__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) **= ax ^{2} + 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 = x^{2}
+ 4x - 5

Solution: Set y equal to 0, to find the
x-intercepts. Now you have x^{2} + 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.

ax^{2} + 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.

Click
here to go to the next page of graphing parabolas.

Click here
to go to sketching parabolas.

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to the index.