For both the MCQ and FRQ sections. We will cover Units 1, 2, and 3. Unit 4 is not assessed on the exam, so it is not included.

How to Use This Guide

This is a review guide. That is, this guide will be of little to no use if you have not learned the content. It mainly serves to provide examples that can be used as a sort of “stem” for responses during the AP exam.

Note: This guide uses integer examples to explain concepts; however, these are simply examples, not the actual formula you would have to input. Please use these examples to learn how to apply concepts, not to copy them verbatim during the exam.

On the free-response section (FRQ), your answer could be 100% mathematically correct, yet still earn zero points for that part because the way the answer was presented was vague or not specific enough. In the multiple-choice section (MCQ), it is common to have multiple answer choices point to the same numerical answer, but each may have a distinct rationale for that answer. Occasionally, there may not even be a numerical answer. It is important to be able to determine which description is correct, and apply it during the exam.

Part 1: General Exam Knowledge and Advice

The exam is 2 hours and 55 minutes long.

Section I is Multiple Choice

42 questions, 1 hour 45 minutes, 62.5% of score

Section II is Free Response

4 questions, 1 hour 10 minutes, 37.5% of score

Each FRQ is worth 6 points, so the entire FRQ section is 24 points.

The exam is a hybrid digital exam: the questions will appear in Bluebook, but students will write their answers (for the FRQ section) in a paper booklet.

You may use a pencil OR a (black or dark blue) pen on the FRQ section. It is a myth that only a pen can be used. If a pencil is used, simply make sure that all your writing is clear, legible, and darkly marked enough so that it can be scanned.

Task Verbs

Nearly every AP exam has “types” of questions. For the AP Precalculus exam, at the very least, the prompts will tell you exactly how each question should be answered. A list of task verbs is shown below:

