What happens to the function f(x)=a(x-p)(x-q) when x=p or x=q?
See solution.
Practice makes perfect
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 ⇒ a( p-p)_0( p-q) = 0 [2em]
x = q ⇒ a( q-p)( q-q)_0 = 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 = p+q2.
Furthermore, since the axis of symmetry passes through the vertex, we know that the x-coordinate of the vertex is p+q2 as well. Finally, we can find the y-coordinate of the vertex by evaluating the function at p+q2.
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= p+q2.
Its vertex is located at the point ( p+q2 , f( p+q2 )).
If a>0, its graph opens up and it has a minimum at f( p+q2 ).
If a<0, its graph opens down and it has a maximum at f( p+q2 ).
