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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 56d
Chapter 1, Problem 56d

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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Start with the base function \( f(x) = \sqrt{x} \).
Apply a horizontal shift to the right by 1 unit to get \( \sqrt{x-1} \). This accounts for the \( x-1 \) inside the square root.
Apply a vertical stretch by a factor of 3 to get \( 3\sqrt{x-1} \). This is done by multiplying the entire function by 3.
Apply a vertical shift downward by 5 units to get \( 3\sqrt{x-1} - 5 \). This is done by subtracting 5 from the entire function.
The transformed function is \( g(x) = 3\sqrt{x-1} - 5 \). Use a graphing utility to verify the transformation visually.

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Key Concepts

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

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant to the input (x) shifts the graph horizontally, while adding a constant to the output (f(x)) shifts it vertically. Understanding these transformations is crucial for manipulating the graph of a function like f(x) = √x to achieve the desired form of g(x).
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Intro to Transformations

Vertical and Horizontal Shifts

Vertical shifts occur when a constant is added or subtracted from the function, affecting its position along the y-axis. Horizontal shifts are achieved by adding or subtracting a constant from the input variable, affecting the x-axis position. In the transformation from f(x) = √x to g(x) = 3√(x-1) - 5, the term (x-1) indicates a horizontal shift to the right by 1 unit, while the -5 indicates a vertical shift downward by 5 units.
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Intro to Transformations

Scaling and Stretching

Scaling refers to multiplying the function by a constant, which affects the steepness of the graph. A vertical scaling, such as multiplying by 3 in g(x) = 3√(x-1) - 5, stretches the graph away from the x-axis, making it steeper. Understanding how scaling impacts the graph is essential for accurately transforming f(x) into g(x) and predicting the resulting shape of the graph.
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Intro to Transformations
Related Practice
Textbook Question

Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.


(ƒ o g) (x) = x⁴ + 6x² + 9

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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)=ƒ(2x−4)g(x) = ƒ(2x - 4)

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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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Textbook Question

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.


(g o ƒ ) (x) = x²⸍³ + 3

Textbook Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

g(x)=−3x2g(x)=-3x^2

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