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AP Precalculus Exam Review

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AP Precalculus Exam Review - Complete Interactive Lesson

Part 1: Exam Format, Pacing and the Calculator Split

🎓 AP Precalculus Exam Review

Part 1 of 7 — Exam Format, Pacing and the Calculator Split

Topic Overview

Part	Review Topic
1	Exam Format, Pacing and the Calculator Split
2	Unit 1: Rates of Change and Polynomial Functions
3	Unit 1: Rational Functions, Transformations and Models
4	Unit 2: Sequences, Exponential and Logarithmic Functions
5	Unit 3: Trigonometric Functions and Sinusoidal Models
6	Unit 3: Inverse Trigonometry, Equations, Identities and Polar Functions
7	Final Mixed Set: Part A and Part B Practice

The 2027 AP Precalculus Exam

Section	Part	Questions	Time	Calculator	Pace
I: Multiple Choice (62.5%)	A	29	65 min	none	about 2 min 15 s per question
I: Multiple Choice	B	13	40 min	graphing calculator	about 3 min per question
II: Free Response (37.5%)	A	Q1, Q2	35 min	graphing calculator	Function Concepts; Non-Periodic Modeling
II: Free Response	B	Q3, Q4	35 min	none	Periodic Modeling; Symbolic Manipulations

Every multiple-choice question has four choices and there is no penalty for a wrong answer: answer every question.

What Is Tested

Unit	Content	Approximate weight
1	Polynomial and Rational Functions (rates of change, polynomials, rational functions, transformations, modeling)	about 35%
2	Exponential and Logarithmic Functions (sequences, exponential and log functions, composition and inverses, data modeling, semi-log plots)	about 32%
3	Trigonometric and Polar Functions (trig functions and graphs, sinusoidal models, inverses, equations, identities, polar coordinates)	about 33%
4	Parameters, vectors, matrices	not assessed

The three assessed units carry roughly equal weight. There is no safe unit to skip.

The Calculator Split

Without a calculator (Part A, Q3, Q4)	With a graphing calculator (Part B, Q1, Q2)
unit-circle values: $\cos\frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$	regression: linear, quadratic, cubic, exponential, logarithmic, sinusoidal
log properties and exact solving: $\log_2 x + \log_2(x - 2) = 3$	numeric solving: where does the model equal $50$ ?
factoring, identities, equivalent forms	evaluating $\ln 7$ , $\sin 2.3$ , $1.299^{9}$
end behavior and asymptotes with limit notation	graphing to count solutions or find extrema
reading tables and graphs for rates of change	finding residuals and comparing models

Key Fact: Part A is the long part: 29 questions in 65 minutes. Two passes beat one slow pass: answer what you know, mark what you don't, come back.

Format check. 🎯

Test-day rules. 🔽

Key Takeaways — Part 1

Section I: 29 questions in 65 min without a calculator, then 13 in 40 min with one; Section II: Q1–Q2 with a calculator, Q3–Q4 without, 35 min each
Units 1, 2, 3 are about 35 / 32 / 33%; Unit 4 is not assessed
Four choices, no penalty: answer everything, two passes through Part A
Know which skills are exact, by hand and which are calculator skills

Part 2: Unit 1: Rates of Change and Polynomial Functions

🎓 AP Precalculus Exam Review

Part 2 of 7 — Unit 1: Rates of Change and Polynomial Functions

Exam Weight: Unit 1 is about 35% of the exam. This part covers CED 1.1–1.6 and 1.10; Part 3 covers 1.7–1.9 and 1.11–1.14.

