Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 50

Chapter 3, Problem 50

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) means that the graph of f(x) is compressed horizontally 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) means that the graph of f(2x) is scaled vertically 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), first apply the horizontal compression to f(x), then apply the vertical scaling to the resulting graph.

Step 5: Plot the transformed points. For each point (x, y) on the graph of f(x), calculate the new coordinates as (x/2, y/2) and plot these points to create the graph of g(x).

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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 1/2, affecting the overall 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. For g(x) = (1/2) f(2x), the '2' inside the function indicates that the graph of f(x) is compressed horizontally by a factor of 2, meaning that points on the graph will be closer together compared to the original function.

Vertical Scaling

Vertical scaling modifies the output values of a function, impacting the graph's height. In g(x) = (1/2) f(2x), the factor of 1/2 indicates a vertical compression, meaning that the output values of f(2x) are halved. This results in the graph being pulled closer to the x-axis, reducing its overall height while maintaining the same x-coordinates.

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Related Practice

Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

Textbook Question

Graph using intercepts: 2x - 5y - 10 = 0

Textbook Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 1)² + y² = 25

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Textbook Question

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

f(x) = 2x, g(x) = x+7

Textbook Question

In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)