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**Introduction**

The AP® Precalculus FRQ section plays a crucial role in determining your overall score on the AP® Precalculus exam. However, if you’re searching for effective strategies to tackle these challenging AP® Precalculus FRQ, you’re in the right place.

In this article, we’ll break down the different types of the AP® Precalculus FRQ, share essential tips from College Board, and explain the key task verbs that appear on the AP® Precalculus exam. Additionally, we’ll take a deep dive into the AP® Precalculus released FRQ to help you get a head start on your preparation.

So, keep reading to uncover everything you need to know to excel on the AP® Precalculus exam!

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**Breaking Down the AP® Precalculus FRQ**

Undoubtedly, the AP® Precalculus FRQ assess a wide range of mathematical skills. Accordingly, understanding the different types of the AP® Precalculus FRQ can help you better prepare for the exam. Below is a breakdown of the four main types you’ll encounter:

### FRQ 1: Function Concepts (Graphing Calculator)

This question presents functions in multiple forms: graphically, numerically, and analytically. Additionally, it includes three parts and covers topics such as function composition, inverse functions, zeros of a function, and end behavior. Moreover, you might be asked to identify the correct function type to construct a model. A graphing calculator is useful for analyzing the functions in this section.

### FRQ 2: Modeling a Non-Periodic Context (Graphing Calculator)

This question centers around a real-life situation. In the first part, you will build a system of equations to model the context using function types like polynomial, piecewise-defined, exponential, or logarithmic. Next, you will calculate and interpret average rates of change, along with their units. Finally, in Part C, you will justify conclusions based on the model’s assumptions or limitations. A graphing calculator plays a crucial role in solving this question as well.

### FRQ 3: Modeling a Periodic Context (No Calculator)

This question focuses on real-life contexts modeled by sinusoidal functions. Initially, you will identify key points on the graph and its midline for two cycles. Afterward, you will determine the parameters of the sinusoidal function, working with its amplitude, period, phase shift, and vertical shift. Finally, you will answer questions about the function’s behavior and the rate of change over a specific interval. Notably, this section must be completed without a calculator.

### FRQ 4: Symbolic Manipulations (No Calculator)

This question presents various functions—exponential, logarithmic, trigonometric, and inverse trigonometric. In two parts, you will solve equations using these functions. Furthermore, the third part will require you to rewrite given expressions in equivalent forms using algebraic methods and rules for exponents and logarithms. As such, showing your work clearly is essential, especially since calculators are not permitted.

Altogether, mastering these question types and knowing when to rely on your graphing calculator are key strategies for success on the AP® Precalculus exam.

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**Tips from College Board on AP® Precalculus FRQ**

Success on the AP® Precalculus FRQ requires both skill and strategy. Accordingly, the College Board provides helpful tips to ensure you’re well-prepared for the AP® Precalculus FRQ.

### Keep a close eye on the clock.

Time management is crucial during the exam. Before you dive in, take a few moments to review both questions in Part A and Part B to plan your approach. Use your time wisely, aiming to answer every part of every question. Remember, all your work should be written in the designated Section II: Free Response booklet.

During the second part of the timed section, when you move to Part B, you’re allowed to continue working on Part A’s questions, but calculators are off-limits during this time. If you make a mistake, simply cross out the incorrect work instead of spending valuable time erasing it—crossed-out work won’t be graded.

### Always show your work, even with a calculator.

The exam readers want to see your thought process, not just the final answer. Thus, if supporting work is requested and you don’t provide it, you may not receive credit—even if the answer is correct. Make sure to clearly outline your steps, especially when you’re using a graphing calculator.

### Use your graphing calculator effectively for Part A.

Your graphing calculator isn’t just there for basic calculations—it’s essential for producing graphs, evaluating functions, solving equations, and more. Be sure to set your calculator to radian mode and keep intermediate calculations as precise as possible. Any decimal answers should be accurate to three decimal places unless the question specifies otherwise.

Maximize your calculator’s potential by storing important information, such as values for constants or equations. This will help maintain precision and ensure accuracy in your final answers.

### Attempt every part of each question.

Every AP® Precalculus FRQ is broken down into multiple parts (like (a), (b), and (c)). Because each part is graded independently, it’s vital to attempt every one. For instance, even if you don’t score any points on part (a), you can still get full credit for parts (b) and (c). Don’t give up if your answer to an earlier part is wrong—later parts might still be salvageable.

