Interpreting Quadratic Functions in Factored Form

To draw the graph of the given function, we will follow four steps.

1. Identify and plot the $x\text{-}$intercepts.
2. Find and graph the axis of symmetry.
3. Find and plot the vertex.
4. Draw the parabola through the vertex and the points where the $x\text{-}$intercepts occur.

Let's go through these steps one at a time.

Identify and Plot the $x\text{-}$intercepts

Let's start by recalling the factored form of a quadratic function. $$\begin{array}{c}f(x)=a(x-p)(x-q)\end{array}$$ In this form, where $a$ $\ne$ $0,$ the $x\text{-}$intercepts are $p$ and $q.$ Let's consider the given function. $$\begin{array}{c}f(x)=\text{-}4(x-7)(x-3)\end{array}$$

We can see that $a=\text{-}4,$ $p=7,$ and $q=3.$ Therefore, the $x\text{-}$intercepts occur at $(7,0)$ and $(3,0).$

Find and Graph the Axis of Symmetry

The axis of symmetry is halfway between $(p,0)$ and $(q,0).$ Since we know that $p=7$ and $q=3,$ the axis of symmetry of our parabola is halfway between $(7,0)$ and $(3,0).$ $$\begin{array}{c}x=\frac{p+q}{2}\Rightarrow x=\frac{7+3}{2}=\frac{10}{2}=5\end{array}$$ We found that the axis of symmetry is the vertical line $x=5.$

Find and Plot the Vertex

Since the vertex lies on the axis of symmetry, its $x\text{-}$coordinate is $5.$ To find the $y\text{-}$coordinate, we will substitute $5$ for $x$ in the given equation. The $y\text{-}$coordinate of the vertex is $16.$ Therefore, the vertex is the point $(5,16).$

Draw the Parabola

Finally, we will draw the parabola as a curve passing through the vertex and the $x\text{-}$intercepts.