Sure, let's break down the transformations applied to the function [tex]\( f(x) = -2(x - 1)^2 + 3 \)[/tex]:
1. Horizontal Shift:
- The term [tex]\( (x - 1) \)[/tex] inside the squared term indicates a horizontal shift.
- Specifically, [tex]\( (x - 1) \)[/tex] shifts the graph of the basic [tex]\( x^2 \)[/tex] function to the right by 1 unit.
2. Reflection and Vertical Stretch:
- The coefficient [tex]\(-2\)[/tex] outside the squared term affects the graph in two ways:
- Reflection: The negative sign in [tex]\(-2\)[/tex] reflects the graph over the x-axis.
- Vertical Stretch: The factor 2 stretches the graph vertically by a factor of 2.
3. Vertical Shift:
- The constant term [tex]\( +3 \)[/tex] at the end shifts the graph vertically.
- Specifically, the [tex]\( +3 \)[/tex] shifts the graph upwards by 3 units.
To summarize, the transformations applied to the function [tex]\( f(x) = -2(x - 1)^2 + 3 \)[/tex] are:
1. A horizontal shift to the right by 1 unit.
2. A reflection over the x-axis and a vertical stretch by a factor of 2.
3. A vertical shift upwards by 3 units.
