Introduction

Are you curious about the new AP® Precalculus course and what it has to offer? Then you’re in the right spot. Firstly, in this AP® Precalculus overview, we’ll explore the curriculum of the course, providing insights into its comprehensive structure. Additionally, we’ll delve into some essential prerequisites to ensure you’re fully prepared. Furthermore, we’ll discuss the format of the exam, offering crucial information for both students and teachers. So, let’s dive in and discover how AP® Precalculus is shaping the future of high school mathematics

Here’s What You Should Know Going into AP® Precalculus

First, when entering AP® Precalculus, it’s important for you to have a strong foundation in several key areas. Here’s what you should be comfortable with:

Algebraic Concepts:

Initially, you should be proficient in algebraic manipulations, such as solving equations and inequalities, simplifying expressions, and understanding functions. Spend some time working with linear, quadratic, and polynomial equations. Factoring quadratic trinomials and using the quadratic formula are also necessary prerequisites.

Functions:

A clear understanding of the concept of a function is crucial. Be familiar with different types of functions, how to graph them, and interpret their graphs. This includes linear, quadratic, basic polynomial functions, and piecewise-defined functions. Know how to work with a function represented graphically, numerically, analytically, and verbally.

Graphing Skills:

Furthermore, be comfortable with graphing functions and interpreting graphs. Understand key features of graphical concepts like slope, intercepts, and asymptotes.

Trigonometry Basics:

While a deep understanding of trigonometry is not expected, familiarity with the basics, such as the sine, cosine, and tangent functions, and the ability to work with right triangles, will be beneficial.

Systems of Equations and Inequalities:

Also, you should know how to solve systems of equations and inequalities, both algebraically and graphically.

Exponents and Radicals:

Understanding the properties of exponents and radicals, and being able to manipulate expressions involving them, is important.

Basic Geometry Knowledge:

Basic knowledge of geometric shapes, their properties, and the Pythagorean theorem will be useful.

Complex Numbers:

Familiarity with complex numbers and performing arithmetic operations with them will give you a leg up before taking this course.

Problem-Solving Skills:

Moreover, you should be able to approach and solve problems logically and be comfortable with mathematical reasoning and critical thinking.

Mathematical Communication: Lastly, the ability to clearly express mathematical ideas in writing and verbally will be an asset in this course.

Is AP® Precalculus Hard?

AP® Precalculus is a challenging but rewarding course that requires dedication and consistent practice. Primarily, It builds upon your existing knowledge and prepares you for more advanced topics in mathematics, including calculus. Whether AP® Precalculus is hard to you depends on a few factors, such as your background in mathematics, your study habits, and how comfortable you are with abstract concepts. Here’s a breakdown to help you gauge what to expect:

Mathematical Foundation:

Initially, AP® Precalculus builds on Algebra II and introduces more advanced concepts. You’re off to a good start if you have a strong grasp of algebra, functions, and basic trigonometry.

New Concepts:

Also, you’ll be introduced to new topics like advanced functions, trigonometry, and the beginnings of calculus. Expect to encounter concepts that are more abstract and complex than in previous math classes.

Problem-Solving and Analysis:

The course emphasizes analytical thinking and problem-solving. You’ll be required to understand and apply mathematical concepts, not just memorize formulas.

Workload and Pace:

AP® courses typically have a faster pace and a heavier workload compared to regular high school courses. Be prepared for regular homework, projects, and tests.

Preparation for AP® Exam:

A significant part of the course will be geared towards preparing for the AP® exam, which includes understanding the exam format and practicing AP-style questions.

Study Habits:

To add on, your success in AP® Precalculus will also depend on your study habits. Regular study, keeping up with homework, and seeking help when needed are key.

Interest and Attitude:

Finally, your interest in mathematics and your attitude towards learning new and challenging material will greatly influence how hard you find the course.

In summary, AP® Precalculus can be challenging, but it’s definitely manageable with the right preparation and mindset. If you enjoy math, are willing to put in the effort, and have done well in your previous math courses, you’ll likely find it a rewarding and enriching experience. Make sure to come back to this AP® precalculus overview to refresh your understanding throughout the year!

