To find the equation of a sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis, we start with the basic sine function:
\[ f(x) = \sin(x) \]
- Horizontal Compression by a Factor of 3: This means we will adjust the function to make the wave oscillate more quickly. The adjustment for horizontal compression is done by multiplying the \(x\) variable by the factor. Thus, for a horizontal compression by a factor of 3, we replace \(x\) with \(3x\):
\[ f(x) = \sin(3x) \]
- Reflection Across the x-axis: To reflect the function across the x-axis, we multiply the entire function by -1. This gives us:
\[ f(x) = -\sin(3x) \]
Putting it all together, the final equation for the sine function that has a horizontal compression by a factor of 3 and a reflection across the x-axis is:
\[ f(x) = -\sin(3x) \]