Introduction

Hello! This is a formula review for AP Precalculus covering the main formulas, identities, relationships, and connections from each unit. Everything here is worth knowing for the AP exam in May. Happy studying!

Unit 1: Polynomial and Rational Functions

Average Rate of Change (AROC)

$\frac{f(b)-f(a)}{b-a}$

where $a$ and $b$ represent the boundaries of an interval on the $x$-axis.

Alternatively,

$\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$

This formula measures the average change in the output for every one-unit change in the input. For a linear function, the average rate of change is constant and equals the slope.

See Unit 1.2: Rates of Change for more on this topic.

Difference Quotient

$\frac{f(x+h)-f(x)}{h}$,

where $h$ represents the horizontal change in $x$.

This is a specific form of AROC, where it is specifically the average rate of change from $x$ to $x+h$.

The Quadratic Formula

For $a{x}^2+bx+c=0$,

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

The expression inside the square root is the discriminant: $\Delta =b^2-4ac$

The discriminant determines the number and type of real solutions:

See Khan Academy: Quadratic formula explained (article). 

Vertex of a Quadratic

For $f(x)=a{x}^2+bx+c$, the $x$-coordinate of the vertex is $x=-\frac{b}{2a}$, and the corresponding $y$-coordinate is $f\left(-\frac{b}{2a}\right)$.

Thus, the vertex is $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$.

Vertex Form: $f(x)=a(x-h{)}^2+k$, where $(h,k)$ is the vertex.

See Prepscholar: Vertex Form: What is it? How do you calculate it?

Factored Form of a Polynomial

(a.k.a. Linear Factorization Theorem)

A polynomial with zeros $r_1,r_2,\dots ,r_n$ can be written as

$f(x)=a(x-r_1)(x-r_2)\dots (x-r_n)$, where $a$ is the leading coefficient.

If $f(r)=0$, then $(x-r)$ is a factor of $f(x)$.

See Unit 1.5: Polynomial Functions and Complex Zeros.

Remainder Theorem

When $f(x)$ is divided by $(x-c)$, the remainder is $f(c)$.

Here, $c$ is a constant number.

See Cuemath: Remainder Theorem

Binomial Theorem

$(a+b{)}^n$=$\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k$,

where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$.

The factorial is defined by $n!=n(n-1)(n-2)\cdots (2)(1)$, $0!=1$.

Here,

$a$ and $b$ are the first and second terms in the binomial, respectively.

$\Sigma$ is the Greek letter Sigma. It’s an operator (like plus and minus) that tells you to add all terms together.

$k$ is the term counter. It starts at zero and goes up by one until it reaches $n$.

Alternatively, Pascal’s Triangle can be used to expand binomials of the form $(a+b{)}^n$:

See Unit 1.11: Equivalent Representations of Polynomial and Rational Expressions

Rational Functions

For $r(x)=\frac{p(x)}{q(x)}$:

Holes (Removable Discontinuities): Occurs at $x=c$ where both $p(c)=0$ and $q(c)=0$ cancel out.

Vertical Asymptotes: Occurs at the zeros of the simplified denominator where $q(x)=0$.

Horizontal Asymptotes:

See Unit 1.7: Rational Functions and End Behavior

Limits and Function Behavior

Left-Hand Limit

$\underset{x\to c^{-}}{\lim }f(x)$: The value $f(x)$ approaches as $x$ gets closer to $c$ from the left ($x\le c$).

Right-Hand Limit

$\underset{x\to c^{+}}{\lim }f(x)$: The value $f(x)$ approaches as $x$ gets closer to $c$ from the right ($x\ge c$).

Two-Sided Limit

$\underset{x\to c}{\lim }f(x)=\mathbf{L}⟺\underset{x\to c^{-}}{\lim }f(x)=\underset{x\to c^{+}}{\lim }f(x)=\mathbf{L}$

If the left-hand and right-hand limits do not match, the limit does not exist (DNE).

Horizontal Asymptotes

If $\underset{x\to \infty }{\lim }f(x)=\mathbf{L}$ or $\underset{x\to -\infty }{\lim }f(x)=\mathbf{L}$, then the line $y=\mathbf{L}$ is a horizontal asymptote.

End Behavior

Polynomial End Behavior is decided by the leading term $a_n{x}^n$:

$\underset{x\to \pm \infty }{\lim }(a_n{x}^n+\cdots +a_{1}x+a_0)=\underset{x\to \pm \infty }{\lim }(a_n{x}^n)$

Removable Discontinuities (Holes)

If a function $f(x)$ has a hole at $x=c$ with coordinates $(c,L)$, $\underset{x\to c}{\lim }=L$, even if $f(c)$ is undefined.

Vertical Asymptotes

If $\underset{x\to c}{\lim }=\pm \infty$, then $c$ is a vertical asymptote.

See Unit 1.6: Polynomial Functions and End Behavior 

Unit 2: Exponential and Logarithmic Functions

Arithmetic Sequences and Linear Growth

See Unit 2.1: Changes in Arithmetic and Geometric Sequences.

Geometric Sequences and Exponential Growth

See Unit 2.1: Changes in Arithmetic and Geometric Sequences.

Exponential Functions

See Unit 2.3: Exponential Functions.

