To identify the transformations for the function \( f(x) = 3\cos(-x + \frac{\pi}{3}) - 2 \), we can analyze each component of the function.
-
Vertical Stretch: The coefficient 3 indicates a vertical stretch by a factor of 3.
-
Reflection: The negative sign before x indicates a reflection over the y-axis. This is because when you have \( \cos(-x) \) rather than \( \cos(x) \), it reflects the graph horizontally.
-
Horizontal Shift: The term \(-x + \frac{\pi}{3}\) can be rewritten as \(-1(x - \frac{\pi}{3})\), which indicates a horizontal shift. The fact that it is \( -x + \frac{\pi}{3} \) means there is a shift to the right by \(\frac{\pi}{3}\).
-
Vertical Shift: The \(-2\) indicates a vertical shift downwards by 2 units.
Putting this all together, the transformations for the function are:
- Vertical stretch by a factor of 3
- Reflection over the y-axis
- Horizontal shift right by \(\frac{\pi}{3}\)
- Vertical shift down by 2
Thus, the correct response should be:
vertical stretch by a factor of 3, reflection over the y-axis, horizontal shift right by \(\frac{\pi}{3}\), and vertical shift down 2.