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

g(x) = ½ f(x)

검증된 단계별 안내

Step 1: Observe the graph of y = f(x). It is a horizontal line segment from (1, -3) to (4, -3). This means that the function f(x) has a constant value of -3 for all x in the interval [1, 4].

Step 2: The new function g(x) = ½ f(x) involves scaling the values of f(x) by a factor of ½. Since f(x) = -3, g(x) = ½ * (-3) = -3/2 for all x in the interval [1, 4].

Step 3: To graph g(x), keep the x-values the same (from 1 to 4), but adjust the y-values to reflect the scaled value of -3/2. This means the new graph will be a horizontal line segment at y = -3/2.

Step 4: Plot the points (1, -3/2) and (4, -3/2) on the graph. These points represent the endpoints of the horizontal line segment for g(x).

Step 5: Draw a horizontal line segment connecting the points (1, -3/2) and (4, -3/2). This is the graph of g(x) = ½ f(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 operations such as scaling, translating, or reflecting. In this case, the function g(x) = ½ f(x) represents a vertical compression of the function f(x) by a factor of ½, meaning that all y-values of f(x) are halved, resulting in a graph that is closer to the x-axis.

Graphing Linear Functions

Graphing linear functions involves plotting points that satisfy the function's equation and connecting them to form a straight line. For the function f(x) shown in the graph, which is a horizontal line at y = -3 between x = 1 and x = 4, understanding how to plot these points is essential for accurately representing the transformed function g(x).

Horizontal and Vertical Scaling

Horizontal and vertical scaling are techniques used to stretch or compress the graph of a function. Vertical scaling, as seen in g(x) = ½ f(x), compresses the graph vertically, affecting the y-values while keeping the x-values unchanged. This concept is crucial for predicting how the graph of g will appear compared to f.

Use the graph of y = f(x) to graph each function g.g(x) = ½ f(x)