AP® Precalculus

Formulas and identities to know

The exam gives no formula sheet. These are the formulas and identities to know by heart, what each is for and when to use it.

Three habits that save points

Use the numbers you are given. A reason earns the point when it cites specific values of the function (two inputs with the same output), not a rule such as "it fails the horizontal line test".
Three decimal places, never partway. On a calculator question, write the equation you solve, store every value, and round or truncate only the final answer, to three places after the decimal point.
Radian mode and exact values. Every angle is in radians. Without a calculator, give exact values (fractions, roots, π, e, logarithms) and combine to one logarithm or one trigonometric term when asked.

Rates of change

Formula	What it is	When you use it	Learn it in
$\frac{f(b) - f(a)}{b - a}$	The average rate of change of $f$ on $[a, b]$ : the slope of the secant line through $(a, f(a))$ and $(b, f(b))$ .	Any rate over an interval, from a formula, a table or a graph. Name the function and the interval, and give the units (output units per input unit).	Unit 1 Rates of ChangePractice it: Rate of change
$y$ $= f(a)$ $+ \frac{f(b) - f(a)}{b - a}(x - a)$	The secant line through $(a, f(a))$ and $(b, f(b))$ , in point-slope form.	To estimate $f(x)$ for $x$ between $a$ and $b$ : the estimate is a point on this line. Concave down on $[a, b]$ gives an underestimate, concave up an overestimate.	Unit 1 Rates of Change in Linear and Quadratic FunctionsPractice it: Rate of change
$\text{rates of change increasing}$ $\Rightarrow \text{concave up}$	Concavity from average rates over consecutive equal-length intervals; rates decreasing means concave down.	Reading concavity from a table or a graph. Say "the rate of change of $f$ is increasing", not a calculus phrase.	Unit 1 Rates of Change in Linear and Quadratic FunctionsPractice it: Rate of change
$\text{constant second differences}$ $\Rightarrow \text{quadratic}$	Over equal input steps, constant differences of the outputs mean linear; constant differences of those differences mean quadratic.	Choosing or justifying a linear or quadratic model from a table: name the equal steps and give the differences.	Unit 1 Rates of Change in Linear and Quadratic FunctionsPractice it: Function lab

Polynomial and rational functions

Formula	What it is	When you use it	Learn it in
$(x - c)^k$	A factor that makes $c$ a zero of multiplicity $k$ . Even $k$ : the graph touches the $x$ -axis at $c$ ; odd $k$ : it crosses.	Sketching a polynomial from its factors, or reading the factors from a graph.	Unit 1 Polynomial Functions and Complex ZerosPractice it: Polynomials and rationals
$a + bi \ \text{and}\ a - bi$	Complex zeros of a polynomial with real coefficients come in conjugate pairs; a polynomial of degree $n$ has $n$ zeros, counted with multiplicity.	Finding every zero, or the least degree a polynomial can have from the zeros you are given.	Unit 1 Polynomial Functions and Complex Zeros
$f(-x)$ $= f(x),$ $\quad f(-x)$ $= -f(x)$	Even functions (the first: symmetric about the $y$ -axis) and odd functions (the second: symmetric about the origin).	Symmetry questions, and filling in a graph or table from half of it.	Unit 1 Polynomial Functions and Complex ZerosPractice it: Function lab
$\lim_{x \to \infty} p(x) \ \text{from}\ a_n x^n$	End behavior of a polynomial comes from its leading term $a_n x^n$ : the sign of $a_n$ and whether $n$ is even or odd.	Writing end behavior in limit notation, for both $x \to \infty$ and $x \to -\infty$ .	Unit 1 Polynomial Functions and End BehaviorPractice it: Function lab
$\lim_{x \to \infty} \frac{ax^n + \cdots}{bx^n + \cdots}$ $= \frac{a}{b}$	Same degree on top and bottom: a horizontal asymptote at $y = \frac{a}{b}$ . Lower degree on top: $y = 0$ . Higher on top: no horizontal asymptote (one more gives a slant asymptote).	End behavior of a rational function, written as a limit.	Unit 1 Rational Functions and End BehaviorPractice it: Polynomials and rationals
$\lim_{x \to c^-} r(x)$ $= \pm\infty,$ $\quad \lim_{x \to c^+} r(x)$ $= \pm\infty$	A vertical asymptote at $x = c$ , where the denominator is 0 and the numerator is not (after cancelling common factors).	Behavior near a vertical asymptote: give both one-sided limits, each with its sign.	Unit 1 Rational Functions and Vertical AsymptotesPractice it: Polynomials and rationals
$\frac{(x - c)g(x)}{(x - c)h(x)}$	A factor $x - c$ that cancels completely leaves a hole at $x = c$ , at height $\frac{g(c)}{h(c)}$ , not a vertical asymptote.	Telling a hole from an asymptote: factor first, then use the value the outputs approach.	Unit 1 Rational Functions and HolesPractice it: Polynomials and rationals

