Graphing Quadratic Polynomials

Date: 02/07/2001 at 17:58:35
From: Amy Gregory
Subject: Quadratic Function

I've been working on questions like y = x^2 - 8x + 15. I have to put 
the equation on some kind of graph, but I don't know how. Can you 
please help me?

Thank you very much,
Amy

Date: 02/08/2001 at 02:20:44
From: Doctor Ian
Subject: Re: Quadratic Function

Hi Amy,

When you want to graph a function like

     y = x^2 - 8x + 15

there are two ways to do it: the hard way, and the easy way. 

The hard way is to start picking values for x, and computing 
corresponding values for y:

     x = 0, y = 0^2 - 8(0) + 15 = 15

     x = 1, y = 1^2 - 8(1) + 15 =  8

     x = 2, y = 2^2 - 8(2) + 15 =  3

and so on. The problem with this is that it can take forever, and 
sometimes the function wiggles in between the points you choose, so 
you end up drawing the wrong graph. 

The easy way is to factor the equation:

     y = x^2 - 8x + 15

       = (x - a)(x - b)             where a + b = 8, ab = 15

       = (x - 3)(x - 5)

What does this tell us? Well, for starters, it tells us that y must 
be zero whenever x = 3 or x = 5. (Do you see why?) So we have two 
interesting points right away: (3,0) and (5,0).

A quadratic function like this is always shaped like a parabola. The 
big question is: Does it open toward the top of the graph, or toward 
the bottom? Well, we know that a parabola is symmetric. That means 
that the vertex (the lowest point of a parabola that opens up, or the 
highest point of a parabola that opens down) has to be halfway between 
x = 3 and x = 5 - that is, it has to be at x = 4. So we can evaluate 
the function at x = 4:

     y = 4^2 - 8(4) + 15

       = 16 - 32 + 15

       = 31 - 32

       = -1

So the function has to look like this:

               |                           A = (3,0)
               |                           B = (5,0)
            |--|--|--|--A--|--B--|--|--|   C = (4,-1)
               |
               |           C

We can complete the graph by sketching a parabola through these three 
points:

               |     .           .
               |
               |
               |      .         .
               |                            
               |       .       .              
            |--|--|--|--A--|--B--|--|--|    
               |         .   .
               |           C

Note that the 'easy' way requires you to _know_ more than the 'hard' 
way.   

A lot of what you're supposed to be learning about graphs is how to 
guess what you're going to see before you begin plotting. In 
particular, the degree of a polynomial function tells you how many 
times it will bend:

     A linear polynomial (y = ax + b) doesn't bend at all.  

     A quadratic polynomial (y = ax^2 + bx + c) bends one time.  

     A cubic polynomial (y = ax^3 + bx^2 + cx + d) bends twice. 

And so on. Once you know how many times it has to bend, then if you 
can find the points at which the function crosses the x-axis, it 
becomes easy to draw the function. That's what I did here. 

(Being able to do this kind of thing, by the way, is why they make 
such a big deal out of factoring.) 

I hope this helps. Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/