Rates of Change (1.1–1.4)

Idea	What to compute or say
Average rate of change on $[a, b]$	$\dfrac{f(b) - f(a)}{b - a}$ : slope of the secant line, with units
Linear function	rate of change is constant (first differences equal)
Quadratic function	rate of change changes at a constant rate (second differences equal)
Increasing / decreasing	sign of the rate of change on the interval
Concave up / down	the rate of change is increasing / decreasing
Point of inflection	where concavity changes
From a table	compare successive average rates of change on equal-width intervals

Polynomial Functions (1.5–1.6, 1.10)

Fact	Detail
Degree $n$	at most $n$ real zeros; at most $n - 1$ local extrema; $n$ th differences constant for equally spaced inputs
End behavior	decided by the leading term $a_nx^n$ : even degree, same direction both ways ( $+$ if $a_n > 0$ ); odd degree, opposite directions
Zero of multiplicity $m$	odd $m$ : graph crosses; even $m$ : graph touches and turns; $m \ge 2$ : the graph flattens there
Between consecutive zeros	exactly one local extremum when the zeros are simple; a local maximum or minimum occurs where the function changes from increasing to decreasing or back
Non-real zeros	come in conjugate pairs when the coefficients are real; a degree- $n$ polynomial has $n$ complex zeros counted with multiplicity
Even / odd function	$f(-x) = f(x)$ : symmetric about the $y$ -axis; $f(-x) = -f(x)$ : symmetric about the origin
Global vs local	a polynomial of even degree has a global maximum or minimum; odd degree has neither

Key Fact: "The rate of change is decreasing" does not mean the function is decreasing. $f(x) = \sqrt{x}$ increases while its rate of change decreases.

Unit 1 practice (no calculator). 🎯

Quick computations. ✍️

1) The average rate of change of $f(x) = x^3$ on the interval $[1, 2]$ :

2) For equally spaced inputs, a table of values has constant third differences (and non-constant second differences). The degree of the polynomial:

3) The number of distinct real zeros of $p(x) = (x^2 + 1)(x - 2)(x + 5)$ :

Key Takeaways — Part 2

Average rate of change $= \frac{\Delta y}{\Delta x}$ with units; constant first differences mean linear, constant second differences mean quadratic
Increasing/decreasing is about the function; concave up/down is about its rate of change
End behavior comes from the leading term; multiplicity decides cross (odd) or touch (even)
Non-real zeros come in conjugate pairs; a degree- $n$ polynomial has at most $n$ real zeros and $n - 1$ local extrema

Part 3: Unit 1: Rational Functions, Transformations and Models

🎓 AP Precalculus Exam Review

Part 3 of 7 — Unit 1: Rational Functions, Transformations and Models

Rational Functions (1.7–1.9, 1.11)

Feature	How to find it	How to state it
Vertical asymptote $x = a$	denominator zero and numerator nonzero at $a$ (after cancelling common factors)	$\lim_{x \to a^+} r(x) = \pm\infty$
Hole at $x = a$	a factor $(x - a)$ common to numerator and denominator	$\lim_{x \to a} r(x) =$ the cancelled function's value at $a$
Horizontal asymptote	degree top $<$ bottom: $y = 0$ ; equal: ratio of leading coefficients; top $>$ bottom: none	$\lim_{x \to \pm\infty} r(x) = c$
Slant asymptote	degree top $=$ degree bottom $+ 1$ : the quotient from polynomial long division	$r(x) \approx mx + b$ for large $\lvert x \rvert$
Zeros	zeros of the numerator that are not holes	$x$ -intercepts
Rational inequality	sign chart over the zeros and the undefined points; never multiply by a denominator of unknown sign	interval notation, excluding undefined points

Transformations (1.12)

Form	Effect on the graph of $f$
$f(x) + k$	shift up $k$
$f(x - h)$	shift right $h$
$a\,f(x)$	vertical stretch by $\lvert a \rvert$ ; reflect over the $x$ -axis if $a < 0$
$f(bx)$	horizontal compression by $\dfrac{1}{\lvert b \rvert}$ ; reflect over the $y$ -axis if $b < 0$
Order	inside changes act on inputs (horizontal), outside changes act on outputs (vertical); apply stretches and reflections before shifts

Choosing a Model (1.13–1.14)

Signal in the data or context	Model
constant first differences, constant rate	linear
constant second differences, one turning point	quadratic
two turning points, a point of inflection	cubic
a limiting value as the input grows	rational (horizontal asymptote), or exponential approaching a horizontal asymptote
the rule changes at certain inputs (fees, tax brackets, speed limits)	piecewise-defined
domain restricted by the context	state the domain with the model: lengths positive, counts whole numbers

Key Fact: For a hole, cancel first, then evaluate: $r(x) = \dfrac{(x - 2)(x + 2)}{(x - 2)(x + 1)}$ has a hole at $\left(2, \dfrac{4}{3}\right)$ and a vertical asymptote only at $x = -1$ .