AP® Precalculus Curriculum 2025

Above all, let’s delve into the heart of the AP® Precalculus course: the curriculum. This AP® Precalculus overview would be incomplete without giving you an inside look at the topics covered. Below is a detailed table that outlines the key topics and concepts covered in 2025. This comprehensive overview will give you a clear picture of what to expect and the areas of focus throughout the course.

Mathematical Practices

Firstly, the Mathematical Practices. There are three mathematical practices that students should focus on and build on as they make their way through the course. The table below provides details about these and their approximate weight that they are assessed on the exam:

Mathematical Practice	Description	Key Words	Percentage
Practice 1: Procedural and Symbolic Fluency	Algebraically manipulate functions, equations, and expressions	Solve, simplify, evaluate, manipulate, identify	39-48%
Practice 2: Multiple Representations	Translate mathematical information between representations	Graph, transform,  interpret, factor	20-27%
Practice 3: Communication and Reasoning	Communicate with precise language, and provide rationales for conclusions	Explain, justify, describe, support	32-39%

Course Content

Secondly, on to the main event: the Course Content. The AP® Precalculus curriculum is broken up into four units. The first three units, along with their exam weighting are shown in the table below. These units are included in the exam. Meanwhile, the fourth unit is NOT tested on the exam and is only included for teachers based on state or local standard requirements:

Course Content Table

Unit	Topics	Resources
Unit 1: Polynomial and Rational Functions (30-40%)	Understanding function relationships Analyzing rates of change Polynomial functions and their characteristics Rational functions and their characteristics Function transformations Constructing function models for applications	Practice on Albert: Unit 1 | Polynomial and Rational Functions External Resources: End Behavior of Polynomial Transformations of Functions Rational Functions
Unit 2: Exponential and Logarithmic Functions (27-40%)	Arithmetic and geometric sequences and their connections to linear and exponential functions Exponential functions and their properties and applications Logarithmic functions and their properties and applications Function composition and inverse functions Solving exponential and logarithmic equations and inequalities	Practice on Albert: Unit 2 | Exponential and Logarithmic Functions External Resources: Exponential Functions Composite and Inverse Functions Properties of Logarithms
Unit 3: Trigonometric and Polar Functions (30-35%)	Modeling periodic phenomena Graphing trigonometric functions and their transformations Understanding key features of all trigonometric and inverse trigonometric functions Solving trigonometric equations and inequalities Applications of sinusoidal functions Polar coordinates and graphs of polar functions	Practice on Albert: Unit 3 | Trigonometric and Polar Functions External Resources: Graphs of Trig Functions Polar Coordinates Inverse Trigonometric Functions
Unit 4:  Functions Involving Parameters, Vectors, and Matrices (Not assessed on the AP® Exam)	Parametric functions Using parametric functions to model planar motion Implicitly defined functions and conic sections Vectors and matrices Applications of vectors and matrices	Practice on Albert: Unit 4 | Functions Involving Parameters, Vectors, and Matrices External Resources: Parametric Functions Conic Sections Matrices and Matrix Operations

Format of the AP® Precalculus Exam

Furthermore, understanding the format of the AP® Precalculus Exam is crucial for effective preparation and success. In summary, we’ll break down the components of the exam, including the types of questions and the allocation of time, to give you a clear roadmap for what to expect on test day.

In brief, the AP® Precalculus exam has a total of 44 questions over a testing period of 3 hours. It is broken into multiple-choice and free-response questions, each with calculator and no calculator parts.

Section	Question Type	Number of Questions	Exam Weighting	Timing
I	Multiple-choice questions
	Part A: No calculator	28	43.75%	80 minutes
	Part B: Graphing calculator required	12	18.75%	40 minutes
II	Free-response questions
	Part A: Graphing calculator required	2	18.75%	30 minutes
	Part B: No calculator	2	18.75%	30 minutes

Section I Multiple-Choice

Section I of the AP® Precalculus test consists of 40 multiple-choice questions split into a non-calculator portion (Part A) and a portion where a graphing calculator may be required (Part B). Each question has four possible answer choices (A, B, C, or D). This section is 2 hours long and counts for 62.5% of your final score.

