Question

the function g is defined as follows.
$g(x) = -2x^2 + 4$
if the graph of g is translated vertically downward by 4 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

A vertical translation downward by \( k \) units of a function \( y = g(x) \) gives \( y = g(x)-k \). Here, \( k = 4 \).

Step2: Apply the translation to \( g(x) \)

Given \( g(x)=-2x^{2}+4 \), translating it downward by 4 units means \( f(x)=g(x)-4 \).
Substitute \( g(x) \) into the equation: \( f(x)=(-2x^{2}+4)-4 \).

Step3: Simplify the expression

Simplify \( (-2x^{2}+4)-4 \): \( -2x^{2}+4 - 4=-2x^{2} \).

Answer:

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