Interpreting Quadratic Functions in Factored Form

We want to write the equation of the quadratic function that passes through the points $(0,0),$ $(2,30),$ and $(7,0).$ Note that the $y\text{-}$coordinate of two of the points is $0.$ Therefore, we know that $0$ and $7$ are the $x\text{-}$intercepts of the parabola.

$\mathbf{x}$	$\mathbf{y}$
$0$	$0$
$2$	$30$
$7$	$0$

Consider the factored form of a quadratic function. $$\begin{array}{c}y=a(x-p)(x-q)\end{array}$$ In this form, $p$ and $q$ are the $x\text{-}$intercepts. Therefore, we know that, for our equation, it is $p$ $=$ $0$ and $q$ $=$ $7.$ $$\begin{array}{c}y=a(x-0)(x-7)\iff y=ax(x-7)\end{array}$$ Since the parabola passes through the point $(2,30),$ we can substitute $2$ for $x$ and $30$ for $y$ in the above equation, and solve for $a.$ Knowing that $a=\text{-}3,$ we can write the full equation of the parabola. $$\begin{array}{c}y=\text{-}3x(x-7)\end{array}$$