Question

write a quadratic function in standard form whose graph is shown. the function is f(x) = □.

Explanation:

Step1: Write the factored - form of quadratic function

Since the x - intercepts are x = - 2 and x = 6, the factored form of the quadratic function is \(f(x)=a(x + 2)(x - 6)\).

Step2: Substitute the vertex coordinates

The vertex of the parabola is (4,-3). Substitute x = 4 and y=-3 into \(f(x)=a(x + 2)(x - 6)\).
\(-3=a(4 + 2)(4 - 6)\)
\(-3=a(6)(-2)\)
\(-3=-12a\)

Step3: Solve for a

Divide both sides of the equation \(-3=-12a\) by - 12.
\(a=\frac{-3}{-12}=\frac{1}{4}\)

Step4: Expand the factored - form to standard form

\(f(x)=\frac{1}{4}(x + 2)(x - 6)\)
\(f(x)=\frac{1}{4}(x^{2}-6x + 2x-12)\)
\(f(x)=\frac{1}{4}(x^{2}-4x - 12)\)
\(f(x)=\frac{1}{4}x^{2}-x - 3\)

Answer:

\(f(x)=\frac{1}{4}x^{2}-x - 3\)