### Answer the full question being asked.

Read carefully! If the question asks for the maximum value of a function, don’t just find the x-value where it occurs; make sure to also identify the maximum itself. Similarly, if you’re asked to explain or justify your answer, make sure your explanation is clear. You can refer to the function’s properties, definitions, or representations from the question—anything that demonstrates your mathematical reasoning.

### Practice makes perfect.

Familiarize yourself with sample FRQs from the AP® Precalculus Course and Exam Description. Notably, each FRQ type has a predictable structure. For example, the “Modeling a Periodic Context” question always involves graphing and labeling points on a sinusoidal model, while the “Symbolic Manipulations” question will follow a familiar set of directions. Thus, practicing these formats will help you feel more confident on exam day.

**Task Verbs in AP® Precalculus FRQ**

In order to successfully answer the AP® Precalculus FRQ, you must understand the task verbs. Each task verb gives you specific instructions on what kind of response is required. Let’s break down the most common task verbs you’ll encounter:

### Construct/Write a Function, Expression, Equation, or Model

When asked to construct or write, you are expected to create an analytical representation that aligns with a given scenario, data set, or specific criteria. Sometimes, this may involve using technology like a graphing calculator, but at other times, you’ll need to do it without.

### Describe

To describe means to create a verbal explanation that accurately reflects the given scenario, data set, or function. Essentially, you’re converting mathematical concepts into clear, written words.

### Determine/Find/Identify

These verbs signal that you need to apply appropriate mathematical methods or processes to reach an answer. Be prepared to calculate, analyze, or identify key values or components within the problem.

### Estimate/Compare

When asked to estimate or compare, you’ll use a function to find approximate values or to make comparisons between different results. In these cases, precision may not be as critical as identifying relative values or trends.

### Explain/Give a Reason/Provide a Rationale/Justify

These verbs require you to delve deeper into your reasoning. You’ll need to explain why a particular solution works by using information from the scenario or the function representation. This step often involves connecting your solution back to the context of the problem.

### Express/Indicate

Here, you are asked to provide a result in a specific form. Often, this might include reporting values with the correct units, so make sure to pay attention to the details.

### Interpret

To interpret means to explain the connection between a mathematical solution and its real-world meaning. Frequently, this involves discussing units or describing how the solution fits within the context of the problem.

### Plot and Label/Sketch and Label

These verbs direct you to develop a graphical representation that aligns with the scenario, data, or criteria given. Make sure your graph is accurate and includes clear labels for important points or values.

### Rewrite

When asked to rewrite, you’ll need to transform a mathematical expression into an equivalent form. This could involve algebraic manipulation, such as simplifying an expression or converting it into a different format.

### Solve

This task verb is straightforward: apply the appropriate methods to find a solution to an equation or inequality. Make sure your steps are clear, and your solution is accurate.

Ultimately, understanding these task verbs and responding to them correctly is essential for success on the AP® Precalculus FRQ. Each verb guides you toward a specific type of response, so paying close attention to them will help you tackle the questions more effectively.

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**AP® Precalculus FRQ 2024**

All of the AP® Precalculus 2024 FRQ have been released, offering a perfect opportunity to practice and refine your skills. In this section, we will walk you through detailed solutions to the AP® Precalculus released FRQ, helping you understand the concepts behind each problem. Whether you’re preparing for the exam or simply looking to improve your precalculus knowledge, these step-by-step explanations will give you the confidence you need to succeed.

**AP® Precalculus FRQ 2024 #1 **

**Part A**

**(i) Let the function h(x) be defined as h(x) = g(f(x)) . Find the value of h(3) as a decimal approximation, or indicate that it is not defined.**

To solve this:

First, identify the value of f(3) . Based on the information given, the point (3, 1) lies on the graph of f , meaning f(3) = 1 .

Next, substitute f(3) into the function g . The function g(x) is defined as g(x) = 2.916 \cdot (0.7)^x .

Substitute f(3) = 1 into g(f(3)) , so:

g(1) = 2.916 \cdot (0.7)^1 = 2.916 \cdot 0.7 .

This results in g(1) = 2.041 .

Therefore:

h(3) = g(f(3)) = 2.041 .

**(ii) Find all values of x for which f(x) = 1 , or indicate that there are no such values.**

We are given that the points (-3, 1) , (0, 1) , and (3, 1) lie on the graph of f . This means that:

f(x) = 1 at x = -3 , x = 0 , and x = 3 .