Unit 1 practice (no calculator). 🎯 Questions 1 and 2 use the graph above.

Choosing a model. 🔽

Key Takeaways — Part 3

Cancel common factors first: a cancelled factor is a hole, a remaining denominator zero is a vertical asymptote
Horizontal asymptotes come from comparing degrees; state every asymptote with limit notation
Inside changes are horizontal ( $f(bx)$ compresses by $\frac{1}{\lvert b \rvert}$ , $f(x - h)$ shifts right); outside changes are vertical
Model choice follows differences, turning points, limiting values and rule changes; state the domain in context

Part 4: Unit 2: Sequences, Exponential and Logarithmic Functions

🎓 AP Precalculus Exam Review

Part 4 of 7 — Unit 2: Sequences, Exponential and Logarithmic Functions

Exam Weight: Unit 2 is about 32% of the exam (CED 2.1–2.15).

Sequences (2.1–2.2)

Type	Rule	Function it matches
Arithmetic, common difference $d$	$a_n = a_k + d(n - k)$	linear: constant rate of change
Geometric, common ratio $r$	$g_n = g_k \cdot r^{\,n - k}$	exponential: constant ratio

Exponential Functions (2.3–2.6)

Fact	Detail
$f(x) = a \cdot b^x$	$a = f(0)$ ; $b$ is the ratio of outputs for a unit increase in $x$ ; growth if $b > 1$ , decay if $0 < b < 1$
Percent change	$b = 1 + r$ : a base of $1.299$ is a $29.9\%$ increase per unit; $0.85$ is a $15\%$ decrease
From two points $(x_1, y_1), (x_2, y_2)$	$b = \left(\dfrac{y_2}{y_1}\right)^{1/(x_2 - x_1)}$ , then solve for $a$
Horizontal asymptote	$y = 0$ for $a \cdot b^x$ ; $y = k$ for $a \cdot b^x + k$ ; stated with $\lim_{x \to -\infty}$ (growth) or $\lim_{x \to \infty}$ (decay)
Equivalent forms	$a \cdot b^{x} = a \cdot \left(b^{1/k}\right)^{kx}$ , so $8^{x/3} = 2^x$ and $(1.06)^{x}$ is $(1.06^{12})^{x/12}$ per year of months
Always concave up	the rate of change of $a \cdot b^x$ ( $a > 0$ ) is increasing whether the function grows or decays

Composition and Inverses (2.7–2.8)

Fact	Detail
$(f \circ g)(x) = f(g(x))$	evaluate the inside function first; the domain needs $x$ in the domain of $g$ and $g(x)$ in the domain of $f$
Inverse	swaps inputs and outputs: $f(a) = b \iff f^{-1}(b) = a$ ; graph reflects over $y = x$ ; domain and range swap
Exists when	$f$ is one-to-one (strictly increasing or strictly decreasing suffices)
Exponential and log	$\log_b x$ is the inverse of $b^x$ : $b^{\log_b x} = x$ and $\log_b(b^x) = x$

Logarithmic Functions (2.9–2.13)

Fact	Detail
Definition	$\log_b x = y \iff b^y = x$ ( $b > 0$ , $b \ne 1$ , $x > 0$ )
Properties	$\log_b(MN) = \log_b M + \log_b N$ ; $\log_b\dfrac{M}{N} = \log_b M - \log_b N$ ; $\log_b M^p = p\log_b M$
Change of base	$\log_b x = \dfrac{\ln x}{\ln b}$
Graph	domain $x > 0$ , vertical asymptote $x = 0$ ( $\lim_{x \to 0^+} \log_b x = -\infty$ for $b > 1$ ), increasing and concave down, no horizontal asymptote
Solving	isolate the exponential or the log, convert, then check the domain