$\begin{array}{lll}\text{Task Verb}& \text{What it means}& \text{How much to write}\\ \text{Determine, Find, Identify}& \begin{array}{l}\text{Use an appropriate}\\ \text{method/process to answer}\\ \text{the question.}\end{array}& \begin{array}{l}\text{Show set-up/steps and}\\ \text{circle/box the final answer.}\\ \text{Make sure the final answer is}\\ \text{legible.}\\ \text{For Identify, you can usually}\\ \text{just state the answer (if you}\\ \text{have to write English, it}\\ \text{should typically be a}\\ \text{sentence max), but if the}\\ \text{method isn’t obvious, show it}\\ \text{anyway.}\\ \text{Tip: Oftentimes, AP readers}\\ \text{will just check your answer}\\ \text{and see if you have work.}\\ \text{This is not always true, but}\\ \text{you should take it upon}\\ \text{yourself to make sure your}\\ \text{answer, at the very least, is}\\ \text{clear.}\end{array}\\ \text{Solve}& \begin{array}{l}\text{Use an appropriate method to}\\ \text{find the solution(s) to an}\\ \text{equation or inequality, while}\\ \text{showing your method (steps)}\\ \text{in reaching that solution(s).}\end{array}& \begin{array}{l}\text{Show each step that leads to}\\ \text{your answer, then circle/box}\\ \text{all solutions. Ensure your}\\ \text{solution falls within the given}\\ \text{interval or domain.}\end{array}\\ \text{Construct, Write}& \begin{array}{l}\text{Create a function,}\\ \text{expression, equation, or}\\ \text{model that is consistent with}\\ \text{the given scenario or data.}\end{array}& \begin{array}{l}\text{Simply state the}\\ \text{mathematical model (you will}\\ \text{usually not have to show}\\ \text{work). Note that you may}\\ \text{occasionally have to define}\\ \text{your own variables.}\end{array}\\ \text{Rewrite}& \begin{array}{l}\text{Using appropriate methods,}\\ \text{find an equivalent analytical}\\ \text{form of an expression.}\end{array}& \begin{array}{l}\text{Show the steps that lead up}\\ \text{to your final solution and give}\\ \text{your final expression in the}\\ \text{requested form.}\end{array}\\ \text{Express, Indicate}& \begin{array}{l}\text{Give your answer in the}\\ \text{requested form, and (if}\\ \text{applicable) include units in}\\ \text{the answer.}\end{array}& \begin{array}{l}\text{State your answer in the form}\\ \text{that was requested. Include}\\ \text{units if applicable.}\end{array}\\ \text{Estimate, Compare}& \begin{array}{l}\text{Use a function representation}\\ \text{(i.e., a graph, table, or}\\ \text{equation) to find approximate}\\ \text{values and/or compare}\\ \text{results.}\end{array}& \begin{array}{l}\text{State your estimation or}\\ \text{comparison, along with the}\\ \text{values you used to reach}\\ \text{your answer (i.e.,}\\ f\left(2\right)=5,g\left(6\right)=7\to g\left(6\right)>f\left(2\right)\end{array}\\ \text{Describe}& \begin{array}{l}\text{Give a verbal description}\\ \text{consistent with the scenario,}\\ \text{data set, or function.}\end{array}& \begin{array}{l}\text{One to two sentences that}\\ \text{accurately answer the}\\ \text{question. Make sure to use}\\ \text{your vocabulary (i.e.,}\\ \text{concavity)}\end{array}\\ \text{Interpret}& \begin{array}{l}\text{Describe the connection}\\ \text{between a mathematical}\\ \text{result and its real-world}\\ \text{meaning/context. Include}\\ \text{units if applicable.}\end{array}& \begin{array}{l}\text{Typically one sentence that}\\ \text{describes a relationship}\\ \text{between the answer and}\\ \text{some context (usually}\\ \text{real-world context).}\end{array}\\ \begin{array}{l}\text{Explain, Justify, Give a}\\ \text{reason, Provide a rationale}\end{array}& \begin{array}{l}\text{Uses information from the}\\ \text{given scenario (or function) to}\\ \text{provide reasoning that}\\ \text{supports a solution or}\\ \text{conclusion.}\end{array}& \begin{array}{l}\text{A claim supported by}\\ \text{calculations or evidence. The}\\ \text{explanation alone (not}\\ \text{including steps) should be}\\ \text{around 1-2 sentences.}\end{array}\\ \begin{array}{l}\text{Plot and label, Sketch and}\\ \text{label}\end{array}& \begin{array}{l}\text{Draw a graph that’s}\\ \text{consistent with the given}\\ \text{scenario, data set, or criteria}\end{array}& \begin{array}{l}\text{Sketch a clean graph with}\\ \text{key features labeled}\\ \text{(intercepts, extreme,}\\ \text{asymptotes, midlines, points}\\ \text{used to create the graph,}\\ \text{etc.)}\end{array}\end{array}$

Universal Justification Format

Every justification on the AP Precalculus exam can be answered with the same C.E.R. format that’s so commonly used in other classes.

Be Specific

After writing a justification, ask yourself if your response uniquely answers the problem given. This piece of advice is more to be safe.

The third response names specific numbers and functions from the actual problem. On the AP exam, you would change it based on YOUR problem.

Remember, this is mainly out of good practice. However, based on the context of the problem, it is possible to get points as long as your answer is clear enough.

Standard rules

Part 2: Common Justifications

Rate of Change and the Four Model Families

Rate of change is arguably one of the most tested concepts in AP Precalculus. Specifically, when deciding which function best models the data, knowing how to calculate the average rate of change (and its average rate of change) is essential. Typically, you are given a table and asked which kind of function fits.

However, before you begin calculating anything, it is important to ensure that your inputs are over consecutive, equal-length, input-value intervals. The reason that phrase matters is because the reasoning that "the outputs go up by 3 each time" is meaningless unless the inputs are spaced evenly. Comparing outputs at $x=1,2,3$ is a correct comparison, but comparing outputs at $x=1,2,17$ is not.

