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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 56b

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(2x1)g(x) = 2ƒ(2x -1)

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1
Start with the function \( f(x) = \sqrt{x} \).
Identify the transformation inside the function: \( f(2x - 1) \). This involves a horizontal scaling and a horizontal shift.
The expression \( 2x - 1 \) indicates a horizontal compression by a factor of \( \frac{1}{2} \) and a shift to the right by \( \frac{1}{2} \).
Apply the vertical transformation: Multiply the function by 2, resulting in \( 2f(2x - 1) \), which vertically stretches the graph by a factor of 2.
Combine these transformations to obtain \( g(x) = 2\sqrt{2x - 1} \). Use a graphing utility to verify the transformations visually.

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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 context, the function ƒ(x) = √x is transformed into g(x) = 2ƒ(2x - 1) by applying specific operations. Understanding how these transformations affect the graph's position and shape is crucial for accurately sketching the new function.
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Intro to Transformations

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of transformations that move the graph of a function without changing its shape. A horizontal shift occurs when the input variable x is adjusted, while a vertical shift involves scaling the output. In the given function g(x), the term (2x - 1) indicates a horizontal shift to the right by 0.5 units, while the multiplication by 2 scales the output vertically.
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Intro to Transformations

Graphing Utilities

Graphing utilities are tools that allow users to visualize mathematical functions and their transformations. These tools can plot graphs based on equations, helping to verify the accuracy of transformations performed manually. In this question, using a graphing utility to check the transformation from ƒ(x) to g(x) provides a visual confirmation of the changes made to the original function.
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Graphing The Derivative