Question

the graph shows g(x), which is a translation of f(x) = x². 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: Recall Vertex Form of Quadratic

The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the vertical stretch/compression and direction. For \( f(x) = x^2 \), the vertex is \((0, 0)\) and \(a = 1\).

Step2: Identify Vertex of \( g(x) \)

From the graph, the vertex of \( g(x) \) appears to be \((0, 8)\) (since it's a vertical translation of \( f(x) = x^2 \) up 8 units, and no horizontal shift, so \( h = 0 \), \( k = 8 \)). The value of \(a\) remains 1 because there's no vertical stretch or compression (the parabola opens upwards with the same width as \( y = x^2 \)).

Step3: Substitute into Vertex Form

Substitute \( a = 1 \), \( h = 0 \), and \( k = 8 \) into \( g(x) = a(x - h)^2 + k \). This gives \( g(x) = 1(x - 0)^2 + 8 \), which simplifies to \( g(x) = x^2 + 8 \).

Answer:

\( g(x) = x^2 + 8 \) (or in the form \( 1(x - 0)^2 + 8 \))