 4 Spanish and art, 3 French and art, 2 all three. $P(\text{at least one}) = \tfrac{45 - 12 + 2}{40} = \tfrac78$
$\displaystyle P(A \cup B \cup C) = P(A) + P(B) + P(C)$
$\displaystyle - P(A \cap B) - P(A \cap C) - P(B \cap C)$
$\displaystyle + P(A \cap B \cap C)$

12.42 Conditional Probability

The second-draw branches change because the first marble is not put back. Multiply along a path to get one outcome; add outcomes to combine them.

12.42.1
Conditional Probabilities From Venn Diagrams
$P(A \mid B)$ is the probability of $A$ given that $B$ happened. Look only inside circle $B$ : divide the overlap by all of $B$ .
e.g. A only 0.3, both 0.1, B only 0.2: $P(A \mid B) = \tfrac{0.1}{0.1 + 0.2} = \tfrac13$
$\displaystyle P(A \mid B) = \frac{P(A \cap B)}{P(B)}$
12.42.2
Conditional Probabilities From Tables
Use only the row or column of the given condition. Divide the cell by that row or column total, not the grand total.
e.g. 25 students play a sport, and 10 of them are in band: $P(\text{band} \mid \text{sport}) = \tfrac{10}{25} = 0.4$
$\displaystyle P(A \mid B) = \frac{n(A \cap B)}{n(B)}$
12.42.3
The Multiplication Law for Conditional Probability
The probability that both happen is the probability of the first times the probability of the second given the first.
e.g. 5 red and 3 blue marbles, draw 2 without replacement: $P(\text{both red}) = \tfrac58 \cdot \tfrac47 = \tfrac{5}{14}$
$\displaystyle P(A \cap B) = P(A) \cdot P(B \mid A)$
12.42.4
The Law of Total Probability
Split into cases that do not overlap and cover everything, such as $B$ and $B'$ . Find $P(A)$ in each case and add.
e.g. 60 percent of days are sunny. You bike on 90 percent of sunny days and 30 percent of others: $P(\text{bike}) = 0.9(0.6) + 0.3(0.4) = 0.66$
$\displaystyle P(A) = P(A \mid B)P(B)$
$\displaystyle \quad + P(A \mid B')P(B')$
12.42.5
Tree Diagrams for Dependent Events
Each branch is a conditional probability given the path so far, so later branches change. Multiply along a path for that outcome, and add paths for an event. Branches from one point sum to 1.
e.g. 4 red and 2 blue, draw 2 without replacement: $P(\text{one of each}) = \tfrac46 \cdot \tfrac25 + \tfrac26 \cdot \tfrac45 = \tfrac{16}{30} = \tfrac{8}{15}$
12.42.6
Tree Diagrams for Dependent Events: Applications
Use a tree for multi-step situations such as medical tests. To reverse a condition, divide the path you want by the sum of all paths that give the observed result.
e.g. 2 percent have a condition; the test is positive for 90 percent of them and 5 percent of others. $P(\text{has it} \mid +) = \tfrac{0.018}{0.018 + 0.049} \approx 0.27$
$\displaystyle P(B \mid A) = \frac{P(B)\,P(A \mid B)}{P(A)}$
12.42.7
Independent Events
$A$ and $B$ are independent if one happening does not change the probability of the other. Check with either equation.
e.g. $P(A) = 0.5$ , $P(B) = 0.4$ , $P(A \cap B) = 0.2 = 0.5 \cdot 0.4$ , so independent
$\displaystyle P(A \mid B) = P(A)$
$\displaystyle P(A \cap B) = P(A) \cdot P(B)$
12.42.8
Tree Diagrams for Independent Events
The branch probabilities repeat at every stage because earlier results do not affect later ones (for example, drawing with replacement). Multiply along each path.
e.g. Makes 70 percent of free throws, shoots 2: $P(\text{exactly one}) = 0.7(0.3) + 0.3(0.7) = 0.42$
12.42.9
The Addition Law of Probability
For "A or B," add the two probabilities and subtract the overlap so it is not counted twice.
e.g. One card: $P(\text{heart or king}) = \tfrac{13}{52} + \tfrac{4}{52} - \tfrac{1}{52} = \tfrac{4}{13}$
$\displaystyle P(A \cup B) = P(A) + P(B)$
$\displaystyle \quad - P(A \cap B)$
12.42.10
Applying the Addition Law With Event Complements
The addition law works with complements too. Use $P(A') = 1 - P(A)$ , and "at least one" is 1 minus "neither."
e.g. $P(A) = 0.6$ , $P(B) = 0.5$ , $P(A \cap B) = 0.3$ : $P(A' \cap B) = 0.2$ , so $P(A' \cup B) = 0.4 + 0.5 - 0.2 = 0.7$
$\displaystyle P(A' \cup B) = P(A') + P(B)$
$\displaystyle \quad - P(A' \cap B)$
$\displaystyle P(A \cup B) = 1 - P(A' \cap B')$
12.42.11
Mutually Exclusive Events
Events that cannot happen together: $A \cap B = \varnothing$ . The addition law then has no overlap term. Mutually exclusive events with nonzero probabilities are never independent.
e.g. One die: $P(1 \text{ or } 6) = \tfrac16 + \tfrac16 = \tfrac13$
$\displaystyle P(A \cap B) = 0$
$\displaystyle P(A \cup B) = P(A) + P(B)$

