The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

$f (2 (x - 1))$

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Identify the transformations applied to the function $ f(x) $. The expression $ f(2(x - 1)) $ involves a horizontal scaling and a horizontal shift.

Recognize that $ 2(x - 1) $ indicates a horizontal compression by a factor of $ \frac{1}{2} $. This means the graph will be compressed towards the y-axis.

The term $ (x - 1) $ represents a horizontal shift to the right by 1 unit. This means every point on the graph of $ f(x) $ will move 1 unit to the right.

Combine the transformations: First, shift the graph of $ f(x) $ 1 unit to the right, then apply the horizontal compression by a factor of $ \frac{1}{2} $.

Sketch the transformed graph by applying these transformations to key points on the original graph of $ 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 changes made to the graph of a function based on modifications to its equation. Common transformations include vertical and horizontal shifts, stretches, and reflections. For example, the expression f(2(x - 1)) indicates a horizontal shift to the right by 1 unit and a horizontal compression by a factor of 2.

Horizontal Stretch and Compression

Horizontal stretch and compression involve altering the width of the graph of a function. A factor greater than 1 compresses the graph, making it narrower, while a factor between 0 and 1 stretches it, making it wider. In the function f(2(x - 1)), the '2' compresses the graph horizontally, affecting how quickly the function values change as x varies.

Graphing Composite Functions

Graphing composite functions involves plotting the output of one function as the input to another. In this case, f(2(x - 1)) means we first apply the transformation to x, then evaluate the function f at that transformed value. Understanding how to graph composite functions is essential for visualizing the effects of transformations on the original function's graph.

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>f(2(x−1))f(2(x - 1))