Graphing Parabolas

About now, some of you may be wondering, what's a parabola?

A parabola is: a graph of a quadratic equation (function) in the form
y (or f(x) in a function, but I am using y) = ax2 + bx + c, where the domain for x is all real numbers, and a ≠ 0.

A domain is the replacement set for the variable in question.

Now a parabola will look like a U, either right-side-up or upside-down.  Although it is curved, there will be a vertex.  This is going to be either the minimum point, or the maximum one.

This is an example:

Since you have solved quadratic equations before, you know that when you solve for x (and y = 0), you get two real number solutions, no real number solutions, or one real number solution.  The solutions to that equation are the x-intercepts of the equation's graph (because y = 0).  If you changed y from 0 to other things, you would eventually get infinitely many two x-coordinate points, and just one one x-coordinate point (the vertex, because the x coordinates are the SAME).  If you got one solution, consider yourself lucky, because you already found the vertex.  For any other value of y besides that of the vertex, you will have 0 or 2 solutions.

For an example, let's find the x-intercepts of y = x2 + 4x - 5
Solution:  Set y equal to 0, to find the x-intercepts.  Now you have x2 + 4x - 5 = 0.  The trinomial can be factored into (x + 5)(x - 1).  Therefore the solutions are x = -5 and 1.  This are the x-intercepts.

However, one goes about doing this slightly differently when graphing a quadratic equation than when solving one.  It is actually easier.  The x-coordinate of the vertex is going to be the average of two x-coordinates with the same y-coordinate, because the U is symmetrical, with the line of symmetry being the line with the x-coordinate of the vertex.  Find the x-coordinate of the vertex of the equation above.
(-5 + 1)/2 = -4/2 = -2

Now that we have the x-coordinate, we have to find y.  To do this, substitute the value you got for x into the equation to get y.  So now find y, and then list your vertex's coordinates.
(-2)2 + 4(-2) - 5 = y
4 - 8 - 5 = y
-9 = y
(-2, -9)

But we don't always want to do that rigorous work to find the vertex, nor the x-coordinates of any other point.  So the formula to find the vertex of a quadratic equation is: -b/2a

Why?  Because of this:
Take one of the most basic forms of a quadratic equation.
ax2 + bx = 0
x(ax + b) = 0
x = 0 or ax + b = 0
x = 0 or x = -b/a
The two x intercepts are 0 and -b/a.
Find the average of them: (0 + -b/a)/2 = (-b/a)/2 = -b/2a

Isn't it cool how it all works?

Once you've found that, just find the y-coordinate the way we did before.
So you can find the vertex of a quadratic equation.  So what?  Can you graph the rest?  Go to the next page to learn how.