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

Recall 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 factored form of our function. $$\begin{array}{c}f(x)=(x-2)(x+2)\iff f(x)=1(x-2)(x-(\text{-}2))\end{array}$$

We can see that $a=1,$ $p=2,$ and $q=\text{-}2.$ Therefore, the $x\text{-}$intercepts occur at $(2,0)$ and $(\text{-}2,0).$

Find and Graph the Axis of Symmetry

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

Find and Plot the Vertex

Since the vertex lies on the axis of symmetry, its $x\text{-}$coordinate is $0.$ To find the $y\text{-}$coordinate, we will substitute $0$ for $x$ in the given equation. The $y\text{-}$coordinate of the vertex is $\text{-}4.$ Therefore, the vertex is the point $(0,\text{-}4).$

Draw the Parabola

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