Ap Pre Calc Frq 2 Practice

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Mastering the AP Precalculus FRQ: Your Ultimate Practice Guide

The AP Precalculus Free-Response Questions (FRQs) are a crucial component of the exam, designed to assess your comprehensive understanding of the course's key concepts and your ability to apply them in problem-solving scenarios. Mastering these FRQs requires dedicated practice and a strategic approach. This guide provides you with the necessary tools and practice examples to excel in this section.

Understanding the AP Precalculus FRQ Section

The AP Precalculus exam includes two distinct types of FRQs:

• FRQ Type 1 (With Calculator): Two questions that allow the use of a graphing calculator.
• FRQ Type 2 (Without Calculator): Two questions that do not allow the use of a calculator.

Each FRQ is graded on a scale of 0 to 9 points, contributing significantly to your overall AP score. The questions are designed to evaluate your skills in:

• Applying Mathematical Concepts: Using precalculus principles to solve problems.
• Procedural Fluency: Executing mathematical procedures accurately.
• Conceptual Understanding: Demonstrating a deep understanding of the underlying concepts.
• Mathematical Reasoning: Justifying your solutions with clear and logical explanations.
• Communication: Effectively communicating your mathematical ideas.

Key Topics Covered in FRQs

The FRQs cover a wide range of topics from the AP Precalculus curriculum. Here's a breakdown of the key areas:

• Polynomial and Rational Functions: Analyzing, graphing, and solving equations involving polynomial and rational functions.
• Exponential and Logarithmic Functions: Understanding exponential growth and decay, logarithmic scales, and solving related equations.
• Trigonometric Functions: Working with trigonometric identities, unit circle, trigonometric equations, and applications like modeling periodic phenomena.
• Parametric Equations and Polar Coordinates: Representing curves using parametric equations, converting between rectangular and polar coordinates, and analyzing polar graphs.
• Vectors: Performing vector operations, understanding vector components, and applying vectors to solve problems.
• Complex Numbers: Performing operations with complex numbers and understanding their geometric representation.
• Modeling with Functions: Creating and analyzing mathematical models using various function types.

Strategies for Tackling FRQs

A strategic approach is essential for maximizing your score on the FRQs. Here’s a step-by-step guide:

1. Read the Question Carefully: Understand exactly what the question is asking. Identify key information and any constraints.
2. Plan Your Approach: Before diving into calculations, outline the steps you will take to solve the problem.
3. Show Your Work: Even if you use a calculator, clearly show each step of your solution process. Partial credit is awarded for correct steps, even if the final answer is incorrect.
4. Justify Your Answers: Provide clear and concise explanations for your solutions. Use mathematical reasoning to support your claims.
5. Check Your Work: If time permits, review your solutions to ensure accuracy and completeness. Look for any potential errors in your calculations or reasoning.
6. Manage Your Time: Allocate your time wisely. Don't spend too much time on a single question. If you're stuck, move on and come back to it later.

Practice FRQ Type 1 (With Calculator)

Question 1

A Ferris wheel with a radius of 30 feet is rotating at a constant rate. A rider enters the Ferris wheel at its lowest point, which is 5 feet above the ground. The Ferris wheel completes one full rotation every 60 seconds.

(a) Write a function h(t) that models the height of the rider above the ground as a function of time t in seconds.

(b) What is the height of the rider above the ground after 15 seconds?

(c) At what time t, during the first rotation, is the rider 50 feet above the ground?

(d) The Ferris wheel speeds up and now completes one full rotation every 45 seconds. Write a new function g(t) that models the height of the rider above the ground with the new rotation speed.

Solution

(a) Since the Ferris wheel starts at its lowest point, we can model the height using a cosine function with a negative coefficient. The amplitude of the cosine function is the radius of the Ferris wheel, which is 30 feet. The vertical shift is the center of the Ferris wheel's rotation, which is the radius plus the initial height, so 30 + 5 = 35 feet. The period of the Ferris wheel is 60 seconds. The general form of the function is:

h(t) = Acos(Bt) + C

Where:

• A = -30 (negative because it starts at the minimum)
• B = 2π/60 = π/30 (related to the period)
• C = 35 (vertical shift)

Therefore, the function is:

h(t) = -30cos((π/30)t) + 35

(b) To find the height after 15 seconds, substitute t = 15 into the function:

h(15) = -30cos((π/30)15) + 35 h(15) = -30cos(π/2) + 35* h(15) = -30(0) + 35* h(15) = 35

The height of the rider above the ground after 15 seconds is 35 feet.

(c) To find the time when the rider is 50 feet above the ground, set h(t) = 50 and solve for t:

50 = -30*cos((π/30)t) + 35 15 = -30cos((π/30)*t) -1/2 = cos((π/30)*t)

The angle whose cosine is -1/2 is 2π/3. Therefore:

(π/30)*t = 2π/3 t = (2π/3) * (30/π) t = 20

The rider is 50 feet above the ground at t = 20 seconds.

(d) With the new rotation speed, the period is now 45 seconds. The only change in the function is the value of B:

B = 2π/45

Therefore, the new function is:

g(t) = -30cos((2π/45)t) + 35

Question 2

A population of bacteria is growing exponentially. At time t = 0 hours, the population is 500. At time t = 3 hours, the population is 1200.

(a) Write a function P(t) that models the population of bacteria as a function of time t.

(b) What is the population of bacteria at time t = 5 hours?

(c) At what time t will the population reach 5000?

(d) Suppose a different population starts with 800 bacteria and doubles every 2 hours. Write a function Q(t) that models this population.

