Horizontal asymptote y = ratio of leading coefficients

Slant (oblique) asymptote — use polynomial long division

No horizontal or slant asymptote

Is Unit 4 on the AP Precalculus Exam?

No. Unit 4 — Functions Involving Parameters, Vectors, and Matrices — is included in the College Board course framework and is taught in class, but College Board does not assess it on the AP Precalculus Exam. Teachers may cover it for the course grade or as a bridge to later math.

How is the AP Precalculus Exam structured?

There are two sections. Section I is 40 multiple-choice questions in 2 hours (28 no-calculator, then 12 with a graphing calculator), worth 62.5%. Section II is 4 free-response questions in 1 hour (2 with a calculator, 2 without), worth 37.5%. Total testing time is about 3 hours.

Can you use a calculator on AP Precalculus?

On part of it. A graphing calculator is required for Part B of the multiple choice (12 questions) and Part A of the free response (2 questions). The other parts — Part A multiple choice and Part B free response — are completed without a calculator.

How is the AP Precalculus Exam scored?

Like other AP exams, it is scored 1 to 5. The multiple-choice section (62.5%) and free-response section (37.5%) combine into a composite score that College Board converts to the 1-5 scale. A 3 or higher is generally considered passing for college credit, depending on the college.

What is the difference between AP Precalculus and AP Calculus?

AP Precalculus builds the function toolkit — polynomial, rational, exponential, logarithmic, trigonometric, and polar functions — that calculus depends on. AP Calculus AB and BC then study limits, derivatives, and integrals. Mastering precalculus functions makes the jump to calculus far smoother.

How should I use this AP Precalculus study guide?

Work through the three assessed units in order, then take each unit's checkpoint quiz to find gaps and drill them with our free practice questions and flashcards. Study Unit 4 separately for your course, knowing it will not appear on the AP Exam. Revisit flagged sections before test day.

Is this AP Precalculus study guide free?

Yes — the full guide, the unit checkpoints, the glossary, the practice questions, and the flashcards are 100% free, with no account required.

What does the multiplicity of a zero tell you about a graph?

Multiplicity is how many times a factor repeats. At a zero of ODD multiplicity the graph crosses the x-axis; at a zero of EVEN multiplicity it touches and turns around (bounces) without crossing. Multiplicity 3 crosses with a flattening, inflecting shape. This is a core Unit 1 polynomial skill. The factor $(x - a)^m$ contributes a zero at $x = a$ with multiplicity $m$. Odd multiplicity changes the function's sign across the zero (crosses); even multiplicity keeps the sign the same (bounces).

How do you find the end behavior of a polynomial?

End behavior is controlled by the leading term alone. For even degree, both ends point the same way; for odd degree, they point opposite ways. A positive leading coefficient lifts the right end up, a negative one sends it down. This is a Unit 1 skill. As $x \to \pm\infty$, every lower-degree term becomes negligible next to the leading term. So a degree-4 polynomial with a positive leading coefficient rises on both ends, while a degree-3 with a negative one rises on the left and falls on the right.

How do you find the asymptotes of a rational function?

Vertical asymptotes occur where the denominator is zero but the numerator is not. The horizontal asymptote comes from comparing degrees: numerator degree less than denominator gives $y = 0$; equal degrees give the ratio of leading coefficients; numerator one degree higher gives a slant asymptote. This is a Unit 1 rational-function skill. If a factor cancels in both numerator and denominator, that input is a hole, not a vertical asymptote. Find a slant asymptote by polynomial long division when the numerator's degree is exactly one more than the denominator's.

What is the average rate of change of a function?

The average rate of change over an interval $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$ — the slope of the secant line through the two endpoints. It tells you how fast the output changes per unit of input on average. This idea anchors function analysis across the whole course. Only linear functions have a constant average rate of change over every interval. When the rate of change is itself increasing, the graph is concave up; when it is decreasing, the graph is concave down.

What is the difference between exponential and linear growth?

Linear growth ADDS a constant amount over equal intervals; exponential growth MULTIPLIES by a constant factor. Over equal-length input intervals, a linear function changes by equal differences while an exponential function changes by equal ratios — so exponentials eventually outpace any line. This is a defining Unit 2 distinction. In $f(x) = a \cdot b^x$, $a$ is the initial value and $b$ is the per-unit growth factor. A growth factor of $b = 1.5$ means a 50% increase each step; a factor between 0 and 1 means decay.

