To determine the set of transformations needed to graph [tex]\( f(x) = -2 \sin(x) + 3 \)[/tex] from the parent sine function, we should analyze each component of the function and how it changes the graph of [tex]\( \sin(x) \)[/tex].
1. Reflection Across the [tex]\( x \)[/tex]-axis:
- The negative sign in front of the coefficient of the sine function, [tex]\( -2 \)[/tex], indicates a reflection across the [tex]\( x \)[/tex]-axis. This transformation flips the graph upside down.
2. Vertical Stretching by a Factor of 2:
- The coefficient 2 in front of [tex]\(\sin(x)\)[/tex] indicates a vertical stretch by a factor of 2. This means that the amplitude of the sine function is multiplied by 2, making the peaks and troughs twice as far from the [tex]\( x \)[/tex]-axis as they are in the parent function.
3. Vertical Translation 3 Units Up:
- The +3 at the end of the function denotes a vertical translation upward by 3 units. This moves the entire graph upward by 3 units on the [tex]\( y \)[/tex]-axis.
Combining these transformations together, we have:
1. A reflection across the [tex]\( x \)[/tex]-axis.
2. A vertical stretching by a factor of 2.
3. A vertical translation upward by 3 units.
Therefore, the correct set of transformations is:
- Reflection across the [tex]\( x \)[/tex]-axis,
- Vertical stretching by a factor of 2,
- Vertical translation 3 units up.
Thus, the correct choice is:
Reflection across the [tex]\( x \)[/tex]-axis, vertical stretching by a factor of 2, vertical translation 3 units up.
