Suppose the graph of f(x) is given. Describe verbally how the graphs of the following functions can be obtained from the graph of f(x). Please be advised that you must use at least one of the following key words: shift, flip or reflect, compress or shrink, stretch. d) f(2x) e) f(-x) f) - f(x)

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Transformation of Functions

### Problem Statement:
Suppose the graph of \( f(x) \) is given. Describe verbally how the graphs of the following functions can be obtained from the graph of \( f(x) \). Please be advised that you must use at least one of the following key words: shift, flip or reflect, compress or shrink, stretch.

a) \( f(2x) \)  
b) \( f(-x) \)  
c) \( -f(x) \)

### Solution and Explanation:

#### a) \( f(2x) \):
To obtain the graph of \( f(2x) \), you need to compress the original graph horizontally by a factor of 2. This means that each \( x \)-coordinate on the graph of \( f(x) \) is divided by 2.

#### b) \( f(-x) \):
To obtain the graph of \( f(-x) \), you need to reflect the original graph across the y-axis. This involves taking every point \( (x, y) \) on the graph of \( f(x) \) and mapping it to \( (-x, y) \).

#### c) \( -f(x) \):
To obtain the graph of \( -f(x) \), you need to reflect the original graph across the x-axis. This involves taking every point \( (x, y) \) on the graph of \( f(x) \) and mapping it to \( (x, -y) \).

These transformations allow us to understand how changes in the function's equation affect its graph.
Transcribed Image Text:## Transformation of Functions ### Problem Statement: Suppose the graph of \( f(x) \) is given. Describe verbally how the graphs of the following functions can be obtained from the graph of \( f(x) \). Please be advised that you must use at least one of the following key words: shift, flip or reflect, compress or shrink, stretch. a) \( f(2x) \) b) \( f(-x) \) c) \( -f(x) \) ### Solution and Explanation: #### a) \( f(2x) \): To obtain the graph of \( f(2x) \), you need to compress the original graph horizontally by a factor of 2. This means that each \( x \)-coordinate on the graph of \( f(x) \) is divided by 2. #### b) \( f(-x) \): To obtain the graph of \( f(-x) \), you need to reflect the original graph across the y-axis. This involves taking every point \( (x, y) \) on the graph of \( f(x) \) and mapping it to \( (-x, y) \). #### c) \( -f(x) \): To obtain the graph of \( -f(x) \), you need to reflect the original graph across the x-axis. This involves taking every point \( (x, y) \) on the graph of \( f(x) \) and mapping it to \( (x, -y) \). These transformations allow us to understand how changes in the function's equation affect its graph.
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