Question

the function h is defined as follows.
( h(x) = 4x^2 + 7 )
if the graph of h is translated vertically downward by 9 units, it becomes the graph of a function f.
find the expression for ( f(x) ).
note that the aleks graphing calculator may be helpful in checking your answer.

Explanation:

Step1: Recall vertical translation rule

For a function \( y = h(x) \), a vertical translation downward by \( k \) units gives \( y = h(x)-k \). Here, \( k = 9 \) and \( h(x)=4x^{2}+7 \).

Step2: Apply the translation

Substitute into the rule: \( f(x)=h(x)-9=(4x^{2}+7)-9 \).

Step3: Simplify the expression

Simplify \( (4x^{2}+7)-9 \) to get \( 4x^{2}+7 - 9=4x^{2}-2 \).

Answer:

\( f(x)=4x^{2}-2 \)