$\begin{array}{lll}\text{Model}& \text{Change in outputs}& \text{What must be constant}\\ \text{Linear}& \begin{array}{l}\text{Outputs change by the same}\\ \text{amount between inputs.}\end{array}& \begin{array}{l}\text{The first differences:}\\ f\left(x+1\right)-f\left(x\right)\end{array}\\ \text{Quadratic}& \begin{array}{l}\text{Outputs change by different}\\ \text{amounts, but those changes}\\ \text{themselves change by a}\\ \text{constant.}\end{array}& \begin{array}{l}\text{The second differences (the}\\ \text{differences of the differences).}\\ \text{This is basically the differences}\\ \text{of}\\ f\left(x+1\right)-f\left(x\right)\text{ (the constant must be}\\ \text{non-zero).}\end{array}\\ \text{Exponential}& \begin{array}{l}\text{Outputs are multiplied by the}\\ \text{same factor each input.}\end{array}& \begin{array}{l}\text{The ratio}\\ \frac{f\left(x+1\right)}{f\left(x\right)}\end{array}\\ \text{Logarithmic}& \begin{array}{l}\text{Outputs increase (or decrease)}\\ \text{but by smaller and smaller}\\ \text{amounts. It is the same as the}\\ \text{exponential relationship but}\\ \text{with the roles input and output}\\ \text{swapped.}\end{array}& \begin{array}{l}\text{Over equal-length output}\\ \text{intervals, the inputs are}\\ \text{proportional, meaning that the}\\ \text{ratio of the inverses is constant}\\ \text{over equal output increments.}\end{array}\end{array}$

Sentence stems

When answering these questions on FRQs, it is imperative that you state the domain in which you are measuring the differences. For example, if you’re finding the constant differences between $x=0$ to $x=5$, then you must include “over the interval $x=0$ to $x=5$” in your response. This is not optional. On the exam, you must include the interval.

Increasing and Decreasing

Occasionally, you will require the formal definition of increasing and decreasing. A function $f\left(x\right)$ is increasing on an interval, if $x_1<x_2<x_3$ , then $f\left(x_1\right)<f\left(x_2\right)<f\left(x_3\right)$ and so on for the entire interval. For decreasing, it is simply the inequality flipped. In layman’s terms, it simply means whether the function values rise or fall as $x$ increases.

On another note, it does not matter whether you label an interval where a function is increasing with closed or open brackets/parentheses, provided any included endpoints are in the domain and the function is increasing on the resulting interval. That is, it does not matter whether you say $\left(a,b\right)$ or $\left[a,b\right]$ when trying to provide an answer to the interval over which a function is increasing.

Concavity

Concavity refers to whether the rate of change is increasing or decreasing.

There is a second way to attribute concavity: if the average rate of change over consecutive equal-length input intervals is increasing, the graph is concave up; if it is decreasing, the graph is concave down.

A point of inflection is where concavity switches from concave up to concave down or vice versa. In other words, it is when the rate of change goes from increasing to decreasing (or vice versa)

The four combinations

A function can be increasing or decreasing, and concave up or concave down.

$\begin{array}{lll}\text{Combination}& \text{Rate of change}& \text{Graph}\\ \text{Increasing, concave up}& \text{Positive and getting larger}& \text{Rising and getting steeper}\\ \text{Increasing, concave down}& \text{Positive and getting smaller}& \text{Rising but flattening out}\\ \text{Decreasing, concave up}& \begin{array}{l}\text{Negative but getting larger}\\ \text{(closer to zero)}\end{array}& \begin{array}{l}\text{Falling but leveling off (hook}\\ \text{shape)}\end{array}\\ \text{Decreasing, concave down}& \begin{array}{l}\text{Negative and getting smaller}\\ \text{(more negative)}\end{array}& \begin{array}{l}\text{Falling and steepening (cliff}\\ \text{shape)}\end{array}\end{array}$

Average Rate of Change and Secant Lines

The average rate of change (AROC) of a function $f$ over an interval $x=a$ to $x=b$ is given by $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. It is the slope of the secant line through the points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$.