Therefore, the values of x for which f(x) = 1 are:

x = -3, 0, 3 .

**Part B**

**(i) Find all values of x , as decimal approximations, for which g(x) = 2 , or indicate that there are no such values.**

Given g(x) = 2.916 \cdot (0.7)^x , we need to solve for x when g(x) = 2 . Set the equation:

2.916 \cdot (0.7)^x = 2

To solve for x , divide both sides of the equation by 2.916 :

(0.7)^x = \frac{2}{2.916} \approx 0.68587

Take the natural logarithm (ln) of both sides to solve for x :

\ln((0.7)^x) = \ln(0.68587)

By logarithm properties:

x \ln(0.7) = \ln(0.68587) .

Solve for x :

x = \frac{\ln(0.68587)}{\ln(0.7)} \approx 1.057

Thus, x \approx 1.057 . This is the only solution because an exponential function is one-to-one.

**(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.**

The function g(x) = 2.916 \cdot (0.7)^x involves exponential decay, as the base 0.7 is less than 1. As x increases, (0.7)^x approaches 0.

Therefore, the limit as x \to \infty of g(x) is:

\lim\limits_{x \to \infty} g(x) = 0

**Part C**

**(i) Determine if f has an inverse function.**

The function f(x) does not have an inverse function.

**(ii) Give a reason for your answer based on the definition of a function and the graph of y = f(x) .**

Looking at the information provided, f(x) contains the points (-3, 1) , (0, 1) , and (3, 1) . Since these points share the same y -value, the graph of f fails the horizontal line test. Specifically, a horizontal line at y = 1 crosses the graph at multiple points ( x = -3, 0, 3 ). Thus, ** f does not have an inverse function** because it is not one-to-one.

**AP® Precalculus FRQ 2024 #2**

**Part A**

**(i) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for G(t) = a + b \ln(t + 1) .**

We are given two data points:

- On the initial day of sales ( t = 0 ), G(0) = 40 thousand units were sold.
- Ninety-one days later ( t = 91 ), G(91) = 76 thousand units were sold.

Using the formula for G(t) , we can set up two equations based on these data points.

For t = 0 :

G(0) = a + b \ln(0 + 1) = a + b \ln(1) = a + 0 = a

For t = 91 :

G(91) = a + b \ln(91 + 1) = 40 + b \ln(92) = 76

So, our two equations are:

40 = a + b \ln(1)

76 = a + b \ln(92)

**(ii) Find the values for a and b as decimal approximations.**

Simplifying the first equation first gives us:

40 = a + b(0)

40 = a

Now, let’s solve for b using the equation:

40 + b \ln(92) = 76

Subtract 40 from both sides:

b \ln(92) = 36

Now, divide by \ln(92) :

b = \frac{36}{\ln(92)} \approx \frac{36}{4.522} \approx 7.961

Thus, the values are:

- a = 40
- b \approx 7.961

**Part B**

**(i) Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from t = 0 to t = 91 days. Express your answer as a decimal approximation. Show the computations that lead to your answer.**

The average rate of change is calculated by finding the change in G(t) over the change in t :

\text{Average Rate of Change} = \frac{G(91) - G(0)}{91 - 0}

Substitute the given values:

\text{Average Rate of Change} = \frac{76 - 40}{91} = \frac{36}{91} \approx 0.396

Thus, the average rate of change is approximately 0.396 thousand units per day.

**(ii) Use the average rate of change found in (i) to estimate the number of units of the video game sold, in thousands, on day t = 50 . Show the work that leads to your answer.**

To estimate the number of units sold on day t = 50 using the average rate of change, we set up the following equation:

G(50) \approx G(0) + (\text{Average Rate of Change}) \times 50

Substitute the known values:

G(50) \approx 40 + 0.396 \times 50 = 40 + 19.8 = 59.8

Thus, the estimated number of units sold on day t = 50 is approximately 59.8 thousand units.

**(iii) Let A_t represent the estimate of the number of units of the video game sold, in thousands, using the average rate of change found in (i). For A_{50} , found in (ii), it can be shown that A_{50} < G(50) . Explain why, in general, A_t < G(t) for all t , where 0 < t < 91 .**

The estimate A_t is based on a constant average rate of change, which assumes a linear increase in sales over time. However, the actual function G(t) = 40 + 7.961 \ln(t + 1) represents logarithmic growth and so the graph is concave down. Therefore, the linear estimate A_t will be a secant line that is below the actual graph of G(t) . So, A_t < G(t) for all t , where 0 < t < 91 .

