Given three points, we are able to find the equation of the parabola that passes through those points. This will help us find the equation of the parabola that passes through all of our vertices.
Let's look at our given vertex points (0,1), (-1,0) and (1,0)
We know the general equation of a parabola is
By substituting our points in for x and y, we get a system of three equations that need to be solved:
1.
2.
3.
Right away we can see that c=1 from the first equation. By solving our second equation for a, we have:
And since we know c=1,
Now, lets solve our third equation for b.
And since we know c=1,
Finally, let's use these solutions to solve for both a and b.
And therefore, by substitution,
So we have our three values
for a, b, and c.
We can plug these values into the parabola equation, in order the get the equation of the parabola that joins our vertex points.
So, the resulting equation is:
