In Exercises 55–59, use the graph of to graph each function g.
g(x) = -f(2x)

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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.

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.

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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