Estimations with AROC

On rare occasions, you may be asked to find the IROC (instantaneous slope) at a point. In these cases, you must estimate by using the AROC of two nearby points. For example, if they were to ask for the slope at $x=7$, you could find the AROC of $x=6$ and $x=8$.

More commonly, you will be asked to use the average rate of change over $\left[a,b\right]$ to estimate a function’s value at some point. You may then be asked whether your estimation is greater or lower than the true value.

Created using desmos.com

In order to receive the points for correct reasoning, you must

A response that earns full points could look like this: “Because the graph of $f$ is concave down on the interval from $x=0$ to $x=8$, the secant line through $\left(0,f\left(0\right)\right)$ and $\left(8,f\left(8\right)\right)$ will lie below the graph of $f$. Thus the estimate using the secant line is an underestimate.”

End Behavior/Limit Notation

The end behavior of a function describes the output values as the input values approach $\infty$ or $-\infty$. There are four conditions that compose proper limit notation:

Put together, it should look something like $\underset{x\to \infty }{\lim }f\left(x\right)=0$ or $\underset{x\to 4}{\lim }g\left(x\right)=-\infty$.

One Sided Limits

Sometimes, the limiting behavior for a function is different as both sides of the function approach a value differently. In this case, we can use one-sided limits! They are similar to regular limits except they have a small $+$ or $-$ next to the value the function is approaching, like so: $\underset{x\to a^{+}}{\lim }f\left(x\right)=0$ $\underset{x\to a^{-}}{\lim }f\left(x\right)=0$

For $\underset{x\to a^{+}}{\lim }f\left(x\right)$, see how there is a small $+$ sign? That indicates that the limit is showing the value that $f\left(x\right)$ is approaching from the right. Similarly, $\underset{x\to a^{-}}{\lim }f\left(x\right)$ is indicating that the limit is for when the function approaches $a$ from the left.

Polynomials/Rational functions

For polynomials, you can determine end behavior just through the leading term. If the lead coefficient is negative, the limit as $x\to \infty$ will equal $-\infty$. If the degree is even, then the function has the same end behavior in the $\infty$ and $-\infty$ directions. If the degree is odd, then the function has opposite end behavior in the $\infty$ and $-\infty$ directions.

As for rational functions, you must compare the degrees of the numerator to the denominator.

Zeros and Multiplicity

Say we have a polynomial $p$ given by $p\left(x\right)=\left(x-a\right){\left(x-b\right)}^2{\left(x-c\right)}^3$, where $a$, $b$, and $c$ are all random distinct integers.

Looking at the graph below, we can see that at $a$ and $c$, the graph of $p$ will pass through the $x$-axis. However, $p$ at $x=b$ will “bounce” off.

When you have an odd multiplicity, the graph will cross the $x$-axis, and the function will change signs as you pass through that zero.

When you have an even multiplicity, the graph will touch the $x$-axis without crossing, and the function will not change signs as you pass through that zero.

To justify some answer using this, you can simply bring up the multiplicity of zero, state the definition (that if the multiplicity of the zero is ____, the graph will _____ the $x$-axis), then state your conclusion.

Inverse functions

A function has an inverse only if every output correlates to exactly one input. (aka the horizontal line test, however, this is not a valid explanation for the exam).

Remember that when a function has an inverse, the domain and range will be swapped; the domain of $f\left(x\right)$ will become the range of its inverse, $g\left(x\right)$, and the range of $f\left(x\right)$ will become the domain of $g\left(x\right)$. Due to this, a function may not be fully invertible due to it being constrained by domain and range properties.

