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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 55b
Chapter 1, Problem 55b

Use shifts and scalings to transform the graph of ƒ(x)=x2ƒ(x)=x^2 into the graph of g. Use a graphing utility to check your work.


g(x)=ƒ(2x−4)g(x) = ƒ(2x - 4)

Verified step by step guidance
1
Start with the function \( f(x) = x^2 \). This is a basic parabola opening upwards with its vertex at the origin (0,0).
Identify the transformation needed to obtain \( g(x) = f(2x - 4) \). This involves a horizontal scaling and a horizontal shift.
First, consider the expression \( 2x - 4 \). Factor out the 2 to get \( 2(x - 2) \). This indicates a horizontal compression by a factor of 2 and a shift to the right by 2 units.
Apply the horizontal compression: Replace \( x \) with \( 2x \) in \( f(x) \) to get \( f(2x) = (2x)^2 = 4x^2 \). This compresses the graph horizontally by a factor of 2.
Apply the horizontal shift: Replace \( x \) with \( x - 2 \) in \( f(2x) \) to get \( f(2(x - 2)) = 4(x - 2)^2 \). This shifts the graph 2 units to the right.

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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 involves altering the graph of a function through shifts, stretches, and reflections. In this case, the function f(x) = x² is transformed into g(x) = f(2x - 4) by applying horizontal and vertical shifts and scalings. Understanding how these transformations affect the graph is crucial for accurately sketching the new function.
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Intro to Transformations

Horizontal Shifts

Horizontal shifts occur when the input of a function is adjusted by adding or subtracting a constant. For g(x) = f(2x - 4), the term '−4' indicates a shift to the right by 4 units. This concept is essential for determining how the graph of the original function moves along the x-axis.
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Intro to Transformations

Vertical Scaling

Vertical scaling refers to stretching or compressing the graph of a function by multiplying the input by a constant factor. In g(x) = f(2x - 4), the '2' in front of x indicates a horizontal compression by a factor of 1/2. This concept helps in understanding how the shape of the graph changes in relation to the original function.
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Relations & Functions Example 1
Related Practice
Textbook Question

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


c. Find a formula for A(x)


ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

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Textbook Question

Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\(\sqrt{x}\)  into the graph of g. Use a graphing utility to check your work.

g(x)=3x−1−5g(x)=3\(\sqrt{x-1}\)-5

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Textbook Question

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


a. Find A(2) .


ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Textbook Question

Shifting a graph Use a shift to explain how the graph of ƒ(x)=−x2+8x+9ƒ(x)=\(\sqrt{-x^2+8x+9}\) is obtained from the graph of g(x)=25−x2g(x)=\(\sqrt{25-x^2}\) . Sketch a graph of ƒ.

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Textbook Question

Use shifts and scalings to transform the graph of ƒ(x)=x2ƒ(x)=x^2 into the graph of g. Use a graphing utility to check your work.

g(x)=6ƒ(x−23)+1g(x)=6ƒ(\(\frac{x-2}{3}\))+1

Textbook Question

Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\(\sqrt{x}\)  into the graph of g. Use a graphing utility to check your work.

g(x)=2ƒ(2x−1)g(x) = 2ƒ(2x -1)

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