Identify all of the transformations for the function f(x) = 3cos(−x + π3)−2

Question

. Check all that apply(4 points)
Responses

reflection over the x-axis
reflection over the x-axis

vertical shift down 2 units
vertical shift down 2 units

horizontal shift to the right π3
horizontal shift to the right pi over 3

reflection over the y-axis
reflection over the y-axis

Vertical stretch by 3
Vertical stretch by 3

vertical shift up 2 units
vertical shift up 2 units

horizontal shift to the left π3
horizontal shift to the left pi over 3

Answer 1

Let's analyze the function \( f(x) = 3\cos(-x + \frac{\pi}{3}) - 2 \) to identify the transformations.

1. Vertical Stretch by 3: The coefficient of 3 before the cosine function indicates a vertical stretch by a factor of 3. (Check)

2. Reflection over the x-axis: The negative inside the argument of the cosine function (\(-x\)) does not reflect the function over the x-axis. Instead, it indicates a horizontal reflection that could be interpreted differently. Hence, this transformation is not applicable.

3. Vertical Shift Down 2 Units: The term \(-2\) at the end of the function indicates a vertical shift downward by 2 units. (Check)

4. Reflection over the y-axis: The negative sign in front of \(x\) (\(-x\)) indicates a reflection across the y-axis. This implies a horizontal reflection. (Check)

5. Horizontal Shift to the Right \(\frac{\pi}{3}\): To determine the horizontal shift, we can rewrite the cosine function: \( -x + \frac{\pi}{3} = -(x - \frac{\pi}{3}) \), which indicates a shift to the right by \(\frac{\pi}{3}\) (due to how the argument of the cosine is arranged). (Check)

6. Vertical Shift Up 2 Units: There is no indication of an upward vertical shift in this function; therefore, this transformation is not applicable.

7. Horizontal Shift to the Left \(\frac{\pi}{3}\): Following the same reasoning as the horizontal shift to the right, there is no leftward shift present. Therefore, this transformation is not applicable.

Summary of Transformations:

• Vertical stretch by 3: Check
• Vertical shift down 2 units: Check
• Reflection over the y-axis: Check
• Horizontal shift to the right \(\frac{\pi}{3}\): Check

The transformations that apply are:

1. Vertical stretch by 3
2. Vertical shift down 2 units
3. Reflection over the y-axis
4. Horizontal shift to the right \(\frac{\pi}{3}\)