Describe the transformation of the graph of f(x)=sinx+2

Question

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−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 f of negative x.

Option 2 is the graph of −f(−x)
.

Answer 1

To describe the transformation of the graph of \( f(x) = \sin(x) + 2 \) to the graph labeled Option 2, we can analyze the options provided:

1. Option 2 is the graph of −f(x): This means that you would take the function \( f(x) \), negate it, and graph it. For \( f(x) = \sin(x) + 2 \), this would result in \( -(\sin(x) + 2) = -\sin(x) - 2 \). This does not match the expected transformation.

2. Option 2 is the graph of negative f times x: This implies a transformation that does not directly relate to a standard graphical transformation of \( f(x) \).

3. Option 2 is the graph of f(x−7): This describes a horizontal shift to the right by 7 units. It does not align with a graph transformation of \( f(x) = \sin(x) + 2 \).

4. Option 2 is the graph of f of open paren x minus 7 close paren: This is similar to Option 3 and describes a rightward shift.

5. Option 2 is the graph of f(−x): This describes a reflection over the y-axis, which would not match the original graph of \( f(x) = \sin(x) + 2 \).

6. Option 2 is the graph of f(−x): It appears as a repeat but is considered reflected about the y-axis.

7. Option 2 is the graph of −f(−x): This is a combination of reflecting the function over the y-axis and then reflecting that reflection over the x-axis.

Given none of these describe a simple vertical shift of \( f(x) = \sin(x) + 2 \) upwards by 2, which results in the function being shifted upwards, that indicates that the most likely transformation described by Option 2 would be incorrect with the options presented.

However, if we are to analyze transformations closely related to vertical movements and flips, none fit the scenario perfectly. The graph of \( f(x) = \sin(x) + 2 \) is itself moved up by 2 units from the parent \( \sin(x) \) graph.

To conclude, it appears that the transformation described in Option 2 does not properly relate to \( f(x) \) in terms of the standard transformations like vertical shifts, horizontal shifts, or reflections as detailed above based solely on the provided options.

If you have further context or visual cues, that would help narrow down the most accurate response based on the transformations.