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```Date: 12/10/2003 at 00:24:29
From: Terri

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

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

Thank you very much. You are truly a genius!
```
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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