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.

Answer 1

To create a sine function that reflects the specified transformations, we start with the basic sine function, which is given by:

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

1. **Horizontal compression by a factor of 3**:
This transformation affects the period of the sine function. The period of the sine function \(\sin(x)\) is \(2\pi\). With a horizontal compression by a factor of 3, the new period becomes:

\[
\text{New Period} = \frac{2\pi}{3}
\]

To achieve this effect, we replace \(x\) with \(3x\) in the function:

\[
g(x) = \sin(3x)
\]

2. **Reflection across the x-axis**:
Reflecting across the x-axis means negating the function. This is achieved by multiplying the entire sine function by -1:

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

Putting it all together, the equation for the sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis is:

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

Thus, the final answer is:

\[
\boxed{-\sin(3x)}
\]