To solve the problem of transforming the parent function [tex]\( y = -5x - 8 \)[/tex] as described, we'll go through the required steps one by one:
1. Reflect the function over the x-axis: This will negate the [tex]\( y \)[/tex]-values of the function. If the original function is [tex]\( y = -5x - 8 \)[/tex], after reflecting it over the x-axis, every value of [tex]\( y \)[/tex] will be multiplied by -1. So, the function [tex]\( y = -5x - 8 \)[/tex] becomes [tex]\( y = 5x + 8 \)[/tex].
2. Shift the function down by 2 units: Shifting the function down means we subtract 2 from the [tex]\( y \)[/tex]-value of the function after reflection. The function after reflection is [tex]\( y = 5x + 8 \)[/tex]. Shifting this function down by 2 units, we get:
[tex]\[ y = 5x + 8 - 2 \][/tex]
Simplifying this, we have:
[tex]\[ y = 5x + 6 \][/tex]
Therefore, the final transformed function is:
[tex]\[ y = 5x + 6 \][/tex]
None of the given options [tex]\( f_2(x) = 5x - 8 \)[/tex], [tex]\( f_2(x) = -5x - 8 \)[/tex], [tex]\( f_2(x) = -5x + 8 \)[/tex], or [tex]\( f_2(x) = 5x + 10 \)[/tex] matches our result.
Hence, the correct answer is none of the other answers provided.
The transformed function after reflecting over the x-axis and shifting down by 2 units is:
[tex]\[ \boxed{y = 5x + 6} \][/tex]
