Using Roots to Generate an Equation
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