Question

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

Explanation:

Step1: Identify the vertex form of a quadratic function

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

Step2: Find the value of \( a \)

We know that the graph passes through the point \((5, -6)\). Substitute \( x = 5 \) and \( f(x) = -6 \) into the equation \( f(x) = a(x - 3)^2 + 2 \):

$$ -6 = a(5 - 3)^2 + 2 $$

Simplify the equation:

$$ -6 = a(2)^2 + 2 \\ -6 = 4a + 2 $$

Subtract 2 from both sides:

$$ -6 - 2 = 4a \\ -8 = 4a $$

Divide both sides by 4:

$$ a = \frac{-8}{4} = -2 $$

Step3: Write the final equation

Substitute \( a = -2 \), \( h = 3 \), and \( k = 2 \) back into the vertex form:

$$ f(x) = -2(x - 3)^2 + 2 $$

We can also expand this to standard form to check:

$$ f(x) = -2(x^2 - 6x + 9) + 2 \\ f(x) = -2x^2 + 12x - 18 + 2 \\ f(x) = -2x^2 + 12x - 16 $$

Answer:

\( f(x) = -2(x - 3)^2 + 2 \) (or \( f(x) = -2x^2 + 12x - 16 \))