To find the vertex of the function defined as [tex]\( g(x) = f(-x) + 3 \)[/tex] given that the vertex of [tex]\( f(x) \)[/tex] is [tex]\( (5, -6) \)[/tex], follow these steps:
1. Reflect the Vertex Over the y-axis:
- The original vertex of [tex]\( f(x) \)[/tex] is at [tex]\( (5, -6) \)[/tex].
- Reflecting a point over the y-axis alters the sign of the x-coordinate while keeping the y-coordinate the same.
- Therefore, reflecting the vertex [tex]\((5, -6)\)[/tex] over the y-axis gives us [tex]\((-5, -6)\)[/tex].
2. Shift the Vertex Up by 3 Units:
- To account for the [tex]\( +3 \)[/tex] in [tex]\( g(x) = f(-x) + 3 \)[/tex], we need to shift the y-coordinate of the reflected vertex [tex]\((-5, -6)\)[/tex] up by 3 units.
- This adjustment is done by adding 3 to the y-coordinate: [tex]\(-6 + 3 = -3\)[/tex].
Thus, the new vertex for the function [tex]\( g(x) = f(-x) + 3 \)[/tex] is at [tex]\( (-5, -3) \)[/tex].