Section II Free-Response

Section II of the ​​AP® Precalculus exam has 4 free-response questions. The first two questions make up Part A, during which a calculator may be required. In contrast, the remaining 2 questions will be completed in Part B without a calculator. Additionally, two questions will incorporate a real-world context or scenario. Furthermore, the other two questions deal with function concepts and algebraically manipulating all types of functions presented in Units 1-3. This section is one hour long and counts for 37.5% of your final score.

What do the AP® Precalculus questions look like?

Curious about the types of questions you’ll encounter on the AP® Precalculus Exam? This section of the AP® precalculus overview offers a sneak peek into the style and nature of questions you can expect, providing insights into the blend of conceptual understanding and problem-solving skills you’ll need to demonstrate.

So, here are some released questions from the Collegeboard website. The multiple-choice questions come from the 2023 AP® Precalculus Practice Exam, and the free-response questions come from the 2023 AP® Precalculus Course and Exam Description.

Section I, Part A (No Calculator)

Source: 2023 AP® Precalculus Practice Exam

This question asks us to apply properties of logarithms to simplify the expression 2\ln x - 3\ln y . Firstly, we should apply the power rule for logarithms to get: ​​

\ln(x^2) - \ln(y^3)

Then, we can apply the quotient rule for logarithms to get:

​ \ln(\dfrac{x^2}{y^3})

So, this gives us the answer A.

Section I, Part B (Calculator Required)

Source: 2023 AP® Precalculus Practice Exam

Firstly, we should graph this function on a graphing calculator to get a picture of the graph. Using Desmos.com, we produced the following image:

Indeed, this image tells us that the correct answer is ‘A’. The maximum height of the tide is 13.8 feet. However, we can also confirm the answer algebraically. We can see that the maximum occurs at  t=0 and  t=12. If we substitute either value for t into the function, we would get: 

h(0) = 6.3\cos(\frac{\pi}{6}0)+7.5

Then, evaluate the cosine:

h(0) = 6.3\cdot 1+7.5=13.8 

So, the maximum is 13.8 feet, which matches answer choice ‘A.’

Click here for more practice with sinusoidal function modeling!

Section II, Part A (Calculator Required)

Source: 2023 AP® Precalculus Course and Exam Description

(A) (i)

Initially, from the information provided, we can see that two data points are given. Those points are ​​ (0,75) \text{ and } (3,70.84) where the points are given as ​​ (t, R(t)). Substituting these points into the equation independently gives us our two equations: ​​

75=a+b\ln(0+1) ​​

Then, the other equation comes from the other value:

70.84=a+b\ln(3+1)

(A) (ii)

Firstly, we can solve the first equation for a: 

75=a+b\ln(0+1)  

Secondly, evaluate the logarithm:

​75=a  

Then, we can use this value to solve the second equation for b: ​​

70.84=75+b\ln(3+1) ​​

Futhermore, simplify the logarithm:

-4.16=b\ln(4)  

Then, solve for b:

​​b=\frac{-4.16}{\ln(4)} \approx -3.001  

(B) (i)

Now, using the information ​​ R(0)=75 \text{ and } R(3)=70.84 , we can calculate the average rate of change as:  ​​

\text{average rate of change} =\dfrac{y_2-y_1}{x_2-x_1}  

Secondly, plug in our values:

= \dfrac{70.84-75}{3-0}  

Then, simplify the numerator and the denominator:

= \dfrac{-4.16}{3}  

Finally, divide the fraction to get our average rate of change:

\approx -1.387 

(B) (ii)

So, for the first three months of the study, the group’s average score decreased by 1.387 points per month.

(B) (iii)

This function model will be a decreasing logarithmic function because of the negative value for b. Also, we know that this means the graph will be concave up. Therefore, rates of change will be increasing in size, so the average rate of change from ​​ t=3 \text{ to } t=p will be greater than the average rate of change found in (i).

(C)

Finally, we need to evaluate the function at different values to see how long it satisfies the following inequality. ​​ |R(b)-R(a)| > 1, where ​​b \text{ and } a are multiples of 12. This will tell us if the model is still predicting the score to be decreasing by at least 1 point each year. At four years, we reach the following:  ​​ |R(48)-R(36)| =0.843< 1. Therefore, it is no longer an appropriate model after three years.