What is a logarithm and how is it related to an exponent?

A logarithm answers the question: to what power must I raise the base to get this number? $\log_b y = x$ means exactly $b^x = y$. The logarithm is the inverse of the exponential, so their graphs reflect across the line $y = x$. This is a core Unit 2 idea. Because $\log_b(b^x) = x$, taking a log of both sides of an exponential equation frees a trapped exponent so you can solve for it. The natural log $\ln$ uses base $e \approx 2.718$; the common log uses base 10.

What are the properties of logarithms?

Three rules condense or expand logs: the product rule $\log_b(MN) = \log_b M + \log_b N$, the quotient rule $\log_b(M/N) = \log_b M - \log_b N$, and the power rule $\log_b(M^k) = k\log_b M$. Use them to combine logs before solving an equation. These Unit 2 properties turn products into sums, quotients into differences, and exponents into coefficients. The change-of-base formula $\log_b x = \dfrac{\ln x}{\ln b}$ lets you evaluate any base on a calculator.

How do you solve an exponential equation?

Isolate the exponential, then take a logarithm of both sides and apply the power rule to bring the exponent down. For $b^x = c$, the solution is $x = \log_b c = \dfrac{\ln c}{\ln b}$. Always check that any solution keeps log arguments positive. This Unit 2 strategy mirrors solving a logarithmic equation: rewrite a log equation in exponential form, or combine into one log and exponentiate. Watch for extraneous solutions — a value that makes the argument of a log zero or negative must be rejected.

What is the unit circle and how do you use it?

The unit circle is a circle of radius 1 centered at the origin. For an angle $\theta$, the point on the circle is $(\cos\theta, \sin\theta)$: cosine is the x-coordinate and sine is the y-coordinate. This definition extends sine and cosine to every angle, not just acute ones. This is the foundation of Unit 3 trigonometry. A full circle is $2\pi$ radians, equal to 360 degrees. Reference angles let you read trig values at any angle from the first quadrant, with the sign set by the quadrant (All Students Take Calculus).

How do you find the amplitude, period, and midline of a sinusoid?

For $y = a\sin(b(x - c)) + d$: the amplitude is $|a|$ (half the max-minus-min), the period is $\dfrac{2\pi}{|b|}$, the phase shift is $c$, and the midline is $y = d$. These four numbers fully describe the wave. This is a Unit 3 modeling skill used for periodic data like tides and daylight hours. Build a model by reading the midline from the average of the max and min, the amplitude from half their difference, and $b$ from $\dfrac{2\pi}{\text{period}}$.

What is the Pythagorean identity?

The Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$, true for every angle $\theta$. It follows directly from the unit circle, where the point $(\cos\theta, \sin\theta)$ is a distance 1 from the origin. This is the most-used Unit 3 identity. It lets you find one trig value from another and is a key tool for verifying other identities. Pair it with the reciprocal, quotient, and even/odd identities to rewrite trig expressions.

How do you convert between polar and rectangular coordinates?

Polar gives a point as $(r, \theta)$ — a distance and an angle. To get rectangular: $x = r\cos\theta$ and $y = r\sin\theta$. Going back: $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(y/x)$, choosing $\theta$ by the quadrant. This is a Unit 3 polar skill. A polar function $r = f(\theta)$ traces curves like circles, roses, and limaçons as $\theta$ sweeps around. Where $r$ is increasing the curve moves away from the pole; where $r$ is decreasing it moves toward the pole.

What is a function transformation?

A transformation shifts, stretches, or reflects a graph. Outside the function affects outputs (vertical): $f(x) + k$ shifts up, $a\,f(x)$ stretches vertically. Inside affects inputs (horizontal) and works in reverse: $f(x - h)$ shifts RIGHT by $h$, $f(bx)$ compresses horizontally. Transformations recur throughout the course — they apply to polynomial, exponential, and trigonometric graphs alike. A negative sign flips the graph: $-f(x)$ reflects across the x-axis, and $f(-x)$ reflects across the y-axis.

What does it mean for two functions to be inverses?

