$f(x)$ is stretched horizontally by a factor of 2 and reflected across the $x$-axis. Which choice shows the correct representation of $f(x)$ after these transformations?
A. $-f(\frac{1}{2} x)$
B. $f(-2 x)$
C. $f(-\frac{1}{2} x)$
D. $-f(2 x)$
Answer (2)
Horizontally stretch f ( x ) by a factor of 2, resulting in f ( 2 1 x ) .
Reflect the stretched function across the x-axis, resulting in − f ( 2 1 x ) .
The correct representation of the transformed function is − f ( 2 1 x ) .
Therefore, the final answer is − f ( 2 1 x ) .
Explanation
Understanding the Transformations We are given a function f ( x ) that undergoes two transformations: a horizontal stretch by a factor of 2 and a reflection across the x-axis. We need to find the correct representation of the transformed function.
Horizontal Stretch A horizontal stretch by a factor of 2 transforms f ( x ) into f ( 2 1 x ) . This is because the input to the function is scaled by a factor of 2 1 to achieve a stretch by a factor of 2.
Reflection across the x-axis A reflection across the x-axis transforms f ( x ) into − f ( x ) . This means the output of the function is multiplied by -1.
Combining the Transformations Applying the horizontal stretch first, we have f ( x ) → f ( 2 1 x ) . Then, applying the reflection across the x-axis, we have f ( 2 1 x ) → − f ( 2 1 x ) . Therefore, the final transformed function is − f ( 2 1 x ) .
Final Answer Comparing the result with the given choices, we see that the correct representation of the transformed function is − f ( 2 1 x ) .
Examples
Imagine you are adjusting the volume of a sound wave. Stretching the function horizontally is like slowing down the playback speed, making the sound last longer. Reflecting the function across the x-axis is like inverting the sound wave, which can create an interesting audio effect. Understanding these transformations helps you manipulate signals and create unique effects in audio engineering and music production.
The correct representation of the transformed function after applying a horizontal stretch by a factor of 2 and reflecting across the x-axis is − f ( 2 1 x ) . Thus, the answer is option A. − f ( 2 1 x ) .
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