Sure, let's solve this step by step.
We start with the given function:
[tex]\[ g(x) = 4x^2 - 16 \][/tex]
Our objective is to perform two transformations on [tex]\( g(x) \)[/tex]:
1. Shift it 9 units to the right.
2. Shift it 1 unit down.
Step 1: Shifting 9 units to the right
To shift the graph of a function [tex]\( g(x) \)[/tex] horizontally to the right by [tex]\( c \)[/tex] units, we replace [tex]\( x \)[/tex] with [tex]\( x - c \)[/tex]. Here, [tex]\( c = 9 \)[/tex], so we replace [tex]\( x \)[/tex] with [tex]\( x - 9 \)[/tex]:
[tex]\[ g(x - 9) = 4(x - 9)^2 - 16 \][/tex]
Step 2: Shifting 1 unit down
To shift the graph of a function vertically downward by [tex]\( c \)[/tex] units, we subtract [tex]\( c \)[/tex] from the entire function. Here, [tex]\( c = 1 \)[/tex], so we subtract 1 from [tex]\( g(x - 9) \)[/tex]:
[tex]\[ h(x) = 4(x - 9)^2 - 16 - 1 \][/tex]
Simplifying the expression:
[tex]\[ h(x) = 4(x - 9)^2 - 17 \][/tex]
Thus, the new equation after the shifts is:
[tex]\[ h(x) = 4(x - 9)^2 - 17 \][/tex]
Comparing this with the given options, the correct answer is:
C. [tex]\( h(x) = 4(x - 9)^2 - 17 \)[/tex]