Inverse functions undo each other: $f(f^{-1}(x)) = x$. Their graphs are reflections across the line $y = x$, and one is found by swapping input and output. Only one-to-one functions (passing the horizontal-line test) have inverse functions. This idea links Units 1 and 2 — most importantly, logarithms are the inverses of exponentials. To find an inverse algebraically, swap $x$ and $y$ in the equation and solve for $y$.

How do you model real-world data with a function?

Choose the function type that matches the pattern of change: linear for constant differences, exponential for constant ratios (fixed percent growth or decay), and sinusoidal for periodic data. Then fit parameters from the data and interpret what each one means in context. Modeling is a thread through every unit on the AP Precalculus Exam. AP problems often ask you to interpret a model's parameters, not just compute. For example, in $y = a \cdot b^x$, $a$ is the initial value and $b$ is the per-unit growth factor.

A College Board Advanced Placement course launched in 2023-24 that prepares students for calculus by studying polynomial, rational, exponential, logarithmic, trigonometric, and polar functions. The AP Exam assesses Units 1-3 only.

Average rate of change

The slope of the secant line over an interval: $\dfrac{f(b) - f(a)}{b - a}$. It measures how fast a function's output changes per unit of input.

How a graph bends. Concave up when the rate of change is increasing (like a cup); concave down when the rate of change is decreasing.

A sum of power functions with whole-number exponents: $f(x) = a_n x^n + \cdots + a_1 x + a_0$. Its degree is the highest exponent.

The highest exponent on the variable in a polynomial. It bounds the number of real zeros (at most the degree) and turning points (at most degree minus one).

How many times a factor repeats at a zero. Odd multiplicity makes the graph cross the x-axis; even multiplicity makes it touch and turn around.

What a graph does as $x \to \pm\infty$. For a polynomial it is set entirely by the leading term (degree and leading coefficient).

A ratio of two polynomials $\dfrac{p(x)}{q(x)}$ with $q(x) \neq 0$. Its asymptotes and holes come from comparing numerator and denominator.

A line $x = a$ the graph approaches but never reaches, occurring where the denominator is zero but the numerator is not.

A horizontal line the graph approaches as $x \to \pm\infty$, found by comparing the degrees of numerator and denominator.

A single missing point on a rational graph, created when a common factor cancels in both numerator and denominator.

A function $f(x) = a \cdot b^x$ with base $b > 0,\ b \neq 1$. Outputs are multiplied by $b$ for each unit increase in $x$.

The base $b$ of an exponential function. A factor greater than 1 means growth; between 0 and 1 means decay. The growth rate is $b - 1$ as a percent.

The inverse of an exponential: $\log_b y = x$ means $b^x = y$. It answers what power the base must be raised to in order to get $y$.

The logarithm base $e$: $\ln x = \log_e x$, where $e \approx 2.718$. It pairs with the natural exponential $e^x$.

A function $f^{-1}$ that undoes $f$: $f(f^{-1}(x)) = x$. Its graph is the reflection of $f$ across the line $y = x$.

An angle measure equal to the arc length on a unit circle. A full circle is $2\pi$ radians, equal to 360 degrees.

A circle of radius 1 centered at the origin. For angle $\theta$, the point on it is $(\cos\theta, \sin\theta)$ — cosine is the x-coordinate, sine the y-coordinate.

A sine or cosine wave $y = a\sin(b(x - c)) + d$, described by its amplitude $|a|$, period $\dfrac{2\pi}{|b|}$, phase shift $c$, and midline $y = d$.

Half the distance between the maximum and minimum of a sinusoid; it equals $|a|$ in $y = a\sin(b(x - c)) + d$.

The horizontal length of one full cycle of a periodic function. For $y = a\sin(bx)$ it is $\dfrac{2\pi}{|b|}$.

The horizontal line $y = d$ about which a sinusoid oscillates, halfway between its maximum and minimum.

The identity $\sin^2\theta + \cos^2\theta = 1$, true for every angle. It follows from the unit circle.

The acute angle between an angle's terminal ray and the x-axis. Trig values at any angle match those of its reference angle, up to a sign by quadrant.