Data Modeling and Semi-log Plots (2.14–2.15)

Fact	Detail
Residual	actual $-$ predicted; a pattern in the residual plot means the model type is wrong
Semi-log plot	$\log y$ against $x$ : exponential data looks linear; slope $= \log b$ , intercept $= \log a$
Log model	$y = a + b\ln x$ : outputs change by equal amounts when the input is multiplied by a constant factor

Key Fact: "Proportional growth" (equal ratios over equal input steps) is the test for an exponential model, exactly as "equal differences" is the test for a linear one.

Unit 2 practice (no calculator). 🎯

Quick computations. ✍️

1) $\log_3 81$ :

2) The solution of $2^x = 20$ , to three decimal places (calculator):

3) In an arithmetic sequence with $a_1 = 7$ and common difference $4$ , the value of $a_{10}$ :

Key Takeaways — Part 4

Arithmetic $\leftrightarrow$ linear (constant difference); geometric $\leftrightarrow$ exponential (constant ratio)
$f(x) = a \cdot b^x$ : $a = f(0)$ , $b = 1 + r$ ; find $b$ from two points with $\left(\frac{y_2}{y_1}\right)^{1/(x_2 - x_1)}$
Inverses swap inputs and outputs; $\log_b x$ undoes $b^x$ ; check the domain after solving a log equation
A linear semi-log plot means exponential data; a patterned residual plot means the wrong model type

Part 5: Unit 3: Trigonometric Functions and Sinusoidal Models

🎓 AP Precalculus Exam Review

Part 5 of 7 — Unit 3: Trigonometric Functions and Sinusoidal Models

Exam Weight: Unit 3 is about 33% of the exam. This part covers CED 3.1–3.8; Part 6 covers 3.9–3.15.

Angles and the Unit Circle (3.1–3.3)

Fact	Detail
Radian measure	$\theta = \dfrac{s}{r}$ ; arc length $s = r\theta$ ; $\pi$ radians $= 180^\circ$
Coterminal angles	differ by multiples of $2\pi$
On the unit circle	the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ ; $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ is the slope of the terminal ray
Special values	$\sin\frac{\pi}{6} = \frac{1}{2}$ , $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ , $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$ ; cosine reverses the order
Signs by quadrant	I: all positive; II: sine; III: tangent; IV: cosine
Reference angle	the acute angle to the $x$ -axis; the same value with the quadrant's sign

Sine, Cosine and Tangent as Functions (3.4, 3.8)

Function	Period	Domain	Range	Key features
$\sin x$	$2\pi$	all reals	$[-1, 1]$	odd; zeros at $k\pi$ ; max $1$ at $\frac{\pi}{2} + 2k\pi$
$\cos x$	$2\pi$	all reals	$[-1, 1]$	even; zeros at $\frac{\pi}{2} + k\pi$ ; max $1$ at $2k\pi$
$\tan x$	$\pi$	$x \ne \frac{\pi}{2} + k\pi$	all reals	odd; vertical asymptotes where $\cos x = 0$ ; always increasing between asymptotes

Sinusoidal Functions (3.5–3.7)

For $f(x) = a\sin\big(b(x + c)\big) + d$ (or with cosine):

Feature	Formula
Amplitude	$\lvert a \rvert = \dfrac{\max - \min}{2}$
Midline	$y = d = \dfrac{\max + \min}{2}$
Period	$\dfrac{2\pi}{\lvert b \rvert}$ , so $b = \dfrac{2\pi}{\text{period}}$
Frequency	$\dfrac{1}{\text{period}} = \dfrac{\lvert b \rvert}{2\pi}$ cycles per unit
Phase shift	$-c$ (the graph of $\sin(b(x + c))$ is shifted left $c$ )
Concavity	concave down above the midline, concave up below it; points of inflection on the midline

Building a model: read max, min and period from the context; midline and amplitude from max and min; $b$ from the period; choose sine or cosine (and a sign) by where the cycle starts; then shift if needed.

