write an equation for the quadratic function $f$ whose graph is shown below.\n$f(x) = \\square$

Explanation:

Step1: Use factored form of quadratic

$ f(x) = a(x - r_1)(x - r_2) $, where $r_1=-9$, $r_2=2$ $ f(x) = a(x + 9)(x - 2) $

Step2: Solve for coefficient $a$

Substitute $(3,2)$ into the equation: $ 2 = a(3 + 9)(3 - 2) $ $ 2 = a(12)(1) $ $ a = \frac{2}{12} = \frac{1}{6} $

Step3: Expand to standard form

Substitute $a=\frac{1}{6}$ and expand: $ f(x) = \frac{1}{6}(x + 9)(x - 2) $ $ f(x) = \frac{1}{6}(x^2 + 7x - 18) $ $ f(x) = \frac{1}{6}x^2 + \frac{7}{6}x - 3 $

Answer:

$ f(x) = \frac{1}{6}(x + 9)(x - 2) $ or $ f(x) = \frac{1}{6}x^2 + \frac{7}{6}x - 3 $