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### Understanding Parabolas

```
Date: 02/04/97 at 16:15:12
From: ALICE
Subject: Slope of a Line

For a parabola with equation y = ax^2 + bx + c, if a is positive, the
parabola opens upward and if a is negative, the parabola opens
downward.

Is this true?  Can you help me to understand?
```

```
Date: 02/04/97 at 19:48:28
From: Doctor Keith
Subject: Re: Slope of a Line

Hi Alice,

Good question!  Lets go through this step by step.  I will try to show
you everything I think you will need to know, but if something is
unclear, please write back and mention where the confusion lies and I
will be happy to explain further.

I used to find trying to memorize what the sign of the first term did
to the parabola a pain till I learned a handy little trick.  We have
an equation:

y = ax^2 + bx + c

We want to know if the the parabola opens up (looks like a bowl) or
opens down (looks like an umbrella) by just looking at the sign of the
leading term (which is a).  Now the fact that this works is best
proven in later math classes (calculus), but we can use the following
trick to see what is going to happen.  Thus you can easily figure it
out any time you need to. This is the trick:

1) Since b and c don't matter, let them be zero

2) Since only the sign of a matters, let a = 1 or a = -1

3) Consider what happens to y as x becomes a large number (either
positive or negative)

4) If y becomes more positive as x becomes large (both + and -), then
the parabola is opening up (in both directions on your line the
parabola is increasing). If y becomes more negative as x becomes
large (both + and -), then the parabola is opening down

Example: y = x^2

Consider positive x:

x  |  1  |  2  |  3  |  4
------------------------------
y  |  1  |  4  |  9  |  16

Consider negative x:

x  |  -1  |  -2  |  -3  |  -4
---------------------------------
y  |   1  |   4  |   9  |  16

You can see in both cases that the y values become more positive as
the x values increase in magnitude (the number regardless of the
sign), so the parabola opens up.

For y = -x^2, consider positive x:

x  |   1  |   2  |   3  |   4
-----------------------------------
y  |  -1  |  -4  |  -9  |  -16

Consider negative x:

x  |  -1  |  -2  |  -3  |  -4
-----------------------------------
y  |  -1  |  -4  |  -9  |  -16

You can see in both cases that the y values become more negative as
the x values increase, so the parabola opens down.

Thus you can see that given the sign of a determines whether the
parabola opens up or down.  You can even get a feeling that only the
sign of a matters since:

->  a parabola with a not equal to 1 or -1 will just be a
scaled version of the example, which will still grow
faster as x gets larger (+ or -)

->  a parabola with b and c not zero will be a parabola
that has a line added, but since the squared term grows
much faster than the linear term, the squared term will
dominate (as x gets really big only x^2 will matter), so
the addition of the b and c will only change where the
parabola is centered, not the direction it opens.

So to summarize:

a>0   =>   parabola opens up (concave up, like a bowl)

a<0   =>   parabola opens down (concave down, like an umbrella)

If you do want to try to memorize the rules, you can think of a
parabola that opens upwards as a smile (it does kind of look like
one).  Since smiles are *positive*, a positive value for a means that
the parabola opens upwards.  Similarly, you can think of a parabola
that opens down as a frown.  Since frowns are *negative*, a negative
value for a means that the parabola opens downwards.

I hope all this helps you see what is going on.  Remember, if you want
to test a rule you can always quickly sketch an example to see if it
makes sense.  While this does not prove it to be true, it can often
help to see what is going on.

-Doctor Keith,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Equations

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