Use the image to answer the question.

Question

Five labeled sinusoidal waves are plotted on a coordinate plane.

Describe the transformation of the graph of f(x)=sinx+2
to the graph labeled Option 2.

(1 point)
Responses

Option 2 is the graph of −f(x)
.
Option 2 is the graph of negative f times x.

Option 2 is the graph of −f(−x)
.
Option 2 is the graph of negative f times negative x.

Option 2 is the graph of f(−x)
.
Option 2 is the graph of f of negative x.

Option 2 is the graph of f(x−7)
.
Option 2 is the graph of f of open paren x minus 7 close paren.

Answer 1

To determine the transformation from \( f(x) = \sin(x) + 2 \) to the graph labeled Option 2, we need to analyze the effects of transformations on the sine function.

1. Identify \( f(x) \):

   • The function \( f(x) = \sin(x) + 2 \) represents a sine wave that has been shifted upward by 2 units.

2. Look at possible transformations:

   • Option 1: \( -f(x) \) indicates a reflection over the x-axis and a shift upward by 2. This makes the amplitude negative.
   • Option 2: \( -f(-x) \) indicates a reflection over the y-axis followed by a reflection over the x-axis (a reflection across the origin).
   • Option 3: \( f(-x) \) indicates a reflection over the y-axis (horizontal flip).
   • Option 4: \( f(x - 7) \) indicates a horizontal shift to the right by 7 units.

3. Conclusion:

   • Without seeing the image, and given that we are comparing Option 2 to \( f(x) = \sin(x) + 2 \), if Option 2 appears as a horizontally flipped version of the original function that also is reflected over the x-axis, then it represents the transformation \( -f(-x) \).

Thus, the most appropriate answer based on the transformations specified would be:

Option 2 is the graph of \( -f(-x) \).