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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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