Key Fact: The graph of a sinusoid is steepest at the midline and flat at the maximum and minimum. Its rate of change is largest in size where it crosses the midline.

Unit 3 practice (no calculator). 🎯 Question 1 uses the graph above.

Exact values and features. ✍️

1) $\sin\left(\dfrac{5\pi}{6}\right)$ :

2) The amplitude of $y = -4\cos(3x) + 2$ :

3) The period of $y = 5\sin\left(\dfrac{\pi}{4}x\right)$ :

Key Takeaways — Part 5

Unit-circle values by reference angle and quadrant sign; $\tan\theta$ is the slope $\frac{\sin\theta}{\cos\theta}$
$\sin$ and $\cos$ have period $2\pi$ ; $\tan$ has period $\pi$ and asymptotes where $\cos x = 0$
Sinusoid: amplitude $\frac{\max - \min}{2}$ , midline $\frac{\max + \min}{2}$ , $b = \frac{2\pi}{\text{period}}$ , shift decided by where the cycle starts
Steepest at the midline, flat at the extremes; concave down above the midline, concave up below

Part 6: Unit 3: Inverse Trigonometry, Equations, Identities and Polar Functions

🎓 AP Precalculus Exam Review

Part 6 of 7 — Unit 3: Inverse Trigonometry, Equations, Identities and Polar Functions

Inverse Trigonometric Functions (3.9)

Function	Domain	Range	Quadrants of the output
$\arcsin x$	$[-1, 1]$	$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$	IV (negative inputs), I (positive)
$\arccos x$	$[-1, 1]$	$[0, \pi]$	I (positive inputs), II (negative)
$\arctan x$	all reals	$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$	IV (negative inputs), I (positive)

$\arcsin(\sin x) = x$ only when $x$ is in the range of arcsine; otherwise find the angle in range with the same sine.

Trigonometric Equations and Inequalities (3.10)

Situation	Method
$\sin\theta = k$ on $[0, 2\pi)$	one solution from the unit circle, the second by symmetry: $\pi - \theta_1$ (sine), $2\pi - \theta_1$ (cosine), $\theta_1 + \pi$ (tangent)
All solutions	add $2\pi k$ (sine, cosine) or $\pi k$ (tangent)
Multiple angle $\sin(2\theta) = k$ on $[0, 2\pi)$	solve for $2\theta$ on $[0, 4\pi)$ , then divide: twice as many solutions
Quadratic in a trig function	factor: $2\sin^2\theta - \sin\theta = \sin\theta(2\sin\theta - 1)$
Two different functions	use an identity to leave one function
Inequality $\sin\theta > k$	solve the equation, then test the arcs of the unit circle between the solutions

Identities and Equivalent Forms (3.11–3.12)

Identity	Form
Pythagorean	$\sin^2\theta + \cos^2\theta = 1$ , $1 + \tan^2\theta = \sec^2\theta$ , $1 + \cot^2\theta = \csc^2\theta$
Sum and difference	$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$ ; $\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$
Double angle	$\sin(2\theta) = 2\sin\theta\cos\theta$ ; $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
Reciprocal and quotient	$\sec\theta = \dfrac{1}{\cos\theta}$ , $\csc\theta = \dfrac{1}{\sin\theta}$ , $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

Polar Coordinates and Polar Functions (3.13–3.15)