12.43 Combinatorics

12.43.1
The Rules of Sum and Product
Product rule: a task done in steps with $m$ ways then $n$ ways can be done $mn$ ways. Sum rule: choosing one option from $m$ choices or $n$ other choices (no overlap) gives $m + n$ ways.
e.g. 4 shirts and 3 pants: $4 \times 3 = 12$ outfits. One of 5 movies or 3 games: $5 + 3 = 8$ choices
$\displaystyle \text{steps (and): } m \times n$
$\displaystyle \text{no overlap (or): } m + n$
12.43.2
Factorials
$n!$ is the product of the whole numbers from $n$ down to 1. By definition $0! = 1$ .
e.g. $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ , and $\tfrac{8!}{6!} = 8 \cdot 7 = 56$
$\displaystyle n! = n(n-1)(n-2)\cdots 1$
$\displaystyle 0! = 1$
12.43.3
Factorials in Variable Expressions
Expand the larger factorial down to the smaller one so they cancel.
e.g. $\tfrac{(n+2)!}{n!} = (n+2)(n+1) = n^2 + 3n + 2$
$\displaystyle n! = n(n-1)!$
$\displaystyle \frac{(n+1)!}{(n-1)!} = (n+1)n$
12.43.4
Ordering Objects
$n$ different objects can be arranged in a row in $n!$ ways. If some objects are identical, divide by the factorial of each repeat count.
e.g. 4 books on a shelf: $4! = 24$ orders. Letters of BOOK: $\tfrac{4!}{2!} = 12$
$\displaystyle n!$
$\displaystyle \frac{n!}{a!\,b!\cdots} \;\text{(repeats)}$
12.43.5
Permutations
Arrange $r$ of $n$ different objects when order matters.
e.g. President, VP, and treasurer from 10 people: ${}_{10}P_3 = 10 \cdot 9 \cdot 8 = 720$
$\displaystyle {}_nP_r = \frac{n!}{(n-r)!}$
12.43.6
Combinations
Choose $r$ of $n$ different objects when order does not matter. It is the permutation count divided by $r!$ .
e.g. A 3-person committee from 10: $\binom{10}{3} = \tfrac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120$
$\displaystyle \binom{n}{r} = {}_nC_r = \frac{n!}{r!\,(n-r)!}$
12.43.7
Computing Probabilities Using Combinatorics
Count the favorable outcomes and the total outcomes with combinations or permutations, then divide.
e.g. 5 boys and 4 girls, pick 3 at random: $P(\text{all girls}) = \tfrac{\binom43}{\binom93} = \tfrac{4}{84} = \tfrac{1}{21}$
$\displaystyle P = \frac{\text{favorable count}}{\text{total count}}$

12.44 Analyzing Data

In a right-skewed histogram the few large values in the tail drag the mean to the right of the median. Here the mean is estimated from bin midpoints; the median is less affected by the tail.

