To analyze the function \( h(x) = |-\frac{8}{7}x| \), let's break it down:
- The absolute value function \( |x| \) is being transformed by a factor of \( -\frac{8}{7} \) for \( x \).
- The negative sign indicates a reflection across the y-axis, because the transformation involves \( -x \).
- The factor \( \frac{8}{7} \) indicates that the function is stretched horizontally.
Now, let's clarify the stretch or compression:
- A normal transformation for a function \( f(kx) \) causes a horizontal compression if \( |k| > 1 \) and a horizontal stretch if \( |k| < 1 \).
- The function term here is \( -\frac{8}{7}x \), which we can treat as \( f(-\frac{8}{7}x) \).
- Since \(\frac{8}{7} > 1\), it implies a compression because of the overall \( -x \) aspect.
Let’s check the options given:
- A horizontal stretch by a factor of \( \frac{8}{7} \) - Incorrect (we have a compression).
- A horizontal compression by a factor of \( \frac{7}{8} \) - Correct.
- A horizontal stretch by a factor of \( \frac{8}{7} \) and a reflection across the y-axis - Incorrect.
- A horizontal compression by a factor of \( \frac{7}{8} \) and a reflection across the y-axis - Correct.
Thus, the correct descriptions of the transformation(s) of the function \( h(x) = |-\frac{8}{7}x| \) are:
- A horizontal compression by a factor of \( \frac{7}{8} \) (option 2 is correct).
- A reflection across the y-axis (part of option 4).
Therefore, the final answer is:
A horizontal compression by a factor of \( \frac{7}{8} \) and a reflection across the y-axis.