Solution

(a) Exponential growth can be modeled by the function:

P(t) = P₀e^(kt)*

Where:

• P₀ is the initial population.
• k is the growth rate.
• t is the time.

We know P₀ = 500. We need to find k. We know that P(3) = 1200. So:

1200 = 500*e^(3k) 1200/500 = e^(3k) 2.4 = e^(3k)

Take the natural logarithm of both sides:

ln(2.4) = 3k k = ln(2.4)/3

Therefore, the function is:

P(t) = 500e^((ln(2.4)/3)t)

(b) To find the population at t = 5 hours, substitute t = 5 into the function:

P(5) = 500e^((ln(2.4)/3)5) P(5) ≈ 500e^(0.29185) P(5) ≈ 500e^(1.459)* P(5) ≈ 5004.302* P(5) ≈ 2151

The population of bacteria at t = 5 hours is approximately 2151.

(c) To find the time when the population reaches 5000, set P(t) = 5000 and solve for t:

5000 = 500*e^((ln(2.4)/3)*t) 10 = e^((ln(2.4)/3)*t)

Take the natural logarithm of both sides:

ln(10) = (ln(2.4)/3)*t t = ln(10) / (ln(2.4)/3) t = (ln(10) * 3) / ln(2.4) t ≈ (2.3026 * 3) / 0.8755 t ≈ 7.886

The population will reach 5000 at approximately t = 7.886 hours.

(d) For the second population, the initial population is 800, and it doubles every 2 hours. So:

Q(t) = 800(2)^(t/2)*

Practice FRQ Type 2 (Without Calculator)

Question 1

Consider the function f(x) = (x² - 4) / (x² - 9)

(a) Find the x-intercepts of f(x).

(b) Find the y-intercept of f(x).

(c) Find the vertical asymptotes of f(x).

(d) Find the horizontal asymptote of f(x).

Solution

(a) To find the x-intercepts, set f(x) = 0 and solve for x:

0 = (x² - 4) / (x² - 9) 0 = x² - 4 x² = 4 x = ±2

The x-intercepts are x = 2 and x = -2.

(b) To find the y-intercept, set x = 0:

f(0) = (0² - 4) / (0² - 9) f(0) = -4 / -9 f(0) = 4/9

The y-intercept is y = 4/9.

(c) To find the vertical asymptotes, set the denominator equal to zero and solve for x:

x² - 9 = 0 x² = 9 x = ±3

The vertical asymptotes are x = 3 and x = -3.

(d) To find the horizontal asymptote, consider the limit as x approaches infinity:

lim (x→∞) (x² - 4) / (x² - 9)

Since the degree of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients:

y = 1/1 = 1

The horizontal asymptote is y = 1.

Question 2

Let v = <2, -3> and w = <-1, 4> be two vectors.

(a) Find 2v - w.

(b) Find the magnitude of v.

(c) Find the dot product of v and w.

(d) Find the angle between v and w.

Solution

(a) To find 2v - w:

2v = <4, -6> 2v - w = <4, -6> - <-1, 4> 2v - w = <4 - (-1), -6 - 4> 2v - w = <5, -10>

(b) To find the magnitude of v:

||v|| = √(2² + (-3)²) ||v|| = √(4 + 9) ||v|| = √13

(c) To find the dot product of v and w:

v · w = (2)(-1) + (-3)(4) v · w = -2 - 12 v · w = -14

(d) To find the angle between v and w, use the formula:

cos θ = (v · w) / (||v|| ||w||)

First, find the magnitude of w:

||w|| = √((-1)² + 4²) ||w|| = √(1 + 16) ||w|| = √17

Now, substitute into the formula:

cos θ = -14 / (√13 * √17) cos θ = -14 / √221 θ = arccos(-14 / √221)

Since you don't have a calculator, leave the answer in terms of arccosine:

θ = arccos(-14 / √221)

Tips for Success

• Review Fundamental Concepts: Ensure you have a strong grasp of the core precalculus topics.
• Practice Regularly: The more you practice, the more comfortable you will become with the types of questions asked.
• Understand Common Mistakes: Identify areas where you typically make mistakes and focus on correcting them.
• Learn from Your Mistakes: Analyze your incorrect answers and understand why you made the mistake.
• Simulate Exam Conditions: Practice under timed conditions to get a feel for the pace of the exam.
• Use Available Resources: Utilize textbooks, online resources, and practice exams to supplement your learning.
• Stay Calm and Focused: Approach the exam with a positive attitude and maintain focus throughout the section.

Advanced Practice Questions

Here are more challenging practice questions to further hone your skills:

FRQ Type 1 (With Calculator)

1. A population of rabbits in a forest is modeled by the function R(t) = 150 + 80sin((π/6)t), where R(t) is the number of rabbits and t is the time in months.
   • (a) What is the maximum and minimum population of rabbits?
   • (b) At what time t, during the first year, does the population reach 200 rabbits?
   • (c) If a pack of wolves is introduced into the forest, the rabbit population decreases by 15% each month. Write a new function S(t) that models the rabbit population with the wolves.

FRQ Type 2 (Without Calculator)

1. Given f(x) = 2x + 3 and g(x) = x² - 1, find:
   • (a) f(g(x))
   • (b) g(f(x))
   • (c) The domain of f(x) / g(x)
   • (d) The range of g(x)

Conclusion

Mastering the AP Precalculus FRQs requires a combination of strong conceptual understanding, procedural fluency, and strategic problem-solving skills. By understanding the structure of the FRQ section, practicing with relevant examples, and following the strategies outlined in this guide, you can significantly improve your performance on the AP exam. Consistent effort and a focused approach will pave the way for success. Good luck!