To determine the transformations needed to convert the parent function [tex]\( f(x) = |x| \)[/tex] to the function [tex]\( f(x) = -|x| + 5 \)[/tex], we need to analyze each component of the transformation. Let's break down the process step by step.
1. Understand the Parent Function:
- The parent function [tex]\( f(x) = |x| \)[/tex] has a V-shaped graph centered at the origin (0, 0). It is symmetrical about the y-axis, and it increases to the left and right of the origin.
2. Transformation Components:
a. Reflection Over the x-axis:
- Reflecting the function [tex]\( f(x) = |x| \)[/tex] over the x-axis will change the function to [tex]\( f(x) = -|x| \)[/tex].
- This inversion takes all positive values of [tex]\( |x| \)[/tex] and makes them negative, effectively flipping the graph upside down.
b. Vertical Shift:
- Adding a constant value to a function translates the entire graph vertically.
- In the given function [tex]\( f(x) = -|x| + 5 \)[/tex], adding the constant 5 to [tex]\( -|x| \)[/tex] shifts the graph upward by 5 units.
3. Resulting Graph:
- The combined transformation first reflects the graph of [tex]\( f(x) = |x| \)[/tex] over the x-axis to get [tex]\( f(x) = -|x| \)[/tex].
- Then, the graph of [tex]\( f(x) = -|x| \)[/tex] is shifted upward by 5 units to obtain [tex]\( f(x) = -|x| + 5 \)[/tex].
4. Summary of Transformations:
- Reflection over the x-axis
- Shift up 5 units
Thus, the correct transformations of the parent function [tex]\( f(x) = |x| \)[/tex] to obtain [tex]\( f(x) = -|x| + 5 \)[/tex] are a reflection over the x-axis and a shift up 5 units.
