Question

the function f is defined as follows.
$f(x) = -4x^2 + 7$
if the graph of f is translated vertically downward by 3 units, it becomes the graph of a function h.
find the expression for $h(x)$
note that the aleks graphing calculator may be helpful in checking your answer.
$h(x) = \square$

Explanation:

Step1: Recall vertical translation rule

For a function \( y = f(x) \), a vertical translation downward by \( k \) units (where \( k>0 \)) gives \( y = f(x)-k \). Here, \( k = 3 \) and \( f(x)=-4x^{2}+7 \).

Step2: Apply the translation

Substitute \( f(x) \) into the translation formula: \( h(x)=f(x)-3 \).
Substitute \( f(x)=-4x^{2}+7 \) into the equation: \( h(x)=(-4x^{2}+7)-3 \).
Simplify the expression: \( h(x)=-4x^{2}+(7 - 3)=-4x^{2}+4 \).

Answer:

\( h(x)=-4x^{2}+4 \)