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Which set of transformations is needed to graph [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] from the parent sine function?

A. Vertical compression by a factor of 2, vertical translation 3 units up, reflection across the [tex]\( y \)[/tex]-axis

B. Vertical compression by a factor of 2, vertical translation 3 units down, reflection across the [tex]\( x \)[/tex]-axis

C. Reflection across the [tex]\( x \)[/tex]-axis, vertical stretching by a factor of 2, vertical translation 3 units up

D. Reflection across the [tex]\( y \)[/tex]-axis, vertical stretching by a factor of 2, vertical translation 3 units down

Sagot :

GinnyAnswer
To transform the parent sine function [tex]\( \sin(x) \)[/tex] into [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex], we need to follow several specific steps. Let's break it down:

1. Reflection across the x-axis: The negative sign in front of the sine function indicates a reflection of the graph across the x-axis. This transformation changes the direction in which the sine wave oscillates.

2. Vertical stretching: The coefficient 2 in front of the sine function indicates a vertical stretch by a factor of 2. This means that the amplitude of the sine wave is doubled.

3. Vertical translation: The addition of 3 outside of the sine function indicates a vertical translation upwards by 3 units. This shifts the entire graph of the sine function 3 units up along the y-axis.

Combining these transformations:

- First, reflect the graph of [tex]\( \sin(x) \)[/tex] across the x-axis to get [tex]\( -\sin(x) \)[/tex].
- Then, stretch this reflected graph vertically by a factor of 2 to get [tex]\( -2 \sin(x) \)[/tex].
- Finally, translate the resulting graph 3 units upward to obtain [tex]\( -2 \sin(x) + 3 \)[/tex].

Thus, the correct set of transformations needed to graph [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] from the parent sine function is:

reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up.
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