12.44.1
Sampling
A sample is part of a population used to learn about the whole. A random sample (simple random, systematic, stratified, or cluster) avoids bias; convenience and voluntary samples often do not represent the population.
e.g. Survey a random 100 of 1,200 students; 35 ride the bus, so estimate $0.35 \times 1200 = 420$ riders
12.44.2
The Mean of a Data Set
Add all the values and divide by how many there are. Write $\mu$ for a population mean and $\bar x$ for a sample mean.
e.g. $3, 5, 7, 9$ : $\bar x = \tfrac{24}{4} = 6$
$\displaystyle \bar x = \frac{\sum x_i}{n}$
12.44.3
Variance and Standard Deviation
Variance is the mean of the squared distances from the mean. Standard deviation is its square root, in the data's units. For a sample, divide by $n - 1$ instead of $n$ .
e.g. $2, 4, 6, 8$ : $\mu = 5$ , squares $9, 1, 1, 9$ , $\sigma^2 = \tfrac{20}{4} = 5$ , $\sigma = \sqrt5 \approx 2.24$ (sample: $s^2 = \tfrac{20}{3}$ )
$\displaystyle \sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$
$\displaystyle \sigma = \sqrt{\sigma^2}$
$\displaystyle s^2 = \frac{\sum (x_i - \bar x)^2}{n - 1}$
12.44.4
Estimating Means and Variances For Grouped Data
Treat every value in a class as the class midpoint $m$ , weighted by its frequency $f$ . The results are estimates.
e.g. Classes 0 to 10, 10 to 20, 20 to 30 with $f = 2, 3, 5$ : midpoints $5, 15, 25$ , $\bar x \approx \tfrac{10 + 45 + 125}{10} = 18$ , $\sigma^2 \approx \tfrac{338 + 27 + 245}{10} = 61$
$\displaystyle \bar x \approx \frac{\sum f m}{\sum f}$
$\displaystyle \sigma^2 \approx \frac{\sum f (m - \bar x)^2}{\sum f}$
12.44.5
Covariance
Measures how two variables move together. Positive means they tend to rise together, negative means one rises as the other falls, and near 0 means no linear pattern. For a sample, divide by $n - 1$ .
e.g. $(1,1), (2,3), (3,5)$ : $\bar x = 2$ , $\bar y = 3$ , products $2, 0, 2$ , $\sigma_{xy} = \tfrac43$ (sample: $2$ )
$\displaystyle \sigma_{xy} = \frac{\sum (x_i - \bar x)(y_i - \bar y)}{n}$
12.44.6
Sums of Squares
$S_{xx}$ and $S_{yy}$ add the squared deviations from the mean; $S_{xy}$ adds the products of paired deviations. The shortcut forms skip computing each deviation. Variance is $S_{xx}/n$ and covariance is $S_{xy}/n$ .
e.g. $x = 1, 2, 3$ : $S_{xx} = 14 - \tfrac{36}{3} = 2$
$\displaystyle S_{xx} = \sum (x - \bar x)^2$
$\displaystyle = \sum x^2 - \frac{(\sum x)^2}{n}$
$\displaystyle S_{xy} = \sum xy - \frac{\sum x \sum y}{n}$

12.45 Correlation and Regression

A residual is the vertical gap between a data point and the fitted line. The least-squares line is the one that makes the squared gaps add up to as little as possible.

