Question

the graph shows g(x), which is a translation of $f(x) = x^2$. write the function rule for g(x).
write your answer in the form $a(x - h)^2 + k$, where a, h, and k are integers or simplified fractions.
$g(x) = $

Explanation:

Step1: Identify the vertex of \( g(x) \)

The parent function \( f(x) = x^2 \) has a vertex at \( (0,0) \). From the graph, the vertex of \( g(x) \) is at \( (-9, 0) \)? Wait, no, looking at the grid, let's check the vertex. Wait, the graph of \( g(x) \) seems to have its vertex at \( (-9, 0) \)? Wait, no, let's look again. Wait, the x - axis: the vertex is at \( x=-9 \), \( y = 0 \)? Wait, no, maybe I misread. Wait, the grid: each square is 1 unit. Let's see, the parabola opens upwards, and the vertex is at \( (-9, 0) \)? Wait, no, maybe \( (-9, 0) \)? Wait, no, let's check the translation. The general form of a translated parabola is \( g(x)=a(x - h)^2 + k \), where \( (h,k) \) is the vertex. For \( f(x)=x^2 \), \( a = 1 \), vertex \( (0,0) \). From the graph, the vertex of \( g(x) \) is at \( (-9, 0) \)? Wait, no, maybe \( (-9, 0) \)? Wait, no, let's see the graph: the vertex is at \( (-9, 0) \)? Wait, no, maybe I made a mistake. Wait, the graph is shifted 9 units to the left. So \( h=-9 \), \( k = 0 \), and \( a = 1 \) (since it's a translation, no vertical stretch or compression). So the function is \( g(x)=(x - (-9))^2+0=(x + 9)^2 \). Wait, let's verify. If \( x=-9 \), \( g(-9)=0 \), which matches the vertex. And it's a translation of \( f(x)=x^2 \), so \( a = 1 \).

Step2: Write the function using the vertex form

The vertex form of a quadratic function is \( g(x)=a(x - h)^2 + k \), where \( (h,k) \) is the vertex and \( a \) is the vertical stretch/compression factor. Since it's a translation (no stretch/compression), \( a = 1 \). The vertex \( (h,k) \) of \( g(x) \) is \( (-9, 0) \), so \( h=-9 \), \( k = 0 \). Substituting into the vertex form: \( g(x)=1\times(x - (-9))^2+0=(x + 9)^2 \). Wait, but let's check another point. Let's take a point on \( g(x) \), say when \( x=-6 \), what's \( y \)? If \( g(x)=(x + 9)^2 \), then when \( x=-6 \), \( g(-6)=(-6 + 9)^2=3^2 = 9 \), but looking at the graph, when \( x=-6 \), the \( y \) - value seems to be 9? Wait, the graph at \( x=-6 \) is at \( y = 9 \), which matches \( (x + 9)^2 \) when \( x=-6 \), \( (-6 + 9)^2=9 \). So that works.

Answer:

\( g(x)=(x + 9)^2 \) (or in the form \( a(x - h)^2 + k \), it's \( 1(x - (-9))^2+0 \), so \( (x + 9)^2 \))