To find the new equation of the function \( H(x) = 4x^2 - 16 \) after it is shifted 7 units to the right and 3 units down, follow these steps:
1. Shift the function 7 units to the right: To shift a function horizontally to the right, replace \( x \) with \( x-7 \). Therefore,
[tex]\[
H(x-7) = 4(x-7)^2 - 16
\][/tex]
2. Shift the function 3 units down: To shift a function vertically down, subtract 3 from the entire function. Therefore,
[tex]\[
H(x-7) - 3 = 4(x-7)^2 - 16 - 3
\][/tex]
3. Simplify the new equation: Combine the constants,
[tex]\[
4(x-7)^2 - 19
\][/tex]
So, the new equation of the function after shifting it 7 units to the right and 3 units down is:
[tex]\[
\boxed{4(x-7)^2 - 19}
\][/tex]
Referencing the given multiple-choice options, the correct answer aligns with option C:
[tex]\[
C. \, H(x) = 4(x-7)^2 - 19
\][/tex]