FREE AP Precalculus Study Guide 2026 + Readiness Score

This free AP Precalculus study guide teaches to the course, launched in 2023-24 — every function family the exam tests, organized the way the course is built.^[1] The single most important thing to know up front: the AP Exam assesses Units 1, 2, and 3 only. Unit 4 is taught in the course but does not appear on the AP Precalculus Exam.^[3]

The three assessed units carry roughly 40% (Unit 1), 30% (Unit 2), and 30% (Unit 3) of the exam. This guide is interactive, not a wall of text: every unit has a built-in checkpoint quiz, hover-able glossary terms, worked examples, and concept questions, so you learn by doing.

Headed to calculus next? Everything here is the foundation for our AP Calculus AB study guide. Read this guide unit by unit, test yourself at each checkpoint, then round out your free AP Precalculus prep with our practice questions and flashcards.

AP Precalculus is one of the 17 AP exams — explore our AP study guides to compare and prep across the whole family.

AP Precalculus Exam Snapshot

AP Precalculus Exam at a glance (2026)
Detail	AP Precalculus Exam
Assessed content	Units 1-3 only (Unit 4 is taught but NOT tested)
Section I	40 multiple-choice questions, 2 hours, 62.5% of the score
Section II	4 free-response questions, 1 hour, 37.5% of the score
Calculator	Required on MCQ Part B (12 Q) and FRQ Part A (2 Q); not on the rest
Total time	About 3 hours
Unit weights	Unit 1 ~40%, Unit 2 ~30%, Unit 3 ~30%
Score scale	1-5 (3 or higher is generally passing for credit)
Launched	2023-24 school year
Publisher	College Board

How the AP Precalculus Exam is built (2026)

Section I · Multiple Choice

40 questions · 2 hours · 62.5% of the score.

Part A: 28 questions, no calculator
Part B: 12 questions, graphing calculator

Section II · Free Response

4 questions · 1 hour · 37.5% of the score.

Part A: 2 questions, graphing calculator
Part B: 2 questions, no calculator

Total time is about 3 hours. Every question draws from Units 1–3 only.

Because Units 1-3 are the only assessed content, spend your AP Exam prep there.^[3] Unit 1 (Polynomial & Rational) is the largest at about 40%, so the function features, zeros, and asymptotes you learn there pay off most:

AP Precalculus Exam weighting (2026 — assessed units)

Unit 1 · Polynomial & Rational40% · ~40% of the exam

Unit 2 · Exponential & Logarithmic30% · ~30% of the exam

Unit 3 · Trigonometric & Polar30% · ~30% of the exam

Unit 4 · Parameters, Vectors & Matrices0% · NOT assessed on the AP Exam

College Board reports unit weights as approximate ranges, so the exact mix shifts slightly each form.^[1] This guide teaches all four units — the three assessed units in depth, then Unit 4 as a clearly-flagged course topic that will not be on the AP Exam.

The four AP Precalculus units — and what the exam actually tests

The AP Precalculus Exam assesses Units 1–3 only. Unit 4 is part of the College Board course framework and is taught in class, but it does not appear on the AP Exam.

Unit 1 · Polynomial & Rational FunctionsAbout 40% of the exam. Rates of change, zeros, multiplicity, end behavior, asymptotes.
Unit 2 · Exponential & Logarithmic FunctionsAbout 30% of the exam. Growth and decay, logs as inverses, log rules, modeling.
Unit 3 · Trigonometric & Polar FunctionsAbout 30% of the exam. The unit circle, sinusoids, identities, and polar graphs.
Unit 4 · Functions Involving Parameters, Vectors & MatricesNot on the AP ExamTaught in the course, but NOT assessed on the AP Exam. Parametric, implicit, conics, vectors, and matrices.

Focus your AP Exam prep on Units 1–3 (roughly 40% / 30% / 30%). Learn Unit 4 for your course grade.

Unit 1 · Polynomial and Rational Functions

About 40% of the exam — the biggest unit. This unit is the language of function analysis: describing how a function changes, finding zeros and their behavior, and reading the asymptotes of rational functions.^[1]

Rates of Change & Function Features

The over an interval $[a, b]$ is $\dfrac{f(b) - f(a)}{b - a}$ — the slope of the secant line. When that rate is itself increasing, the graph is ; when it is decreasing, the graph is concave down. A point where concavity switches is a point of inflection.