Properties of Logarithms

Definition: $y={\log }_b(x)⟺b^y=x(b\ge 0,b\ne 1)$

Fundamental Identities

See Unit 2.12: Logarithmic Function Manipulation. 

Semi-Log Plots and Exponential Models

If $(x,y)$ follows an exponential relationship $y=a\cdot b^x$, then plotting $(x,\ln (y))$ produces a linear relationship:

$\ln (y)=\ln (a)+x\ln (b)$

Slope $m=\ln (b)$, $y$-intercept$=\ln (a)$

See Unit 2.15: Semi-Log Plots 

Unit 3: Trigonometric and Polar Functions

SOH-CAH-TOA

Sin is Opposite over Hypotenuse

Cosine is Adjacent over Hypotenuse

Tangent is Opposite over Adjacent

$\sin (\theta )=\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$

$\theta =\arcsin \left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$

$\cos (\theta )=\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$

$\theta =\arccos \left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$

$\tan (\theta )=\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

$\theta =\arctan \left(\frac{\text{opposite}}{\text{adjacent}}\right)$

Radians and Degrees

Radians $=\left(\frac{\pi }{180°}\right)\times \text{Degrees}$

Degrees $=\left(\frac{180°}{\pi }\right)\times \text{Radians}$

Reciprocal and Quotient Identities

$\tan (x)=\frac{\sin (x)}{\cos (x)}$

$\csc (x)=\frac{1}{\sin (x)}$

$\sec (x)=\frac{1}{\cos (x)}$

$\cot (x)=\frac{1}{\tan (x)}=\frac{\cos (x)}{\sin (x)}$

See Unit 3.11: The Secant, Coseant, and Cotangent Functions. 

Pythagorean Identities

${\sin }^2(x)+{\cos }^2(x)=1$

See Unit 3.10: Trigonometric Equations and Inequalities. 

Cofunction Symmetry (Even/Odd), and Periodicity

Cofunction

$\sin (x)=\cos \left(\frac{\pi }{2}-x\right)$

$\cos (x)=\sin \left(\frac{\pi }{2}-x\right)$

$\tan (x)=\cot \left(\frac{\pi }{2}-x\right)$

Even/Odd (Negative Angle)

$\sin (-x)=-\sin (x)$

$\cos (-x)=\cos (x)$

$\tan (-x)=-\tan (x)$

Supplementary Angle Identities

$\sin (\pi -x)=\sin (x)$

$\cos (\pi -x)=-\cos (x)$

$\tan (\pi -x)=-\tan (x)$

Periodicity for all $n$ 

Given that $n$ is an integer ($n\in \mathbb{Z}$),

$\sin (x+2\pi n)=\sin (x)$

$\cos (x+2\pi n)=\cos (x)$

$\tan (x+\pi n)=\tan (x)$

See Unit 3.7: Sinusoidal Function Context and Data Modeling. 

Sinusoidal Function Transformation Form

For $f(t)=A\sin (B(t-C))+D$ or $f(t)=A\cos (B(t-C))+D$:

See Unit 3.6: Sinusoidal Function Transformations. 

Sum and Difference Formulas

$\sin (\alpha \pm \beta )=\sin (\alpha )\cos (\beta )\pm \cos (\alpha )\sin (beta)$

$\cos (\alpha \pm \beta )=\cos (\alpha )\cos (\beta )\mp \sin (\alpha )\sin (\beta )$

$\tan \left(\alpha \pm \beta \right)=\frac{\tan (\alpha )\pm \tan (\beta )}{1\mp \tan \left(alpha\right)\tan (\beta )}$

Note: The Sum and Difference formula for tangent is unlikely to appear on the AP Exam.

See Unit 3.10: Trigonometric Equations and Inequalities. 

Connection to Double-Angle Formulas

Setting $\beta =\alpha =x$ in the sum formulas gives:

$\sin (x+x)=\sin (2x)=2\sin (x)\cos (x)$

$\cos (x+x)=\cos (2x)={\cos }^2(x)-{\sin }^2(x)$

Using ${\sin }^2(x)=1-{\cos }^2(x)⟹\cos (2x)=2{\cos }^2(x)=1$

Using ${\cos }^2(x)=1-{\sin }^2(x)⟹\cos (2x)=1-2{\sin }^2(x)$

$\tan \left(x+x\right)=\tan \left(2x\right)=\frac{2\tan \left(x\right)}{1-{\tan }^2(x)}$

See Unit 3.12: Equivalent Representations of Trigonometric Functions 

Sum-to-Product and Product-to-Sum

$\cos (x)+\cos (y)=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\cos (x)-\cos (y)=-2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$

$\sin (x)\pm \sin (y)=2\sin \left(\frac{x\pm y}{2}\right)\cos \left(\frac{x\mp y}{2}\right)$

$2\sin (x)\sin (y)=\cos (x-y)-\cos (x+y)$

$2\sin (x)\cos (y)=\sin (x+y)+\sin (x-y)$

$2\cos (x)\cos (y)=\cos (x+y)+\cos (x-y)$

Polar Coordinates and Conversions

Given a point in polar $(r,\theta )$ and rectangular $(x,y)$:

See Unit 3.13: Trigonometry and Polar Coordinates 

Arc Length and Sector Area