Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 43

Chapter 3, Problem 43

Use the graph of y = f(x) to graph each function g. g(x) = (1/2)f(2x)

Verified step by step guidance

Step 1: Understand the transformations applied to the function f(x). The given function g(x) = (1/2)f(2x) involves two transformations: a horizontal compression by a factor of 2 and a vertical scaling by a factor of 1/2.

Step 2: Start with the horizontal compression. The term f(2x) indicates that the graph of f(x) is horizontally compressed by a factor of 2. This means that every x-coordinate of the points on the graph of f(x) is divided by 2.

Step 3: Apply the vertical scaling. The term (1/2)f(2x) indicates that the graph of f(2x) is vertically scaled by a factor of 1/2. This means that every y-coordinate of the points on the graph of f(2x) is multiplied by 1/2.

Step 4: Combine the transformations. To graph g(x), take each point (x, y) on the graph of f(x), apply the horizontal compression to get (x/2, y), and then apply the vertical scaling to get (x/2, y/2).

Step 5: Plot the transformed points on the graph and connect them smoothly to complete the graph of g(x). Ensure that the overall shape of the graph reflects the combined transformations.

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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 stretching, compressing, or shifting. In the case of g(x) = (1/2)f(2x), the function undergoes both a vertical compression by a factor of 1/2 and a horizontal compression by a factor of 2, which affects the shape and position of the graph.

Horizontal Scaling

Horizontal scaling involves changing the input values of a function, which affects how the graph is stretched or compressed along the x-axis. In g(x) = (1/2)f(2x), the '2x' indicates that the function f(x) is being evaluated at twice the rate, resulting in a horizontal compression of the graph by a factor of 2, making it appear narrower.

Vertical Scaling

Vertical scaling modifies the output values of a function, impacting the graph's height. In the function g(x) = (1/2)f(2x), the factor of 1/2 indicates that the output of f(2x) is halved, leading to a vertical compression of the graph. This means that all y-values of the function g will be half of those of f, effectively lowering the graph.

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