Interpreting Quadratic Functions in Factored Form

We want to write the equation of the parabola that passes through the point $(6,5)$ and has $x\text{-}$intercepts $1$ and $7.$

To do so, we will use the intercept 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 can partially write our equation. $$\begin{array}{c}y=a(x-1)(x-7)\end{array}$$ Since the parabola passes through the point $(6,5),$ we can substitute $6$ for $x$ and $5$ for $y$ in the above equation, and solve for $a.$ Knowing that $a=\text{-}1,$ we can write the full equation of the parabola. $$\begin{array}{c}y=\text{-}1(x-1)(x-7)\iff y=\text{-}(x-1)(x-7)\end{array}$$