12.45.1
The Linear Correlation Coefficient
$r$ measures the strength and direction of a linear relationship. Near $1$ or $-1$ is strong, near 0 is weak or none. Correlation does not prove causation.
e.g. $(1,1), (2,3), (3,2)$ : $S_{xx} = 2$ , $S_{yy} = 2$ , $S_{xy} = 1$ , so $r = \tfrac{1}{\sqrt4} = 0.5$
$\displaystyle r = \frac{S_{xy}}{\sqrt{S_{xx}\,S_{yy}}}$
$\displaystyle -1 \le r \le 1$
12.45.2
Residuals and Residual Plots
A residual is actual minus predicted. Positive means the point is above the line. A residual plot with random scatter around 0 means a linear model fits; a curve or other pattern means it does not.
e.g. Line $\hat y = 2x + 1$ , point $(3, 8)$ : $\hat y = 7$ , residual $= 8 - 7 = 1$
$\displaystyle e = y - \hat y$
12.45.3
Selecting a Regression Model
Match the shape: linear for steady change, exponential for steady percent change, quadratic for a curve that turns. Leftover patterns in the residuals mean a poor fit.

1.1 The Number System

The Real Number System

Writing Repeating Decimals as Fractions

Sums and Products of Rational and Irrational Numbers

1.2 Introduction to Complex Numbers

Imaginary Numbers

Quadratic Equations with Purely Imaginary Solutions

Complex Numbers

Adding and Subtracting Complex Numbers

Multiplying Complex Numbers

Solving Quadratic Equations With Complex Roots

The Cyclic Property of the Imaginary Unit

2.3 Polynomials

The Least Common Multiple of Two Monomials

The Least Common Multiple of Two Polynomials

Describing Relationships Using Polynomial Identities

2.4 Quadratic Equations

Introduction to Quadratic Equations

Solving Perfect Square Quadratic Equations

Perfect Square Quadratic Equations with One or No Solutions

The Zero Product Rule for Solving Quadratic Equations

Solving Quadratic Equations Using a Difference of Squares

Solving Quadratic Equations with No Constant Term

Solving Quadratic Equations by Factoring

Solving Quadratic Equations with Leading Coefficients by Factoring

Completing the Square

Completing the Square With Odd Linear Terms

Completing the Square With Leading Coefficients

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations With Leading Coefficients by Completing the Square

The Quadratic Formula

The Discriminant of a Quadratic Equation

Determining Unknown Parameters in Quadratic Equations With One Real Solution

Modeling With Quadratic Equations

Equating Polynomial Coefficients

Vieta's Formulas

2.5 Factoring Polynomials

Further Factoring of Polynomials Using GCFs

Factoring Higher-Order Polynomials as a Difference of Squares

Factoring Cubic Expressions by Grouping

Factoring Sums and Differences of Cubes

Factoring Biquadratic Expressions

2.6 Polynomial Equations

Determining the Roots of Polynomials

Solving Polynomial Equations Using the GCF

Solving Cubic Equations by Grouping

Solving Biquadratic Equations

3.7 Quadratic Functions

Graphing Elementary Quadratic Functions

Vertical Reflections of Quadratic Functions

Graphs of General Quadratic Functions

Roots of Quadratic Functions

The Discriminant of a Quadratic Function

The Axis of Symmetry of a Parabola

The Average of the Roots Formula

The Vertex Form of a Parabola

Writing the Equation of a Parabola in Vertex Form

Domain and Range of Quadratic Functions

Solving Systems of Nonlinear Equations

Finding Intersections of Lines and Quadratic Functions

3.8 Modeling With Quadratic Functions

Modeling Downwards Vertical Motion

Modeling Upwards Vertical Motion

Vertical Motion

Revenue, Cost, and Profit Functions

Constructing Revenue, Cost, and Profit Functions

Maximizing Profit and Break-Even Points

4.9 Functions

The Arithmetic of Functions

Function Composition

Describing Function Composition

Local Extrema of Functions

One-To-One Functions

Introduction to Inverse Functions

Calculating the Inverse of a Function

Inverses of Quadratic Functions

Graphs of Inverse Functions

Domain and Range of Inverse Functions

Invertible Functions

Plotting X as a Function of Y