Question

Determine the equation for a sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis. (1 point) f(x)=-3sin x f(x)=sin ( 1/3 x) f(x)=sin (- 1/3 x) f(x)=sin (-3x)

Answer 1

To determine the equation for a sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis, we need to consider both transformations.

1. **Horizontal Compression**: A horizontal compression by a factor of 3 means that we multiply the inside of the sine function by 3. The standard sine function \( \sin(x) \) will become \( \sin(3x) \).

2. **Reflection Across the x-axis**: A reflection across the x-axis inverses the function, which means we take the negative of it. This transforms \( \sin(3x) \) into \( -\sin(3x) \).

Combining these two transformations, we get:

\[
f(x) = -\sin(3x)
\]

Now, let's evaluate the provided options to see which one matches our derived function:

1. \( f(x) = -3\sin x \) - This has a vertical stretch and a reflection, but not the correct compression.
2. \( f(x) = \sin( \frac{1}{3}x ) \) - This represents a horizontal stretch, not a compression.
3. \( f(x) = \sin( -\frac{1}{3}x ) \) - This has both a horizontal stretch and a reflection across the x-axis.
4. \( f(x) = \sin(-3x) \) - This shows only a reflection across the x-axis but not the horizontal compression.

None of the provided options exactly match \( -\sin(3x) \). However, considering the transformations directly, the closest representation would be:

- The horizontal compression is reflected in the argument of the sine function, making the correct transformed function look like \( -\sin(3x) \), but that was not in the options.

Since the options don't include the derived function, none match the transformation requirement precisely as supposed.