AP Pre-Calculus: Ultimate Study Guide 🚀

Hey there, future AP Pre-Calculus master! This guide is designed to be your go-to resource for acing the exam. Let's dive in and make sure you're feeling confident and ready to rock! 💪

1.1 Change in Tandem

A Quick Refresher on Functions

Okay, let's talk functions—not the party kind, but the mathematical kind! 🥧

Think of it like a machine: you put something in, and you get something specific out. No surprises! 🎁

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• Domain: All possible input values. ⛳
• Range: All possible output values.

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Variables

• Independent Variable: The input that you control or change (the cause). 👍
• Dependent Variable: The output that depends on the input (the effect).

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The Function Rule

The function rule is how you transform inputs into outputs. 📐 It can be shown:

• Graphically: Using a graph.
• Numerically: Using a table or sequence.
• Analytically: Using a formula or equation.
• Verbally: Using words.

Increasing vs. Decreasing Functions

Increasing Functions

• Output values increase as input values increase. 📈
• If a < b, then f(a) < f(b).

Example: f(x) = x or f(x) = $e^x$

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Decreasing Functions

• Output values decrease as input values increase. 📉
• If a < b, then f(a) > f(b).

Example: f(x) = -x or f(x) = 1/x

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Features of Functions

Concavity

• Concave Up: Graph curves upward (like a bowl). Rate of change is increasing. 🥣
• Concave Down: Graph curves downward (like a frown). Rate of change is decreasing. ☹️

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Zeroes

• Where the function's graph intersects the x-axis. 0️⃣
• Also called roots or solutions.
• Found by solving f(x) = 0.

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Practice Question

Practice Questions

Multiple Choice

1. Which of the following statements is true about the function $f(x) = -x^2 + 4x - 3$? a) It is increasing for all real numbers. b) It is decreasing for all real numbers. c) It is increasing on the interval $(-\infty, 2)$ and decreasing on the interval $(2, \infty)$. d) It is decreasing on the interval $(-\infty, 2)$ and increasing on the interval $(2, \infty)$.

2. The graph of a function $g(x)$ is concave down on the interval $(a, b)$. Which of the following must be true about the rate of change of $g(x)$ on this interval? a) The rate of change is constant. b) The rate of change is increasing. c) The rate of change is decreasing. d) The rate of change is zero.

Free Response Question

Consider the function $h(x) = x^3 - 6x^2 + 9x$.

(a) Find the zeros of the function $h(x)$. (3 points) (b) Determine the intervals where $h(x)$ is increasing and decreasing. (4 points) (c) Determine the intervals where $h(x)$ is concave up and concave down. (4 points)

Answer Key

Multiple Choice

1. c)
2. c)

Free Response Question

(a) To find the zeros, set $h(x) = 0$: $x^3 - 6x^2 + 9x = 0$ $x(x^2 - 6x + 9) = 0$ $x(x - 3)^2 = 0$ Zeros: $x = 0$ and $x = 3$ (3 points: 1 for factoring, 1 for each zero)

(b) Find the first derivative: $h'(x) = 3x^2 - 12x + 9$ Set $h'(x) = 0$ to find critical points: 3x^2 - 12x + 9 = 0 $x^2 - 4x + 3 = 0$ $(x - 1)(x - 3) = 0$ Critical points: $x = 1$ and $x = 3$ Test intervals: $(-\infty, 1)$, $(1, 3)$, $(3, \infty)$ Increasing: $(-\infty, 1)$ and $(3, \infty)$ Decreasing: $(1, 3)$ (4 points: 1 for derivative, 1 for critical points, 1 for intervals, 1 for correct increasing/decreasing)

(c) Find the second derivative: $h''(x) = 6x - 12$ Set $h''(x) = 0$ to find inflection points: 6x - 12 = 0 $x = 2$ Test intervals: $(-\infty, 2)$, $(2, \infty)$ Concave down: $(-\infty, 2)$ Concave up: $(2, \infty)$ (4 points: 1 for second derivative, 1 for inflection point, 1 for intervals, 1 for correct concavity)

Final Exam Focus 🎯

• High-Value Topics: Focus on understanding functions, their behavior (increasing/decreasing, concavity), and finding zeros. These are fundamental and appear in many forms.
• Common Question Types: Expect questions that ask you to analyze graphs, find intervals of increasing/decreasing behavior, determine concavity, and solve for zeros.
• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back to it later.
• Common Pitfalls: Be careful with signs and algebraic manipulations. Double-check your work, especially when dealing with derivatives.
• Strategies: Practice, practice, practice! The more you work through problems, the more comfortable you'll become with the concepts. Use this guide as a quick reference to refresh your memory before the exam.

Remember, you've got this! Go into the exam with confidence, and show them what you know! 🎉