Question 22 (5 points)
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[tex]f(x)[/tex] is stretched horizontally by a factor of 2 and reflected across the [tex]x[/tex]-axis. Which choice shows the correct representation of [tex]f(x)[/tex] after these transformations?
A. [tex]-f\left(\frac{1}{2} x\right)[/tex]
B. [tex]f(-2 x)[/tex]
C. [tex]f\left(-\frac{1}{2} x\right)[/tex]
D. [tex]-f(2 x)[/tex]
Answer (2)
A horizontal stretch by a factor of 2 transforms f ( x ) to f ( 2 1 x ) .
A reflection across the x-axis transforms f ( x ) to − f ( x ) .
Applying both transformations in sequence results in − f ( 2 1 x ) .
The correct representation of the transformed function 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 determine 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 x needs to be halved to achieve the same output as the original function, but over a wider range of x values.
Reflection across the x-axis A reflection across the x-axis transforms f ( x ) into − f ( x ) . This negates the output of the function for each input x .
Combining the Transformations Applying the horizontal stretch first, we get f ( 2 1 x ) . Then, reflecting across the x-axis, we get − f ( 2 1 x ) .
Final Answer Therefore, the correct representation of the transformed function is − f ( 2 1 x ) .
Examples
Consider the function representing the trajectory of a ball. Stretching it horizontally by a factor of 2 would mean the ball takes twice as long to complete its trajectory. Reflecting it across the x-axis would invert the trajectory, making the ball move in the opposite vertical direction. Understanding transformations helps in modeling and predicting changes in various real-world scenarios.
The transformations lead to the function being represented as [tex]-figg(\frac{1}{2} xigg)$. The correct answer is A.
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