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:

Identify the vertex of the translated parabola

The graph shows a parabola \(g(x)\) which is a translation of the parent function \(f(x) = x^2\).
By observing the graph, the lowest point (vertex) of the parabola \(g(x)\) is located at \((0, 6)\).

Determine the horizontal and vertical shifts

The parent function \(f(x) = x^2\) has its vertex at \((0, 0)\).
The vertex of \(g(x)\) is at \((0, 6)\).
This represents:

• A horizontal shift \(h = 0\).
• A vertical shift \(k = 6\).

Determine the vertical stretch or compression factor

The general vertex form of a quadratic function is:

$$g(x) = a(x - h)^2 + k$$

Substituting \(h = 0\) and \(k = 6\):

$$g(x) = a(x - 0)^2 + 6 = ax^2 + 6$$

We can find \(a\) by identifying another clear point on the graph.
The graph passes through the point \((1, 7)\) and \((-1, 7)\).
Substitute \((1, 7)\) into the equation:

$$7 = a(1)^2 + 6$$

$$7 = a + 6$$

$$a = 1$$

Write the final function rule

Using \(a = 1\), \(h = 0\), and \(k = 6\), we write the function in the requested form \(a(x - h)^2 + k\):

$$g(x) = 1(x - 0)^2 + 6$$

Simplifying the expression inside the parentheses gives:

$$g(x) = x^2 + 6$$

Answer:

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.

\(g(x) =\) <blank>\(x^2 + 6\)</blank>