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

Verified step by step guidance

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

Key Concepts

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.