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 form of a parabola

The vertex form of a parabola is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the parent function \( f(x) = x^2 \), the vertex is at \((0, 0)\).

Step2: Determine the vertex of \( g(x) \)

From the graph, the vertex of \( g(x) \) is at \((0, -3)\) (since the minimum point of the parabola is on the y - axis at \( y=-3 \)). So \( h = 0 \) and \( k=-3 \).

Step3: Determine the value of \( a \)

Since \( g(x) \) is a translation of \( f(x)=x^2 \) and there is no vertical stretch or compression (the shape of the parabola is the same as \( y = x^2 \)), \( a = 1 \).

Step4: Substitute \( a \), \( h \), and \( k \) into the vertex form

Substitute \( a = 1 \), \( h = 0 \), and \( k=-3 \) into \( g(x)=a(x - h)^2 + k \). We get \( g(x)=1(x - 0)^2-3 \), which simplifies to \( g(x)=x^2 - 3 \).

Answer:

\( g(x)=x^{2}-3 \) (or in the form \( 1(x - 0)^{2}-3 \))