**Part C**

**The makers of the video game reported that daily sales of the video game decreased each day after t = 91 . Explain why the error in the model G increases after t = 91 .**

The model G(t) = a + b \ln(t + 1) assumes continued logarithmic growth, meaning that sales are predicted to keep increasing, though at a slower rate. However, after t = 91 , the makers of the game reported that sales actually began to decrease each day. This deviation from the model’s assumption of continuous growth leads to increasing error because the model no longer accurately reflects the reality of declining sales. As time progresses beyond t = 91 , the model’s predictions will diverge further from the actual sales, causing the error to grow.

**AP® Precalculus FRQ 2024 #**3

**AP® Precalculus FRQ 2024 #**3

**Part A**

**Determine possible coordinates (t, h(t)) for the five points: F, G, J, K, and P.**

Based on the information provided:

- The tire completes a full rotation in 2 seconds, so the period of the sinusoidal function h(t) is 2 seconds.
- The maximum height of point W is 18 inches, and the radius of the tire is 9 inches. Therefore, the midline of the sinusoidal function is at 9 inches.

Given that point F is located at the first maximum, point J is at the first minimum, and points G and K are on the midline, we can use the properties of the sinusoidal function to estimate the coordinates.

**Point F (First Maximum):**

This occurs when the function reaches its maximum height of 18 inches. If we assume the function is increasing after reaching the first minimum at 1/2 seconds, the maximum occurs halfway before the next minimum at t = 5/2 second. Therefore, the coordinates of F are (3/2, 18) .

**Point J (First Minimum):**

This point occurs when h(t) reaches its minimum height of 0 inches (the height when point W is touching the ground). We are told in the prompt that this occurs at t = 1/2, 5/2 seconds. Since J occurs after F , based on our previous answer, the coordinates could be (5/2, 0) .

**Point G (On the Midline):**

Since G is on the midline, the height is 9 inches. Point G occurs when point W is falling towards the minimum. This occurs halfway between F and J . Therefore, the coordinates of G are (2, 9) .

**Point K (On the Midline):**

Point K is also on the midline at h(t) = 9 , but this time it occurs as the function is rising towards the maximum. Therefore, K occurs at t = 3 seconds. The coordinates of K are (3, 9) .

**Point P (Next Maximum):**

Point P represents the next maximum of h(t) . Since the period is 2 seconds, the next maximum occurs 2 seconds after the first maximum. Therefore, the coordinates of P are (7/2, 18) .

Thus, the approximate coordinates for the points are:

- F(3/2, 18)
- G(2, 9)
- J(5/2, 0)
- K(3, 9)
- P(7/2, 18)

**Part B**

**The function h(t) can be written in the form h(t) = a \sin(b(t + c)) + d . Find values of constants a , b , c , and d .**

**Amplitude ( a ):**The amplitude is the distance between the midline and the maximum (or minimum) of the sinusoidal function. Given that the maximum height is 18 inches and the midline is at 9 inches, the amplitude is:

a = 18 - 9 = 9

**Vertical Shift ( d ):**The vertical shift is the value of the midline, which is at 9 inches:

d = 9

**Period ( \frac{2\pi}{b} ):**The period of the sinusoidal function is the time it takes for the tire to complete one full rotation, which is 2 seconds. Therefore, the period is 2 seconds, so we can set up the equation for b :

\frac{2\pi}{b} = 2 \implies b = \pi

**Phase Shift ( c ):**If we consider the new sinusoidal function h(t) = 9 \sin(\pi t) + 9 , we’ll notice that the first minimum occurs at t = 3/2 . According to the information given, the first minimum should be at t = 1/2 . Therefore, the phase shift, c , should shift the function left by 1 second, so:

c = -1

Thus, the function can be written as:

h(t) = 9 \sin(\pi (t - 1)) + 9

**Part C**

**(i) On the interval (t_1, t_2) , which of the following is true about h(t) ?**

Based on the description, t_1 corresponds to point K , which is at (3, 9) , and t_2 corresponds to point P , which is at (7/2, 18) .

Between t_1 and t_2 , the function is increasing from the midline to the maximum. Therefore, the correct answer is:

**a. h is positive and increasing.**

**(ii) Describe how the rate of change of h(t) is changing on the interval (t_1, t_2) .**

As h(t) increases from the midline to the maximum, the rate of change decreases. This is because the graph is concave down on this interval. Thus, the rate of change of h(t) is **positive but decreasing** on the interval (t_1, t_2) .

