Change in Tandem

Alice White
7 min read
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Study Guide Overview
This study guide covers functions, including their domain and range, independent and dependent variables, and different representations (graphical, numerical, analytical, verbal). It also explains increasing and decreasing functions, concavity (up and down), and finding zeros/roots of functions. Practice questions and an answer key are provided.
AP Pre-Calculus: Ultimate Study Guide ๐
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! ๐ช
1.1 Change in Tandem
A Quick Refresher on Functions
Okay, let's talk functionsโnot the party kind, but the mathematical kind! ๐ฅง
A function is a relationship where each input (from the domain) has exactly one output (in the range).
Think of it like a machine: you put something in, and you get something specific out. No surprises! ๐
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- Domain: All possible input values. โณ
- Range: All possible output values.
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Variables
- Independent Variable: The input that you control or change (the cause). ๐
- 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. ๐ 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. ๐
- If a < b, then f(a) < f(b).
Example: f(x) = x or f(x) =
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Decreasing Functions
- Output values decrease as input values increase. ๐
- 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. ๐ฅฃ
- Concave Down: Graph curves downward (like a frown). Rate of change is decreasing. โน๏ธ
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Zeroes
- Where the function's graph intersects the x-axis. 0๏ธโฃ
- Also called roots or solutions.
- Found by solving f(x) = 0.
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Practice Question
Practice Questions
Multiple Choice
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Which of the following statements is true about the function ? a) It is increasing for all real numbers. b) It is decreasing for all real numbers. c) It is increasing on the interval and decreasing on the interval . d) It is decreasing on the interval and increasing on the interval .
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The graph of a function is concave down on the interval . Which of the following must be true about the rate of change of 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 .
(a) Find the zeros of the function . (3 points) (b) Determine the intervals where is increasing and decreasing. (4 points) (c) Determine the intervals where is concave up and concave down. (4 points)
Answer Key
Multiple Choice
- c)
- c)
Free Response Question
(a) To find the zeros, set : Zeros: and (3 points: 1 for factoring, 1 for each zero)
(b) Find the first derivative:
Set to find critical points:
3x^2 - 12x + 9 = 0
Critical points: and
Test intervals: , ,
Increasing: and
Decreasing: (4 points: 1 for derivative, 1 for critical points, 1 for intervals, 1 for correct increasing/decreasing)
(c) Find the second derivative:
Set to find inflection points:
6x - 12 = 0
Test intervals: ,
Concave down:
Concave up: (4 points: 1 for second derivative, 1 for inflection point, 1 for intervals, 1 for correct concavity)
Final Exam Focus ๐ฏ
- 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! ๐

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Question 1 of 11
What is a core characteristic of a function? ๐ค
Each input has multiple outputs
Each output has multiple inputs
Each input has exactly one output
Inputs and outputs are unrelated