Transformations and models

Formula	What it is	When you use it	Learn it in
$g(x) = a f\left(b(x - h)\right) + k$	The graph of $f$ stretched vertically by $a$ , horizontally by $\frac{1}{b}$ , shifted right $h$ and up $k$ . A negative $a$ reflects over the $x$ -axis; a negative $b$ over the $y$ -axis.	Building a function from a parent graph, or reading the changes from a formula. Factor $b$ out before reading the shift.	Unit 1 Transformations of FunctionsPractice it: Function lab
$\text{residual}$ $= \text{actual} - \text{predicted}$	A positive residual means the model predicted too little; a negative one, too much.	Judging a model: residuals scattered around 0 with no pattern mean the model is appropriate.	Unit 2 Competing Function Model ValidationPractice it: Regression lab

Exponential and logarithmic functions

Formula	What it is	When you use it	Learn it in
$a_n$ $= a_0 + dn,$ $\quad g_n$ $= g_0 r^n$	Arithmetic sequences add a common difference $d$ ; geometric sequences multiply by a common ratio $r$ .	Linear and exponential patterns at whole-number steps; a linear function is the continuous arithmetic case, an exponential function the geometric one.	Unit 2 Change in Arithmetic and Geometric SequencesPractice it: Exponential and log lab
$f(x) = ab^x$	An exponential function: initial value $a$ , growth factor $b$ (growth when $b > 1$ , decay when $0 < b < 1$ ). Over equal input steps the outputs have a constant ratio.	Justifying an exponential model: equal-length input intervals and proportional outputs, with the numbers.	Unit 2 Exponential FunctionsPractice it: Exponential and log lab
$b^m b^n$ $= b^{m + n},$ $\quad \left(b^m\right)^n$ $= b^{mn},$ $\quad b^{-n}$ $= \frac{1}{b^n}$	The exponent rules for products, powers and negative exponents.	Rewriting exponential expressions, such as $2^{x + 3} = 8 \cdot 2^x$ , and comparing growth over different time units.	Unit 2 Exponential Function ManipulationPractice it: Symbolic steps
$f(t)$ $= a(1 + r)^t,$ $\quad f(t)$ $= ae^{kt}$	Exponential models with a growth rate $r$ per unit of time, and with the natural base $e$ .	Models from a percent change or from two data points; store the constants, never round partway.	Unit 2 Exponential Function Context and Data ModelingPractice it: Regression lab
$(f \circ g)(x) = f(g(x))$	Composition: $g$ acts first, then $f$ . Usually $f(g(x)) \ne g(f(x))$ .	Composite values from tables or graphs: show the inner value, then the outer.	Unit 2 Composition of FunctionsPractice it: Function lab
$f\left(f^{-1}(x)\right) = x$	An inverse undoes a function; its graph is the reflection over $y = x$ . A function has an inverse only if no two inputs share an output.	Finding an inverse (swap $x$ and $y$ , then solve), and justifying invertibility with specific values.	Unit 2 Inverse FunctionsPractice it: Function lab
$\log_b c = a \Leftrightarrow b^a = c$	The definition of a logarithm, for $b > 0$ , $b \ne 1$ and $c > 0$ . $\ln$ is $\log_e$ ; $\log$ alone is $\log_{10}$ .	Evaluating logarithms exactly and moving between exponential and logarithmic forms.	Unit 2 Logarithmic ExpressionsPractice it: Exponential and log lab
$\log_b\left(b^x\right)$ $= x,$ $\quad b^{\log_b x}$ $= x$	Exponential and logarithmic functions with the same base are inverses.	Undoing one with the other when solving equations.	Unit 2 Inverses of Exponential FunctionsPractice it: Exponential and log lab
$\log_b(mn)$ $= \log_b m$ $+ \log_b n,$ $\quad \log_b\frac{m}{n}$ $= \log_b m - \log_b n$	The product and quotient rules for logarithms, for $m > 0$ and $n > 0$ .	Combining into a single logarithm or expanding one. There is no rule for $\log_b(m + n)$ .	Unit 2 Logarithmic Function ManipulationPractice it: Symbolic steps
$\log_b\left(m^k\right)$ $= k\log_b m,$ $\quad \log_b x$ $= \frac{\ln x}{\ln b}$	The power rule and the change of base rule.	Moving a coefficient into an exponent before combining, and evaluating any logarithm on a calculator.	Unit 2 Logarithmic Function ManipulationPractice it: Symbolic steps
$\log y = (\log b)x + \log a$	Taking the logarithm of $y = ab^x$ gives a line: on a semi-log plot exponential data look linear, with slope $\log b$ .	Recognizing exponential data on a semi-log plot and reading $a$ and $b$ from the line.	Unit 2 Semi-log PlotsPractice it: Exponential and log lab