Fact	Detail
Conversions	$x = r\cos\theta$ , $y = r\sin\theta$ ; $r^2 = x^2 + y^2$ , $\tan\theta = \dfrac{y}{x}$ with the quadrant checked
Negative $r$	$(-r, \theta)$ is the same point as $(r, \theta + \pi)$
Circles	$r = a$ (center at the pole); $r = a\cos\theta$ or $r = a\sin\theta$ (through the pole, diameter $\lvert a \rvert$ )
Roses $r = a\cos(k\theta)$	$k$ petals if $k$ is odd, $2k$ petals if $k$ is even
Limaçons $r = a + b\cos\theta$	inner loop if $\lvert b \rvert > \lvert a \rvert$ ; cardioid if $\lvert a \rvert = \lvert b \rvert$ ; dimpled or convex if $\lvert a \rvert > \lvert b \rvert$
Rate of change	where $r$ is increasing with $\theta$ the point moves away from the pole; where $r$ is decreasing it moves toward the pole (for $r > 0$ )
Complex numbers	$z = a + bi = r(\cos\theta + i\sin\theta)$ with $r = \sqrt{a^2 + b^2}$

Key Fact: For $\sin(2\theta) = k$ on $0 \le \theta < 2\pi$ , the inside angle $2\theta$ runs through two full turns, so expect four solutions when $\lvert k \rvert < 1$ .

Unit 3 practice (no calculator). 🎯

Polar graphs. 🔽

Key Takeaways — Part 6

Inverse trig outputs live in fixed ranges: $\arcsin, \arctan$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ , $\arccos$ in $[0, \pi]$
Solve trig equations with the unit circle and symmetry; a multiple angle $k\theta$ multiplies the number of solutions by $k$
Identities are tools for one function: Pythagorean substitutions and the three forms of $\cos(2\theta)$
Polar: convert with $x = r\cos\theta$ , $y = r\sin\theta$ ; roses have $k$ or $2k$ petals; $r$ increasing means moving away from the pole

Part 7: Final Mixed Set: Part A and Part B Practice

🎓 AP Precalculus Exam Review

Part 7 of 7 — Final Mixed Set: Part A and Part B Practice

How to Use This Set

The questions below mix all three units the way the exam does. The first four are Part A style: no calculator, about 2 minutes 15 seconds each. The last two are Part B style: use a graphing calculator, about 3 minutes each. Time yourself, then review every explanation, including the ones you got right.

The Final Checklist

Before the exam	During Part A (no calculator)	During Part B (calculator)
unit-circle values cold: $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ and their reflections	two passes: bank the sure questions, mark the rest	enter data once, store the regression, reuse it
log properties and the three forms of $\cos(2\theta)$	eliminate choices with end behavior, domain, sign	solve numerically by intersection; read to three decimals
asymptote rules by degree comparison	every answer exact: $\pi$ , radicals, fractions	check that an intersection is inside the asked interval
midline, amplitude, period formulas	for "all solutions", count how many the interval should hold	a model prediction is approximately

Key Fact: The exam rewards recognition: naming the model type from differences or ratios, the asymptote from the degrees, the quadrant from the signs. Each recognition saves a minute.

Part A style: no calculator. 🎯 Four questions, about 9 minutes.

Part B style: graphing calculator. 🎯 Two questions, about 6 minutes. Report decimals to three places.

Last exact checks. ✍️

1) The average rate of change of $f(x) = 2^x$ on the interval $[1, 3]$ :

2) The value of $\cos\pi$ :

3) The value of $\log_2 8 + \log_2 4$ :

Key Takeaways — AP Precalculus Exam Review

Unit	Recognize	Compute	Say
1	model type from differences; asymptotes from degrees; multiplicity from the graph	average rates of change; zeros; transformed points	increasing/decreasing and concave up/down, with limit notation for asymptotes
2	exponential from ratios; log from doubling inputs; linear semi-log plot	$b$ from two points; inverses by swapping; log equations with a domain check	percent change per unit; "approximately" for model outputs
3	quadrant from signs; period from $b$ ; petals from $k$	exact unit-circle values; all solutions in an interval; polar conversions	midline/amplitude/period; "moves toward or away from the pole"

42 multiple-choice in 105 minutes, then 4 free-response in 70; Units 1–3 about 35 / 32 / 33%
Part A and Q3–Q4: exact by hand; Part B and Q1–Q2: calculator, three decimals
Answer every multiple-choice question; attempt every free-response part

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