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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 57
Chapter 3, Problem 57

In Exercises 55–59, use the graph of to graph each function g. Graph showing the transformation of g(x) = -f(2x) with x and y axes labeled.
g(x) = -f(2x)

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1
Identify the original function f(x) from the given graph.
Apply the horizontal compression by a factor of 2 to the function f(x) to get f(2x). This means you will compress the graph horizontally by a factor of 2.
Reflect the graph of f(2x) over the x-axis to get -f(2x). This means you will invert the graph vertically.
Plot the new function g(x) = -f(2x) using the transformations applied in the previous steps.
Verify the transformations by comparing key points from the original graph to the transformed graph.

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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 process of altering the graph of a function through various operations, such as shifting, reflecting, stretching, or compressing. In this case, the function g(x) = -f(2x) involves a reflection across the x-axis and a horizontal compression by a factor of 2. Understanding these transformations is crucial for accurately graphing the new function based on the original function f.
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Domain & Range of Transformed Functions

Reflection Across the X-Axis

Reflecting a function across the x-axis involves changing the sign of the output values. For the function g(x) = -f(2x), this means that for every point (x, f(x)) on the graph of f, the corresponding point on g will be (x, -f(x)). This transformation results in the graph of g being a mirror image of f with respect to the x-axis, which is essential for visualizing the new function.
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Horizontal Compression

Horizontal compression occurs when the input values of a function are scaled by a factor greater than 1, effectively 'squeezing' the graph towards the y-axis. In the function g(x) = -f(2x), the factor of 2 compresses the graph of f horizontally by half. This means that points on the graph of f will be reached more quickly in g, altering the overall shape and behavior of the graph.
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Related Practice
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Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -(x − 2)²

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