Question

the function h is defined as follows.
( h(x) = -4x^2 - 1 )
if the graph of h is translated vertically upward by 3 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 upward by \( k \) units gives \( y = h(x)+k \). Here, \( k = 3 \) and \( h(x)=-4x^{2}-1 \).

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

To find \( f(x) \), we add 3 to \( h(x) \). So \( f(x)=h(x)+3 \). Substitute \( h(x)=-4x^{2}-1 \) into this: \( f(x)=(-4x^{2}-1)+3 \).

Step3: Simplify the expression

Simplify \( (-4x^{2}-1)+3 \) by combining like terms. \( -1 + 3=2 \), so \( f(x)=-4x^{2}+2 \).

Answer:

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