Intercepts of Quadratic Functions

Date: 12/10/2003 at 00:24:29
From: Terri
Subject: Quadratic Functions

In some quadratic functions, why are there two x-intercepts and one y-
intercept?

Date: 12/10/2003 at 07:17:55
From: Doctor Dotty
Subject: Re: Quadratic Functions

Hi Terri,

Thanks for the question.

The following is a graph of a quadratic function with two x-
intercepts and one y-intercept:

      \        |
       \       |
        \      |                                     /
         \     |                                    /
          \    |                                   .'
           \   |                                   '
           '.  |                                  /
            `. |                                 /
             | |                                /
              \|                               /
               .                              /
               |`.                           /
               |  \                         /
               |   \                      ,'
   ------------+----`.------------------,-------------
               |      `.              ,'
               |        `._         ,'
               |           `--._,-''
               |
               |
               |
               |

No function can have more than one y-intercept, because for any
function each input maps onto no more than one output.  If there were
two y-intercepts, it would mean that a single value of x (i.e., x=0)
maps onto two distinct values of y. 

I don't know whether you are familiar with the quadratic equation,

    Whenever  ax^2 + bx + c = 0,

         -b +/- sqrt(b^2 - 4ac)
    x = ------------------------
                  2a


When the quantity b^2 - 4ac is equal to zero, x has only one real 
value, corresponding to the case where the function intercepts the
x-axis in only one place. 

When b^2 - 4ac is less than zero, x has no real values, corresponding
to the case where the function doesn't intercept the x-axis at all. 

But when b^2 - 4ac is greater than zero, x has two real values, each
of which corresponds to one of its x-intercepts. 

Write back if I can be of any more help--on this or anything else.

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

Date: 12/11/2003 at 00:03:38
From: Terri
Subject: Thank you (Quadratic Functions)

Thank you very much. You are truly a genius!