Question

Function
\[g\] can be thought of as a translated (shifted) version of
\[f(x)=x^2\].
A parabola labeled f has a vertex at the point 0, 0. A parabola labeled g has a vertex at the point 3, negative 4.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[\blueD{f}\]
\[\maroonD{g}\]
Write the equation for
\[g(x)\].
\[g(x)=\]

Answer 1

To find the equation for the function \( g(x) \), which is a translated version of the function \( f(x) = x^2 \), we need to determine the effects of the shift on the vertex.

The vertex of the function \( f(x) = x^2 \) is at the point (0, 0). The vertex of the function \( g \) is at the point (3, -4). To translate \( f(x) \) to obtain \( g(x) \), we follow these steps:

1. **Horizontal Shift**: The vertex shifts horizontally from 0 to 3, which means we will replace \( x \) with \( (x - 3) \) in the function \( f(x) \). This accounts for the rightward shift by 3 units.

2. **Vertical Shift**: The vertex also shifts vertically from 0 to -4. To apply this vertical shift, we will subtract 4 from the entire function.

Putting it all together, we create the function \( g(x) \):

\[
g(x) = (x - 3)^2 - 4
\]

Thus, the equation for \( g(x) \) is

\[
g(x) = (x - 3)^2 - 4
\]