Polynomials, Zeros & End Behavior

A of $n$ has at most $n$ real zeros and at most $n - 1$ turning points. The of each zero controls the graph there:

What multiplicity does at a zero
Multiplicity	Graph behavior at the zero
Odd (1, 3, 5, ...)	Crosses the x-axis (multiplicity 3+ flattens as it crosses)
Even (2, 4, ...)	Touches and turns around (bounces) — does not cross

is set by the leading term alone. Use the degree and the sign of the leading coefficient:

Polynomial end behavior — set by degree and leading coefficient

DegreePositive leading coefficientNegative leading coefficient

Even degreeBoth ends rise (↑ ↑)Both ends fall (↓ ↓)

Odd degreeFalls left, rises right (↓ ↑)Rises left, falls right (↑ ↓)

Only the leading term matters at the ends. Inside, the zeros and their multiplicities shape the curve.

Rational Functions & Asymptotes

A is a ratio $\dfrac{p(x)}{q(x)}$ . A sits where the denominator is zero but the numerator is not; if a factor cancels in both, you get a instead. The comes from comparing degrees:

Horizontal & slant asymptote rules
Compare degrees (numerator n, denominator m)	Asymptote
n < m	Horizontal asymptote y = 0
n = m	Horizontal asymptote y = ratio of leading coefficients
n = m + 1	Slant (oblique) asymptote — use polynomial long division
n > m + 1	No horizontal or slant asymptote

Transformations & Inverses

Transformations shift, stretch, or reflect any graph. Changes outside the function act on outputs (vertical): $f(x) + k$ shifts up, $a\,f(x)$ stretches vertically. Changes inside act on inputs (horizontal) and run in reverse: $f(x - h)$ shifts right by $h$ . An undoes the original and reflects across the line $y = x$ .

Checkpoint · Unit 1 · Polynomial & Rational

Question 1 of 10

A degree 4 polynomial with positive leading coefficient has exactly two real zeros, both simple. How many times does its graph touch but not cross the x-axis?

Unit 2 · Exponential and Logarithmic Functions

About 30% of the exam. This unit is about multiplicative change — growth and decay — and the logarithm, which is the tool that undoes an exponential.^[1]

Exponential Functions: Growth & Decay

An $f(x) = a \cdot b^x$ multiplies the output by the $b$ for every unit increase in $x$ . The defining feature: over equal-length input intervals, outputs change by equal ratios, not equal differences.

Logarithms & Their Properties

A is the inverse of an exponential: $\log_b y = x$ means exactly $b^x = y$ . Because they are inverses, their graphs reflect across $y = x$ :

Exponentials and logarithms are inverses

Exponential formb^x = yinput x → output y

Logarithmic formlog_b y = xinput y → output x

A logarithm undoes an exponential. Their graphs reflect across the line y = x, and you solve for a trapped exponent by taking a log of both sides.

Three properties let you condense or expand logarithms before solving:

Logarithm properties you should know cold
Rule	Identity
Product rule	$\log_b(MN) = \log_b M + \log_b N$
Quotient rule	$\log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N$
Power rule	$\log_b(M^k) = k\,\log_b M$
Change of base	$\log_b x = \frac{\ln x}{\ln b}$

Solving Equations & Modeling

To solve an exponential equation, take a of both sides and use the power rule to bring the exponent down. To solve a logarithmic equation, rewrite it in exponential form — then always check that log arguments stay positive.

Checkpoint · Unit 2 · Exponential & Logarithmic

Question 1 of 6

An arithmetic sequence corresponds to which type of function when the term number is the input?

Unit 3 · Trigonometric and Polar Functions

About 30% of the exam. This unit moves from right-triangle trig to the unit circle, sinusoidal models, identities, and a new coordinate system: polar.^[1]

The Unit Circle & Radians

A measures an angle by the arc length it cuts on a unit circle; a full circle is $2\pi$ radians = 360°. On the , the point for angle $\theta$ is $(\cos\theta, \sin\theta)$ :

The unit circle — (cos θ, sin θ)

x-coordinate
cos θ

y-coordinate
sin θ

Pythagorean
sin²θ + cos²θ = 1

A full circle is 2π radians = 360°. Reference angles let you read any angle from the first quadrant.

Sinusoids: Amplitude, Period & Midline

A $y = a\sin(b(x - c)) + d$ is described by four numbers: the $|a|$ , the $\dfrac{2\pi}{|b|}$ , the phase shift $c$ , and the $y = d$ .

