Question

write an equation for the quadratic function $f$ whose graph is shown below.

Explanation:

Step1: Use factored quadratic form

The quadratic has x-intercepts at $x=-9$ and $x=1$, so the function is $f(x)=a(x+9)(x-1)$ where $a$ is a constant.

Step2: Solve for $a$ with $(3,4)$

Substitute $x=3$, $f(x)=4$ into the equation:
$4=a(3+9)(3-1)$
Simplify the right-hand side:
$4=a(12)(2)$
$4=24a$
Solve for $a$:
$a=\frac{4}{24}=\frac{1}{6}$

Step3: Expand to standard form (optional, can keep factored)

Substitute $a=\frac{1}{6}$ back into the factored form and expand:
$f(x)=\frac{1}{6}(x+9)(x-1)$
$f(x)=\frac{1}{6}(x^2 -x +9x -9)$
$f(x)=\frac{1}{6}(x^2 +8x -9)$
$f(x)=\frac{1}{6}x^2 + \frac{4}{3}x - \frac{3}{2}$

Answer:

$f(x)=\frac{1}{6}(x+9)(x-1)$ or $f(x)=\frac{1}{6}x^2 + \frac{4}{3}x - \frac{3}{2}$