To determine the set of transformations needed to graph the function [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] from the parent function [tex]\( \sin(x) \)[/tex], we need to carefully analyze the given function.
1. Start with the reflection across the [tex]\( x \)[/tex]-axis:
- The negative sign in front of the sine function, [tex]\( -2 \sin(x) \)[/tex], indicates a reflection across the [tex]\( x \)[/tex]-axis. This means that the graph of [tex]\( \sin(x) \)[/tex] will be flipped upside down.
2. Apply the vertical stretching:
- The coefficient 2 in front of the sine function, [tex]\( -2 \sin(x) \)[/tex], indicates a vertical stretch by a factor of 2. This means that any point on the graph of [tex]\( \sin(x) \)[/tex] will be stretched vertically, making the peaks twice as high and the valleys twice as low compared to the original function.
3. Apply the vertical translation:
- The constant +3 at the end of the function, [tex]\( -2 \sin(x) + 3 \)[/tex], indicates a vertical translation 3 units up. This means that the entire graph of the modified sine function will be shifted upwards by 3 units.
Considering all these transformations together, the function [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] is obtained through:
- Reflection across the [tex]\( x \)[/tex]-axis,
- Vertical stretching by a factor of 2,
- Vertical translation 3 units up.
Therefore, the correct set of transformations needed to graph [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] from the parent function [tex]\( \sin(x) \)[/tex] is:
[tex]\[
\text{Reflection across the } x\text{-axis, vertical stretching by a factor of 2, vertical translation 3 units up}.
\][/tex]
This corresponds to the third option:
[tex]\[
\text{reflection across the } x\text{-axis, vertical stretching by a factor of 2, vertical translation 3 units up}
\][/tex]