**AP® Precalculus FRQ 2024 #**4

**Part A**

**(i) Solve g(x) = 10 for values of x in the domain of g .**

Given the function g(x) = e^{x + 3} , we need to solve:

e^{x + 3} = 10

To isolate x , take the natural logarithm of both sides:

\ln(e^{x + 3}) = \ln(10)

By logarithmic properties:

x + 3 = \ln(10)

Now, solve for x :

x = \ln(10) - 3

Thus, the solution is:

x = \ln(10) - 3

**(ii) Solve h(x) = \frac{\pi}{4} for values of x in the domain of h .**

Given the function h(x) = \arcsin\left(\frac{x}{2}\right) , we need to solve:

\arcsin\left(\frac{x}{2}\right) = \frac{\pi}{4}

Take the sine of both sides to remove the inverse sine:

\frac{x}{2} = \sin\left(\frac{\pi}{4}\right)

Since \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} , we have:

\frac{x}{2} = \frac{\sqrt{2}}{2}

Now, multiply both sides by 2 to solve for x :

x = \sqrt{2}

Thus, the solution is:

x = \sqrt{2}

**Part B**

**(i) Rewrite j(x) as a single logarithm base 10 without negative exponents in any part of the expression. Your result should be of the form \log_{10}(\text{expression}) .**

Given the function j(x) = \log_{10}(8x^5) + \log_{10}(2x^2) - 9\log_{10}(x) , we start by applying the logarithmic properties:

- Combine the first two logarithms using the product rule:

\log_{10}(8x^5) + \log_{10}(2x^2) = \log_{10}(8x^5 \cdot 2x^2)

This simplifies to:

\log_{10}(16x^7)

- Now, subtract the third term using the quotient rule:

j(x) = \log_{10}\left(\frac{16x^7}{x^9}\right)

Simplify the expression inside the logarithm:

j(x) = \log_{10}\left(\frac{16}{x^2}\right)

Thus, the simplified form of j(x) is:

j(x) = \log_{10}\left(\frac{16}{x^2}\right)

**(ii) Rewrite k(x) as a single term involving \tan(x) .**

Given the function k(x) = \left(\frac{1 - \sin^2(x)}{\sin(x)}\right)\sec(x) , we can simplify it step by step:

- Use the Pythagorean identity: 1 - \sin^2(x) = \cos^2(x) . So the expression becomes:

k(x) = \frac{\cos^2(x)}{\sin(x)} \cdot \sec(x)

- Rewrite \sec(x) as \frac{1}{\cos(x)} :

k(x) = \frac{\cos^2(x)}{\sin(x)} \cdot \frac{1}{\cos(x)}

This simplifies to:

k(x) = \frac{\cos(x)}{\sin(x)}

- Recognize that \frac{\cos(x)}{\sin(x)} = \cot(x) . Thus, the function becomes:

k(x) = \cot(x)

Finally, express \cot(x) in terms of \tan(x) :

k(x) = \frac{1}{\tan(x)}

**Part C**

**The function m(x) = \cos^{-1}(\tan(2x)) . Find all values in the domain of m that yield an output value of 0.**

We need to solve for x when m(x) = 0 . So, we solve:

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

Taking the cosine of both sides:

\tan(2x) = \cos(0) = 1

Now, solve for 2x :

\tan(2x) = 1

The general solution for \tan(\theta) = 1 occurs at:

2x = \frac{\pi}{4} + n\pi where n is an integer.

Solve for x :

x = \frac{\pi}{8} + \frac{n\pi}{2}

Now, determine the domain of m(x) . The function \cos^{-1} is defined for inputs between -1 and 1 , so we restrict the solutions of \tan(2x) accordingly.

Thus, the values of x that satisfy this equation are:

x = \frac{\pi}{8} + \frac{n\pi}{2} for appropriate values of n within the domain of m(x) .

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**Conclusion**

To sum up, preparing for the AP® Precalculus FRQ requires a deep understanding of functions, equations, and trigonometric concepts. Of course, by mastering the different types of the AP® Precalculus FRQ, practicing task verbs, and carefully analyzing past exam questions, you’ll be well-equipped to tackle the AP® Precalculus exam with confidence. Most important, practice makes perfect—working through sample problems like those discussed in this article will help you develop the strategies and skills needed to excel. Good luck, and keep up the hard work on your AP® Precalculus journey!

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