Creating a Quadratic Equation That Matches the following Criteria

[JSmith]

Give an example of a quadratic function with real zero x = -3 that is increasing on (-5, -3). I have been messing around with a graphing calculator and can't think of how to do this... help?

Thanks!!!!

 

[Yogi]

Start with a very simple quadratic, y=x^2 then use transformations to shift it to the left and change the slope if you need.
http://www.purplemath.com/modules/fcntrans.htm

 

[JeffM]

Give an example of a quadratic function with real zero x = -3 that is increasing on (-5, -3). I have been messing around with a graphing calculator and can't think of how to do this... help?

Thanks!!!!

Another way is to use the zero product property.

Note that any quadratic of the form: Q(x) = a(x + 3)(x - 0) = ax^2 + 3ax, 0 < |a|, will have zeroes at - 3 and 0. So Q(x) will have an extremum somewhere between - 3 and 0. Call it e for extremum. If a < 0, Q(x) will have a maximum at e, and Q(x) will be increasing everywhere to the left of e, including (-5, - 3).

So pick a = - 1.

That gives Q(x) = - x^2 - 3x.

 

[soroban]

Hello, JSmith!

Another approach . . .

Give an example of a quadratic function with real zero x = -3 that is increasing on (-5, -3).

First, I sketched a graph that satisfies the requirements.
. . I think I found the simplest.

              |
              |
              *
           *  |  *
    -5   *    |    *
    -+--*-----+-----*----
     . -3     |     3
     . *      |      *
     .        |
     .        |
     .*       |       *
     .        |
     .        |
     .        |
     *        |       -*
              |

The function is: .\(\displaystyle y \:=\: 9-x^2\)