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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 40d
Chapter 1, Problem 40d

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>


f(2(x−1))f(2(x - 1))

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Identify the transformations applied to the function \( f(x) \). The expression \( f(2(x - 1)) \) involves a horizontal scaling and a horizontal shift.
Recognize that \( 2(x - 1) \) indicates a horizontal compression by a factor of \( \frac{1}{2} \). This means the graph will be compressed towards the y-axis.
The term \( (x - 1) \) represents a horizontal shift to the right by 1 unit. This means every point on the graph of \( f(x) \) will move 1 unit to the right.
Combine the transformations: First, shift the graph of \( f(x) \) 1 unit to the right, then apply the horizontal compression by a factor of \( \frac{1}{2} \).
Sketch the transformed graph by applying these transformations to key points on the original graph of \( f(x) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Transformation

Function transformation refers to the changes made to the graph of a function based on modifications to its equation. Common transformations include vertical and horizontal shifts, stretches, and reflections. For example, the expression f(2(x - 1)) indicates a horizontal shift to the right by 1 unit and a horizontal compression by a factor of 2.
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Intro to Transformations

Horizontal Stretch and Compression

Horizontal stretch and compression involve altering the width of the graph of a function. A factor greater than 1 compresses the graph, making it narrower, while a factor between 0 and 1 stretches it, making it wider. In the function f(2(x - 1)), the '2' compresses the graph horizontally, affecting how quickly the function values change as x varies.
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Intro to Transformations

Graphing Composite Functions

Graphing composite functions involves plotting the output of one function as the input to another. In this case, f(2(x - 1)) means we first apply the transformation to x, then evaluate the function f at that transformed value. Understanding how to graph composite functions is essential for visualizing the effects of transformations on the original function's graph.
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Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question

Solve the following equations.

sin3x=22,0≤x<2π\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\]\lt{2\pi}\)

Textbook Question

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

a. ƒ(x+1)ƒ( x + 1)

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(F(x))

Textbook Question

Solve the following equations.

cos3x=sin3x,0≤x<2π\(\cos\)3x=\(\sin\)3x,0\(\leq{x}\]\lt\)2\(\pi\)

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(g(y))