To determine the transformations necessary to change the parent function [tex]\( f(x) = |x| \)[/tex] to the new function [tex]\( f(x) = -|x| - 5 \)[/tex], we can break down the transformations step by step:
1. Reflection over the [tex]\( x \)[/tex]-axis:
- The function [tex]\( f(x) = |x| \)[/tex] becomes [tex]\( f(x) = -|x| \)[/tex] when it is reflected over the [tex]\( x \)[/tex]-axis. This is because applying a negative sign to the entire function reflects it over the horizontal axis. Graphically, each point [tex]\( (x, y) \)[/tex] on the original graph is mirrored to [tex]\( (x, -y) \)[/tex] on the new graph.
[tex]\[
f(x) = |x| \quad \rightarrow \quad f(x) = -|x|
\][/tex]
2. Shift down 5 units:
- The function [tex]\( f(x) = -|x| \)[/tex] becomes [tex]\( f(x) = -|x| - 5 \)[/tex] when it is shifted downwards by 5 units. This transformation involves subtracting 5 from the entire function, effectively moving each point on the graph down by 5 units.
[tex]\[
f(x) = -|x| \quad \rightarrow \quad f(x) = -|x| - 5
\][/tex]
Thus, the transformations needed are:
1. Reflection over the [tex]\( x \)[/tex]-axis
2. Shift down 5 units
Among the given multiple choice options, the correct answer is:
Reflection over the [tex]\( x \)[/tex]-axis, shift down 5 units
Therefore, the correct choice is:
3. Reflection over the [tex]\( x \)[/tex]-axis, shift down 5 units
