In this exercise we are considering a quadratic function where the polynomial that defines it is in factored form,

a (x - p) (x - q) .

Notice that when

x = p

or

x = q,

one of its factors becomes zero.

x = p \Rightarrow a 0 (p - p) (p - q) = 0 x = q \Rightarrow a (q - p) 0 (q - q) = 0

By the Zero Product Property we know that the whole function becomes zero for those cases. Therefore,

x = p

and

x = q

are the zeros of the function, which also means they are the

x - intercepts

of the graph.

Recall that the axis of symmetry is halfway between the $x - intercepts .$ Therefore, we can write its equation as $x = \frac{p + q}{2} .$

Furthermore, since the axis of symmetry passes through the vertex, we know that the $x - coordinate$ of the vertex is $\frac{p + q}{2}$ as well. Finally, we can find the $y - coordinate$ of the vertex by evaluating the function at $\frac{p + q}{2} .$

As we can see in the graph above, the $y - value$ of the vertex is also the maximum or minimum value of the function. If $a > 0$ the parabola opens up and we have a minimum. If $a < 0,$ it opens down and has a maximum instead.

Summary

For any quadratic function of the form $f (x) = a (x - p) (x - q),$ its graph has the characteristics mentioned below.

Its $x - intercepts$ are $p$ and $q,$ which are also the zeros of the function.
Its axis of symmetry is the line $x = \frac{p + q}{2} .$
Its vertex is located at the point $(\frac{p + q}{2}, f (\frac{p + q}{2})) .$
If $a > 0,$ its graph opens up and it has a minimum at $f (\frac{p + q}{2}) .$
If $a < 0,$ its graph opens down and it has a maximum at $f (\frac{p + q}{2}) .$

Exercise