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 the vertex form of a parabola

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

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

From the graph, the vertex of \(g(x)\) is at \((0, 7)\) (since the parabola has its minimum point on the y - axis at \(y = 7\)). So, \(h = 0\) and \(k=7\).

Step3: Determine the value of \(a\)

Since \(g(x)\) is a translation of \(f(x)=x^2\) (no vertical stretch or compression, just a vertical translation), the value of \(a\) remains \(1\) (because the shape of the parabola is the same as \(y = x^2\), it opens upwards with the same width).

Step4: Write the function rule for \(g(x)\)

Substitute \(a = 1\), \(h = 0\), and \(k = 7\) into the vertex form \(g(x)=a(x - h)^2 + k\). We get \(g(x)=1(x - 0)^2+7\), which simplifies to \(g(x)=x^2 + 7\).

Answer:

\(g(x)=x^{2}+7\) (or in the form \(1(x - 0)^{2}+7\))