*Title*

Using Roots to Generate an Equation

*Problem Statement*

A parabola crosses the x-axis at x = 2 and x = -1. Find an equation of the parabola.

Find an equation of a parabola that crosses the x-axis at x = a and x = b.

*Plans to Solve/Investigate the
Problem*

Since we know two points on the x axis of the parabola, we can use the graphing calculator program to show us the equation when the points are plotted.

*Investigation/Exploration of
the Problem*

All parabolas of the form y = ax^{2}
are reflected symmetrically over the y axis.
Therefore, we know that we need a formula that will cross the x axis at
2 and -1. When using the formula (x +
1)(x – 2) = 0 where (x + 1) = A and (x – 2) = B, we know that either A or B
must be equal to zero to make the equation true. If we solve for x in x+1 = 0, we know that x
= -1 and that in x-2 = 0, x = 2. These
two points are where the parabola crosses the x axis.

In general, this formula is
(x-a)(x-b) = 0. By multiplying this with
FOIL, we find x^{2} –bx-ax+ab =0.
That means x^{2 }– bx – ax + ab = 0 and x^{2} – (a+b) x + ab =0. Therefore a= -1 and b=2. Substituting this in the formula x^{ 2}
– x – 2 = y, we can graph this and see
that the parabola intersects the x axis at -1 and 2. These points are the roots of the equation.

*Author & Contact*

Insert name and contact information.

Jill Jackson and Shirley Crawford

jjackson@rockdale.k12.ga.us and scrawford@rockdale.k12.ga.us