Let's analyze the transformations for the function \( f(x) = 3\cos(-x + \frac{\pi}{3}) - 2 \).
-
Vertical Stretch by 3: The coefficient \( 3 \) in front of the cosine function indicates a vertical stretch by a factor of 3. This is correct.
-
Vertical Shift Up 2 Units: The term \(-2\) at the end of the function indicates a vertical shift down, not up. This statement is incorrect.
-
Vertical Shift Down 2 Units: The term \(-2\) indicates a vertical shift down by 2 units. This is correct.
-
Reflection Over the X-Axis: There is no negative sign in front of the whole function, only in front of the \( x \) in the cosine argument, so there is no reflection over the x-axis. This is incorrect.
-
Reflection Over the Y-Axis: The term \(-x\) in the cosine argument indicates a reflection over the y-axis. This is correct.
-
Horizontal Shift to the Right \( \frac{\pi}{3} \): The term \( -x + \frac{\pi}{3} \) can be rewritten as \( -(x - \frac{\pi}{3}) \). Thus, this indicates a shift to the right by \( \frac{\pi}{3} \). This is correct.
-
Horizontal Shift to the Left \( \frac{\pi}{3} \): Since we established that the function shifts to the right by \( \frac{\pi}{3} \), this is incorrect.
Summary of Correct Answers:
- Vertical stretch by 3
- Vertical shift down 2 units
- Reflection over the y-axis
- Horizontal shift to the right \( \frac{\pi}{3} \)
The other responses regarding vertical shift up 2 units, reflection over the x-axis, and horizontal shift to the left \( \frac{\pi}{3} \) are incorrect.