Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 50

3장, 문제 50

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

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

관련 실천

교과서 질문

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)

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교과서 질문

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)

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교과서 질문

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

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교과서 질문

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g(x) = √x −1 (x = 0, 1, 4, 9)

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교과서 질문

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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교과서 질문

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

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

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