The unit circle and sinusoids

Formula	What it is	When you use it	Learn it in
$\sin\theta$ $= \frac{y}{r},$ $\quad \cos\theta$ $= \frac{x}{r},$ $\quad \tan\theta$ $= \frac{y}{x}$	Sine, cosine and tangent of an angle in standard position whose terminal ray passes through $(x, y)$ , with $r = \sqrt{x^2 + y^2}$ . On the unit circle, $(\cos\theta, \sin\theta)$ .	Values from a point or the unit circle; tangent is the slope of the terminal ray.	Unit 3 Sine, Cosine, and TangentPractice it: Unit circle
$\pi \ \text{radians} = 180 \ \text{degrees}$	Converting between radians and degrees. The exam works in radians.	Every angle: set the calculator to radian mode.	Unit 3 Sine, Cosine, and TangentPractice it: Unit circle
$\sin\frac{\pi}{6}$ $= \frac{1}{2},$ $\quad \sin\frac{\pi}{4}$ $= \frac{\sqrt{2}}{2},$ $\quad \sin\frac{\pi}{3}$ $= \frac{\sqrt{3}}{2}$	The special values; cosine has the same values in the opposite order ( $\cos\frac{\pi}{3} = \frac{1}{2}$ ).	Exact values without a calculator, with the sign set by the quadrant.	Unit 3 Sine and Cosine Function ValuesPractice it: Unit circle
$f(\theta)$ $= a\sin\left(b(\theta + c)\right) + d$	A sinusoid: amplitude $\|a\|$ , period $\frac{2\pi}{\|b\|}$ , midline $y = d$ , shifted left by $c$ (right if $c < 0$ ).	Periodic models: $b$ from the period, $d$ from the midline, $a$ from half the distance from maximum to minimum.	Unit 3 Sinusoidal Function TransformationsPractice it: Sinusoid builder
$\tan x = \frac{\sin x}{\cos x}$	Tangent has period $\pi$ and vertical asymptotes where $\cos x = 0$ , at $x = \frac{\pi}{2} + k\pi$ .	Graphs and values of tangent, and rewriting to a single function.	Unit 3 The Tangent FunctionPractice it: Unit circle
$\sin^{-1}x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right],$ $\quad \cos^{-1}x \in [0, \pi],$ $\quad \tan^{-1}x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$	The ranges of the inverse trigonometric functions (the range of $\tan^{-1}x$ leaves out its endpoints, where tangent is undefined). $\sin^{-1}x$ is not $\frac{1}{\sin x}$ .	Inverse trig values and the first solution of a trig equation; use symmetry for the others.	Unit 3 Inverse Trigonometric FunctionsPractice it: Symbolic steps
$\sec x$ $= \frac{1}{\cos x},$ $\quad \csc x$ $= \frac{1}{\sin x},$ $\quad \cot x$ $= \frac{1}{\tan x}$	The reciprocal functions, undefined where the function in the denominator is 0.	Rewriting expressions in sine and cosine, and finding their asymptotes.	Unit 3 The Secant, Cosecant, and Cotangent FunctionsPractice it: Symbolic steps

