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 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.

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 f of negative x.

Answer 1

To determine how the graph of \( f(x) = \sin x + 2 \) transforms into the graph labeled Option 2, we need to analyze the transformations that occur.

1. Vertical Shift: The original function \( f(x) = \sin x + 2 \) is a sine curve that has been shifted vertically upward by 2 units.

2. Reflection and/or Horizontal Transformation: The options describe transformations involving reflections and/or shifts along the x-axis.

Given that you want to know how the graph of \( f(x) = \sin x + 2 \) transforms to match Option 2, please note that:

• Option 2 is the graph of \( -f(-x) \): This would involve reflecting the graph across the x-axis and then reflecting it across the y-axis (or leftward shift). This kombination results in a sine wave that is flipped upside down and reversed left-to-right.

I would suggest that Option 2 is the graph of \( -f(-x) \) because it reflects both horizontally and vertically which makes significant changes to the appearance of the sine wave.

Thus, the correct response is:

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