Question

find the equation of the quadratic function g whose graph is shown below.
\\(g(x)=\square\\)

Explanation:

Step1: Identify vertex form

The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the graph, the vertex is \((5, -3)\), so \( h = 5 \) and \( k = -3 \). Substituting these into the formula, we get \( g(x) = a(x - 5)^2 - 3 \).

Step2: Find the value of \( a \)

We know the point \((8, 6)\) lies on the graph, so substitute \( x = 8 \) and \( g(x) = 6 \) into the equation:
\[
6 = a(8 - 5)^2 - 3
\]
Simplify the equation:
\[
6 = a(3)^2 - 3 \\
6 = 9a - 3
\]
Add 3 to both sides:
\[
9 = 9a
\]
Divide both sides by 9:
\[
a = 1
\]

Step3: Write the final equation

Substitute \( a = 1 \), \( h = 5 \), and \( k = -3 \) back into the vertex form:
\[
g(x) = (x - 5)^2 - 3
\]
We can expand this to standard form if needed, but the vertex form is also acceptable. Expanding \((x - 5)^2 - 3\):
\[
(x^2 - 10x + 25) - 3 = x^2 - 10x + 22
\]

Answer:

\( g(x) = (x - 5)^2 - 3 \) (or \( g(x) = x^2 - 10x + 22 \))