Section II, Part B (No Calculator)

Source: 2023 AP® Precalculus Course and Exam Description

(A) (i)

Firstly, we need to use the properties of logarithms to simplify ​​the function. Applying the power property of logarithms gives us:

g(x)=\ln(x^3)-\ln(\sqrt{x})

Then, applying the quotient rule of logarithms gives us:

g(x)=\ln(\dfrac{x^3}{\sqrt{x}})

(A) (ii)

Subsequently, we’ll need to use a trigonometric identity called the Pythagorean Identity. This identity tells us that ​​ \sin^2(x)+\cos^2(x)=1 . Thus, we can rearrange that equation to match the numerator of our given function: 

​​ \cos^2(x)=1 -\sin^2(x) ​​

Or, we can multiply by -1 to get the following:

-\cos^2(x)=\sin^2(x)-1

So, we can make the substitution into the numerator and simplify from there:

​​ h(x)=\dfrac{\sin^2(x)-1}{\cos(x)}

Then, you can make another substitution:

​​ h(x)=\dfrac{-\cos^2(x)}{\cos(x)}

Finally, simplify the fraction by dividing:

​​ h(x)=-\cos(x)

(B) (i)

Initially, we need to set ​​the function equal to zero:  ​​

2(\sin{x})(\cos{x})-\cos{x})=0 ​​

Then, we should factor out ​​cosine from each term:

\cos{x}(2\sin{x}-1)=0

Thereafter, set each factor equal to zero and solve: ​​

\cos{x}=0

Solving that first equation gives us the following:

x=\dfrac{\pi}{2}  ​​

Then, we need to look at the second equation:

2\sin{x}-1=0

Thus, make sure to isolate the sine function:

​​ \sin{x}=\dfrac{1}{2} ​​

Finally, solving the second equation gives us:

x=\dfrac{\pi}{6} 

So, we have two answers on the interval provided.

(B) (ii)

Firstly, let’s set the function equal to the desired value:

3e=8e^{3x}-e

Then, we can isolate the base:

4e=8e^{3x}

Following, divide both sides by 8:

\dfrac{e}{2}=e^{3x}

Then, take the natural logarithm of both sides:

\ln(\dfrac{e}{2})=\ln(e^{3x})

As a matter of fact, the natural log and exponential function are inverses:

\ln(\dfrac{e}{2})=3x

Furthermore, use logarithm properties to expand the quotient:

\ln(e)-\ln(2)=3x

Then, evaluate the logarithm:

1-\ln(2)=3x

Finally, isolate the variable:

x=\dfrac{1-\ln(2)}{3}  

(C)

Since the domain of this function is all real numbers, we will have an infinite number of periodic solutions. First, set the function equal to the desired value:

\cos(2x)+4 =\dfrac{9}{2}

Then, isolate the cosine function:

\cos(2x) =\dfrac{1}{2}

Then, use inverse cosine on both sides to isolate the variable:

2x=\cos^{-1}(\dfrac{1}{2})

So, this is asking us to identify a specific value for cosine on the unit circle. There are two answers for this, x=\frac{\pi}{3} and x=\frac{5\pi}{3} . Each of these answers repeats every 2\pi because of the period of \cos(x). So, we will add integer multiples of 2\pi to each:

2x=\dfrac{\pi}{3} + 2\pi k  

As a result, for the second answer:

2x=\dfrac{5\pi}{3} + 2\pi k

Then, divide both sides by 2: 

x=\dfrac{\pi}{6} + \pi k  

Lastly, we need to consider the other answer too:

x=\dfrac{5\pi}{6} + \pi k

Those represent all values in the domain of  m(x) that are equal to \frac{9}{2} .

Summary: AP® Precalculus Overview

In conclusion, we unraveled the mysteries of the fresh addition to the library of AP® math courses. We looked at the necessary prerequisites for the course, discovered the intricacies of its curriculum, laid out the exam format, and gained insights into the types of questions posed. This AP® Precalculus overview was tailored to equip you with the confidence and understanding needed to master this new course for the 2024-2025 school year.