Identities & Inverse Trig

The most-used identity is the $\sin^2\theta + \cos^2\theta = 1$ . Inverse trig functions have restricted ranges so each returns a single angle: $\sin^{-1}$ outputs angles in $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$ , $\cos^{-1}$ in $[0, \pi]$ .

Polar Coordinates & Graphs

give a point as $(r, \theta)$ — a distance and an angle. Convert with $x = r\cos\theta$ and $y = r\sin\theta$ . A $r = f(\theta)$ traces curves like circles, rose curves, and limaçons; where $r$ is increasing the curve moves away from the pole.

Checkpoint · Unit 3 · Trigonometric & Polar

Question 1 of 8

What is the range of the cosine function?

Unit 4 · Parameters, Vectors & Matrices (Not Assessed)

⚠️ Unit 4 is part of the College Board AP Precalculus course framework and is taught in class, but it is NOT assessed on the AP Precalculus Exam. Learn it for your course grade and as a bridge to later math — but you will not see it on test day.^[3]

Parametric, Implicit & Conic Functions

A parametric function gives $x = f(t)$ and $y = g(t)$ , tracing a curve as the parameter $t$ varies (with a direction, called orientation). Implicit relations like $x^2 + y^2 = 25$ define a curve without solving for $y$ , and the conic sections — circle, ellipse, parabola, hyperbola — each have a standard equation.

Vectors & Matrices

A $\langle a, b \rangle$ has magnitude $\sqrt{a^2 + b^2}$ and a direction; vectors add component-wise. A is a rectangular array used to model and transform data — multiplying a vector by a matrix can rotate, scale, or reflect it. Again, these are valuable for your course and future classes, but they are not on the AP Exam.

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside official College Board practice and our free tools. For the AP Exam, put your time into Units 1-3, weighting Unit 1 most (it is about 40% of the exam).

Learn Unit 4 separately for your course, knowing it will not be tested. Because AP Precalculus blends conceptual and procedural questions, mixed, spaced practice beats one long cram.

A study loop that actually works

1
Read an assessed unit here
Work through one unit at a time — Unit 1, then 2, then 3 (the only tested units).
2
Take the checkpoint
The quick check at the end of each assessed unit exposes what didn't stick.
3
Drill the gaps
Send your weak unit straight into the free practice questions and flashcards.
4
Take full, timed practice
Sit full practice sets — calculator and no-calculator — to build exam stamina, then review every miss.

AP Precalculus Concept Questions

Common AP Precalculus skills the exam actually measures — at least one per assessed unit (1-3). Tap any card for a short, exam-ready answer backed by an official source (College Board), then test yourself on them as flashcards.

AP Precalculus Glossary

Quick definitions for the terms you’ll see most across AP Precalculus:

