Question

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

Explanation:

Step1: Recall vertical translation rule

For a function \( y = f(x) \), a vertical translation upward by \( k \) units gives \( y = f(x)+k \).
Here, \( f(x)=-3x^{2}+3 \) and \( k = 3 \).

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

To find \( g(x) \), we add 3 to \( f(x) \). So \( g(x)=f(x)+3 \).
Substitute \( f(x)=-3x^{2}+3 \) into the equation:
\( g(x)=(-3x^{2}+3)+3 \)

Step3: Simplify the expression

Simplify \( (-3x^{2}+3)+3 \) by combining like terms.
\( -3x^{2}+(3 + 3)=-3x^{2}+6 \)

Answer:

\( g(x)=-3x^{2}+6 \)