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Change in Tandem

Alice White

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 ๐Ÿš€

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! ๐Ÿ’ช

1.1 Change in Tandem

A Quick Refresher on Functions

Okay, let's talk functionsโ€”not the party kind, but the mathematical kind! ๐Ÿฅง

Key Concept

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! ๐ŸŽ

CNX_Precalc_Figure_01_01_012.jpg

Image Courtesy of Phil Schatz

  • Domain: All possible input values. โ›ณ
  • Range: All possible output values.

CNX_Precalc_Figure_01_02_0082.jpg

Image Courtesy of Lumen Learning

Variables

  • Independent Variable: The input that you control or change (the cause). ๐Ÿ‘
  • Dependent Variable: The output that depends on the input (the effect).

Independent-vs-Dependent-Variable.png

Image Courtesy of Science Notes

The Function Rule

The function rule is how you transform inputs into outputs. ๐Ÿ“ 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. ๐Ÿ“ˆ
  • If a < b, then f(a) < f(b).

Example: f(x) = x or f(x) = exe^x

Graph-Increasing-and-decreasing-function.png

Image Courtesy of Byjus

Decreasing Functions

  • Output values decrease as input values increase. ๐Ÿ“‰
  • If a < b, then f(a) > f(b).

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

main-qimg-e8039930690220adbb2c4295df2681ec-lq.jpeg

Image Courtesy of Quora

Features of Functions

Concavity

  • Concave Up: Graph curves upward (like a bowl). Rate of change is increasing. ๐Ÿฅฃ
  • Concave Down: Graph curves downward (like a frown). Rate of change is decreasing. โ˜น๏ธ

Screenshot 2023-08-07 at 12.51.09 PM.png

Image Courtesy of Massey University

Zeroes

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

Screenshot 2023-08-07 at 12.50.31 PM.png

Image Courtesy of Math Leaks

Practice Question

Practice Questions

Multiple Choice

  1. Which of the following statements is true about the function f(x)=โˆ’x2+4xโˆ’3f(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 (โˆ’โˆž,2)(-\infty, 2) and decreasing on the interval (2,โˆž)(2, \infty). d) It is decreasing on the interval (โˆ’โˆž,2)(-\infty, 2) and increasing on the interval (2,โˆž)(2, \infty).

  2. The graph of a function g(x)g(x) is concave down on the interval (a,b)(a, b). Which of the following must be true about the rate of change of g(x)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)=x3โˆ’6x2+9xh(x) = x^3 - 6x^2 + 9x.

(a) Find the zeros of the function h(x)h(x). (3 points) (b) Determine the intervals where h(x)h(x) is increasing and decreasing. (4 points) (c) Determine the intervals where h(x)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)=0h(x) = 0: x3โˆ’6x2+9x=0x^3 - 6x^2 + 9x = 0 x(x2โˆ’6x+9)=0x(x^2 - 6x + 9) = 0 x(xโˆ’3)2=0x(x - 3)^2 = 0 Zeros: x=0x = 0 and x=3x = 3 (3 points: 1 for factoring, 1 for each zero)

(b) Find the first derivative: hโ€ฒ(x)=3x2โˆ’12x+9h'(x) = 3x^2 - 12x + 9 Set hโ€ฒ(x)=0h'(x) = 0 to find critical points: 3x^2 - 12x + 9 = 0 x2โˆ’4x+3=0x^2 - 4x + 3 = 0 (xโˆ’1)(xโˆ’3)=0(x - 1)(x - 3) = 0 Critical points: x=1x = 1 and x=3x = 3 Test intervals: (โˆ’โˆž,1)(-\infty, 1), (1,3)(1, 3), (3,โˆž)(3, \infty) Increasing: (โˆ’โˆž,1)(-\infty, 1) and (3,โˆž)(3, \infty) Decreasing: (1,3)(1, 3) (4 points: 1 for derivative, 1 for critical points, 1 for intervals, 1 for correct increasing/decreasing)

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

Final Exam Focus ๐ŸŽฏ

  • 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! ๐ŸŽ‰

Question 1 of 11

What is a core characteristic of a function? ๐Ÿค”

Each input has multiple outputs

Each output has multiple inputs

Each input has exactly one output

Inputs and outputs are unrelated