Amplitude: Half the distance between the maximum and minimum of a sinusoid; it equals $|a|$ in $y = a\sin(b(x - c)) + d$ .
AP Precalculus: A College Board Advanced Placement course launched in 2023-24 that prepares students for calculus by studying polynomial, rational, exponential, logarithmic, trigonometric, and polar functions. The AP Exam assesses Units 1-3 only.
Average rate of change: The slope of the secant line over an interval: $\dfrac{f(b) - f(a)}{b - a}$ . It measures how fast a function's output changes per unit of input.
Concavity: How a graph bends. Concave up when the rate of change is increasing (like a cup); concave down when the rate of change is decreasing.
Degree: The highest exponent on the variable in a polynomial. It bounds the number of real zeros (at most the degree) and turning points (at most degree minus one).
End behavior: What a graph does as $x \to \pm\infty$ . For a polynomial it is set entirely by the leading term (degree and leading coefficient).
Exponential function: A function $f(x) = a \cdot b^x$ with base $b > 0,\ b \neq 1$ . Outputs are multiplied by $b$ for each unit increase in $x$ .
Free response question: Section II of the exam: 4 open-ended problems worth 37.5% of the score, requiring shown work and justification.
Growth factor: The base $b$ of an exponential function. A factor greater than 1 means growth; between 0 and 1 means decay. The growth rate is $b - 1$ as a percent.
Hole: A single missing point on a rational graph, created when a common factor cancels in both numerator and denominator.
Horizontal asymptote: A horizontal line the graph approaches as $x \to \pm\infty$ , found by comparing the degrees of numerator and denominator.
Inverse function: A function $f^{-1}$ that undoes $f$ : $f(f^{-1}(x)) = x$ . Its graph is the reflection of $f$ across the line $y = x$ .
Logarithm: The inverse of an exponential: $\log_b y = x$ means $b^x = y$ . It answers what power the base must be raised to in order to get $y$ .
Matrix: A rectangular array of numbers used to model and transform data. Matrices are part of Unit 4, which is taught but not assessed on the AP Exam.
Midline: The horizontal line $y = d$ about which a sinusoid oscillates, halfway between its maximum and minimum.
Multiplicity: How many times a factor repeats at a zero. Odd multiplicity makes the graph cross the x-axis; even multiplicity makes it touch and turn around.
Natural logarithm: The logarithm base $e$ : $\ln x = \log_e x$ , where $e \approx 2.718$ . It pairs with the natural exponential $e^x$ .
Period: The horizontal length of one full cycle of a periodic function. For $y = a\sin(bx)$ it is $\dfrac{2\pi}{|b|}$ .
Polar coordinates: A point given as $(r, \theta)$ : $r$ is the distance from the pole (origin) and $\theta$ is the angle from the polar axis.
Polar function: A relationship $r = f(\theta)$ giving radius as a function of angle, tracing curves such as circles, rose curves, and limaçons.
Polynomial function: A sum of power functions with whole-number exponents: $f(x) = a_n x^n + \cdots + a_1 x + a_0$ . Its degree is the highest exponent.
Pythagorean identity: The identity $\sin^2\theta + \cos^2\theta = 1$ , true for every angle. It follows from the unit circle.
Radian: An angle measure equal to the arc length on a unit circle. A full circle is $2\pi$ radians, equal to 360 degrees.
Rational function: A ratio of two polynomials $\dfrac{p(x)}{q(x)}$ with $q(x) \neq 0$ . Its asymptotes and holes come from comparing numerator and denominator.
Reference angle: The acute angle between an angle's terminal ray and the x-axis. Trig values at any angle match those of its reference angle, up to a sign by quadrant.
Sinusoid: A sine or cosine wave $y = a\sin(b(x - c)) + d$ , described by its amplitude $|a|$ , period $\dfrac{2\pi}{|b|}$ , phase shift $c$ , and midline $y = d$ .
Unit circle: A circle of radius 1 centered at the origin. For angle $\theta$ , the point on it is $(\cos\theta, \sin\theta)$ — cosine is the x-coordinate, sine the y-coordinate.
Vector: A quantity with magnitude and direction, written $\langle a, b \rangle$ . Vectors are part of Unit 4, which is taught but not assessed on the AP Exam.
Vertical asymptote: A line $x = a$ the graph approaches but never reaches, occurring where the denominator is zero but the numerator is not.

Free AP Precalculus Study Materials & Resources

Everything you need to prepare for AP Precalculus is free here — no paywall, no credit card. This guide is the foundation; pair it with the rest of our free AP Precalculus study materials for active recall, timed practice, and last-minute review:

AP Precalculus Practice Test — exam-style questions across the three assessed units, with explanations.
AP Precalculus Flashcards — active-recall decks for function types, identities, log rules, and the unit circle.
AP Calculus AB Study Guide — the next step, where these functions meet limits, derivatives, and integrals.

AP Precalculus Study Guide FAQ

The AP Precalculus Exam assesses only Units 1-3: Unit 1 Polynomial and Rational Functions (about 40%), Unit 2 Exponential and Logarithmic Functions (about 30%), and Unit 3 Trigonometric and Polar Functions (about 30%). Unit 4 is part of the course but is not assessed on the exam.

References

1.College Board. “AP Precalculus Course and Exam Description.” College Board. ↑
2.College Board. “AP Precalculus — Course Overview.” College Board. ↑
3.College Board. “AP Precalculus Exam — AP Students.” College Board. ↑
4.College Board. “About the AP Precalculus Course — AP Central.” College Board. ↑

Sources for the concept answers

Every answer in the AP Precalculus concept questions above is drawn from an official primary source:

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