Identities and equations

Formula	What it is	When you use it	Learn it in
$\sin^2 x + \cos^2 x = 1$	The Pythagorean identity. Dividing by $\cos^2 x$ gives $1 + \tan^2 x = \sec^2 x$ ; by $\sin^2 x$ , $1 + \cot^2 x = \csc^2 x$ .	Rewriting to one function, and finding $\cos x$ from $\sin x$ (the sign from the quadrant).	Unit 3 Equivalent Representations of Trigonometric FunctionsPractice it: Symbolic steps
$\sin(\alpha \pm \beta)$ $= \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$	The sum and difference identities for sine.	Exact values such as $\sin\frac{7\pi}{12}$ , and rewriting a shifted sine.	Unit 3 Equivalent Representations of Trigonometric FunctionsPractice it: Symbolic steps
$\cos(\alpha \pm \beta)$ $= \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$	The sum and difference identities for cosine; the sign in the middle flips.	Exact values and rewriting, as for sine.	Unit 3 Equivalent Representations of Trigonometric FunctionsPractice it: Symbolic steps
$\sin(2x)$ $= 2\sin x\cos x,$ $\quad \cos(2x)$ $= \cos^2 x - \sin^2 x$	The double-angle identities; also $\cos(2x)$ $= 2\cos^2 x - 1$ $= 1 - 2\sin^2 x$ .	Pick the form of $\cos(2x)$ that cancels what is next to it, and solve equations with $2x$ inside.	Unit 3 Equivalent Representations of Trigonometric FunctionsPractice it: Symbolic steps
$x = x_0 + 2k\pi$	Sine and cosine repeat every $2\pi$ (tangent every $\pi$ ), so one solution gives infinitely many.	Every solution in an interval: find the solutions in one period, then add periods while they stay in the interval.	Unit 3 Trigonometric Equations and InequalitiesPractice it: Symbolic steps

Polar coordinates

Formula	What it is	When you use it	Learn it in
$x$ $= r\cos\theta,$ $\quad y$ $= r\sin\theta,$ $\quad r^2$ $= x^2$ $+ y^2$	Converting between polar $(r, \theta)$ and rectangular $(x, y)$ coordinates.	Plotting polar points and rewriting equations; check the quadrant when you find $\theta$ from $\tan\theta = \frac{y}{x}$ .	Unit 3 Trigonometry and Polar CoordinatesPractice it: Polar graphs
$a + bi$ $= r(\cos\theta + i\sin\theta)$	The polar form of a complex number, with $r = \sqrt{a^2 + b^2}$ .	Moving a complex number between rectangular and polar form.	Unit 3 Trigonometry and Polar CoordinatesPractice it: Polar graphs
$\frac{r(\theta_2) - r(\theta_1)}{\theta_2 - \theta_1}$	The average rate of change of $r$ with respect to $\theta$ . If $r > 0$ and $r$ is increasing, the curve moves away from the origin.	How the distance from the origin changes as $\theta$ turns; the units are distance per radian.	Unit 3 Rates of Change in Polar FunctionsPractice it: Polar graphs

Easy to forget

Three decimal places, never partway. Store every constant and intermediate value in the calculator and round only the final answer. A model built from rounded constants often misses the third decimal place. Function Model Construction and Application
Every solution in the interval. A calculator or inverse function gives one solution. Widen the window, use symmetry or add a period, and list every solution in the stated interval, with no extras. Trigonometric Equations and Inequalities
$\sin^{-1} x$ is not $\frac{1}{\sin x}$ . The $-1$ means the inverse function, whose outputs are in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ . The reciprocal is $\csc x$ . Inverse Trigonometric Functions
$\log_b(m + n)$ has no rule. Only products, quotients and powers inside a logarithm can be split. $\ln(x^2) = 2\ln x$ only for $x > 0$ . Logarithmic Function Manipulation
$e^x$ is never 0 or negative. In an equation quadratic in $e^x$ , reject the root where $e^x$ would be 0 or negative, and say why. Exponential and Logarithmic Equations and Inequalities
$b$ is not the period. The period of $\sin(bx)$ is $\frac{2\pi}{b}$ , so $b = \frac{2\pi}{\text{period}}$ . Factor $b$ out before reading the phase shift. Sinusoidal Function Transformations
Both sides of a vertical asymptote. Write each one-sided limit with its sign: $\lim_{x \to c^-}$ and $\lim_{x \to c^+}$ . Rational Functions and Vertical Asymptotes
Invertible needs values. Show two inputs with the same output (not invertible), or explain why every output comes from one input. "Fails the horizontal line test" alone earns nothing. Inverse Functions

Practice with it

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