Function

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 
$$g(x)$$.
$$g(x)=$$

GPT-4o mini · Answer

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 $$ g(x) = (x - 3)^2 - 4 $$