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 = ax2 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 x2 –bx-ax+ab =0.  That means x2 – bx – ax + ab = 0 and  x2 – (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