To justify that a function is not invertible, you must name the specific two inputs that share an output. For example: “Because $f\left(6\right)=f\left(7\right)=67$, the function $f$ is not invertible, as there are two input values that produce the same output value. The resulting inverse would no longer qualify as a function.”

To justify that a function is invertible, you can simply state, “The function $f$ is invertible because each output is unique to one input.”

Residuals

A residual is the actual value of a function minus the predicted value. A residual graph shows the relationship between that difference and the input. If

Examples of proper responses could be: “The residual plot appears randomly distributed and there is no clear pattern. Thus, an exponential model is appropriate for the data.” “The residual plot depicts a clear parabolic pattern, so a linear model is not appropriate.”

MCQ

When the MCQ is clearly asking for reasoning or some sort of explanation (rather than a numerical answer), you should try to predict the answer before you read the answer choices.

Identify the Model from a Table

Compute the differences. If it is constant, then it's linear. If the second differences are constant, it is quadratic. Third, cubic; fourth, quartic, etc. (you usually won't get that far). If the differences didn’t work, check if the ratios are constant; if they are, then it should be an exponential. Remember that for this to work your inputs have to be equally spaced apart.

Concavity/AROC from a Graph/Table

Finding the AROC from a table is self-explanatory. You would take the change in $y$-values and divide them by the change in $x$-values $\frac{\Delta Y}{\Delta X}$ (like how you would take a slope from a table). However, for concavity with equally spaced inputs, you must first find the first differences and check whether they are increasing or decreasing. If they are increasing, then the data must resemble a function that is concave up. If decreasing, then concave down.

Typical Trap Answer Patterns

The typical trap answer deviates from the real answer in a couple of ways:

Simply make sure to note these trap justifications/answers.

FRQ

There are four types of FRQs that will show up in a predictable order.

$\begin{array}{llll}\text{FRQ \#}& \text{Topic}& \text{Calculator}& \begin{array}{l}\text{Difficulty (National}\\ \text{Mean Score out of 6}\\ \text{in 2025)}\end{array}\\ 1& \text{Function Concepts}& \text{Yes}& 3.51\\ 2& \begin{array}{l}\text{Modeling a}\\ \text{Non-Periodic Context}\end{array}& \text{Yes}& 2.16\\ 3& \begin{array}{l}\text{Modeling a Periodic}\\ \text{Context}\end{array}& \text{No}& 3.13\\ 4& \begin{array}{l}\text{Symbolic}\\ \text{Manipulations}\end{array}& \text{No}& 1.93\end{array}$

FRQ 1 (Function Concepts)

For more FRQ practice, look at FiveHive FRQ Guides and AP Precalculus Past FRQs -- CollegeBoard

This question permits a calculator and is worth 6 points.

Generally, it will look something like this:

Part A:

Evaluate a composition (for example, $p\left(x\right)=g\left(f\left(x\right)\right)$ at some $x$) Find input value(s) that yield some output value

Part B:

Solve an equation (can use calculator) State the end behavior (remember to use limit notation)

Part C:

Typically a justification question. It is typically, “does this function have an inverse and why”, or “which type of model best models the data and why.”

Because this is a justification guide, we will only look at part C.

Let's look at the first example

If the function is not invertible, then you would say something like “Because $f\left(-5\right)=f\left(5\right)=7$, the function $f$ is not invertible, as there are two different input values that correspond to the same output value.” If the function is invertible, then you could say something like “Because $f$ is increasing/decreasing over its entire domain, there will only be one input for every output value. Thus, $f$ has an inverse function.”

In the second example, you could say, for example, “Because the output values change by a constant factor of $0.5$ over consecutive, equal-length, input-value intervals from $x=0$ to $x=5$, an exponential model is appropriate.”

FRQ 2 (Modeling a Non-Periodic Context)

This question permits a calculator and is worth 6 points. By far, FRQ 2 is the most unpredictable. It is unlike FRQ 3.

