To determine the new equation after shifting the function [tex]\( g(x) = 4x^2 - 16 \)[/tex] 7 units to the right and 3 units down, we follow these steps:
1. Shift 7 units to the right:
- Shifting to the right involves replacing [tex]\( x \)[/tex] with [tex]\( x - a \)[/tex] in the original function, where [tex]\( a \)[/tex] is the number of units shifted.
- Therefore, for a shift of 7 units to the right, we replace [tex]\( x \)[/tex] with [tex]\( x - 7 \)[/tex]:
[tex]\[
g(x - 7) = 4(x - 7)^2 - 16
\][/tex]
2. Expand and simplify the new equation:
- First, compute [tex]\( (x - 7)^2 \)[/tex]:
[tex]\[
(x - 7)^2 = x^2 - 14x + 49
\][/tex]
- Substituting back into the function:
[tex]\[
4(x - 7)^2 - 16 = 4(x^2 - 14x + 49) - 16
\][/tex]
- Distribute the 4 and simplify:
[tex]\[
= 4x^2 - 56x + 196 - 16 = 4x^2 - 56x + 180
\][/tex]
3. Shift 3 units down:
- Shifting a function down involves subtracting the number of units from the entire equation.
- For a shift of 3 units down, we subtract 3 from [tex]\( 4x^2 - 56x + 180 \)[/tex]:
[tex]\[
4x^2 - 56x + 180 - 3 = 4x^2 - 56x + 177
\][/tex]
4. Express the new function:
- The new function after both shifts (7 units to the right and 3 units down) is:
[tex]\[
h(x) = 4(x - 7)^2 - 19
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{A. \, h(x) = 4(x - 7)^2 - 19} \][/tex]
