To find the expression for the function [tex]\( h(x) \)[/tex] after translating the graph of [tex]\( f(x) \)[/tex] vertically downward by 3 units, follow these steps:
1. Understand the translation: Translating a graph vertically downward by 3 units means you subtract 3 from the function's output.
2. Original function: The original function is given by
[tex]\[
f(x) = x^2 + 8
\][/tex]
3. Applying the translation: To translate [tex]\( f(x) \)[/tex] vertically downward by 3 units, you need to subtract 3 from [tex]\( f(x) \)[/tex]. Therefore, the new function [tex]\( h(x) \)[/tex] will be:
[tex]\[
h(x) = f(x) - 3
\][/tex]
4. Substitute [tex]\( f(x) \)[/tex]: Replace [tex]\( f(x) \)[/tex] in the equation with its given expression:
[tex]\[
h(x) = (x^2 + 8) - 3
\][/tex]
5. Simplify the expression: Simplify the right-hand side of the equation:
[tex]\[
h(x) = x^2 + 8 - 3
\][/tex]
[tex]\[
h(x) = x^2 + 5
\][/tex]
So, the expression for the function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = x^2 + 5
\][/tex]