But generally, it will look something like this:

Part A:

Embedded in some story, you are given some data points. You have to set up some equations and solve for unknown constants (or use a regression).

Part B:

i. Compute the AROC between a specific time interval ii. Interpret the solution from the b(i) within the context of the question

Part C:

i. Write an equation that is appropriate for a predicted value given by the question ii. Explain any domain restrictions with your new equation in the problem’s context.

The first justification question is part B(iii). Recall, you must explain the concavity and make an explicit reference to the secant line. Review the “Estimations with AROC” section.

The second justification question is Part C. Limitations of the model basically refer to the fact that at some point, the mathematical model may not make sense in the real world (for example, if volume ever became negative) An example could be, "According to the model, after $t=c$, the [output variable] becomes negative. Because it is impossible to have a negative amount of [output variable], this is a limitation of the model."

End behavior could be like if a model approaches infinity as the input approaches infinity (or even an asymptote). An example is shown below: "According to the model, as $t$ increases forever, [output variable] increases without bound. However, this is not possible in reality, as [output variable] cannot grow indefinitely, and thus is a limitation of the model." However, it could also simply be asking you to justify the meaning of an end behavior. For example, "As $t$ approaches infinity, the model predicts that [output variable] will approach $c$ [where c is any real number]. That means that over a long period of time, the [dependent variable] will stabilize at $c$" An example of [dependent variable] here could be temperature.

Reasonable domain is similar to the limitations. However, one especially well-known example is a flat paper that has squares cut from its corners, and the question is what the maximum volume of the newly formed box is. In these cases, the length of the square obviously cannot be $0$, nor can it be longer than half the length of the entire box. And of course, it cannot be negative. But a general structure for this type of response could be something like, "A reasonable domain for this model is $0\le t\le c$ [where c is a real number]. The domain begins at $t=0$ because [observation/context], and it ends at $t=c$ because [observation/context]."

An important note, though, is that quite often the reasonable domain is simply whatever domain was given in the data. That is, a model is only accurate in the domain of the data. If the data only spans from $x=0$ to $x=5$, then the model is only accurate to $x=0$ to $x=5$.

FRQ 3 (Modeling a Periodic Context)

This question does not permit a calculator and is worth 6 points. This is by far the most predictable FRQ. It is always a sinusoidal graph and always asks the same question (you can see that the National Mean is significantly higher in FRQ 3 than 2 or 4).

Part A:

You are given a sinusoidal graph with five labeled points ($F$, $G$, $J$, $K$, and $P$). You are also given some real-world context that secretly tells you facts about the sinusoidal, which may include the period, a max/min, the amplitude, the midline, etc. You must give possible coordinates for these points

Part B:

You have to write the sinusoid as an equation in the form $f\left(x\right)=a\sin \left(b\left(x+c\right)\right)+d$.

Part C:

You have to answer some questions regarding the function’s behavior (it is typically a multiple choice question where you are instructed to figure out the correct statement according to the given graph.) You have to describe the concavity (if the graph’s AROC is increasing/decreasing) and whether the rate of change is increasing or decreasing. The question may also ask about the sign of the function values. (ex. The graph of $f\left(x\right)$ is positive/negative). Make sure to clarify what the question is about – the actual graph or the rate of change – because this can alter your answer by quite a bit.

Regarding the behaviors, you must simply state the behavior in question, describe the definition, then explain whether the given sinusoidal function (on that interval) fulfills that definition or not.

FRQ 4 (Symbolic Manipulations)

This question does not permit a calculator and is worth 6 points. This FRQ is for students to demonstrate how well they can manipulate functions to get to a certain answer, and a lot of that is based on algebra. Due to that fact, FRQ #4 does not require any justification, however supporting work is required for all portions.

If you would like a guide on this FRQ, please check the FRQ #4 Guide also on our website.

Practice Section

(Calculator Permitted for All Questions)