Quadratic Function

Date: 05/04/2001 at 16:12:09
From: Trey Garza
Subject: The effects of changing the values of a, b, and c in a 
quadratic function

What are the effects of changing the values of a, b, and c in a 
quadratic function?  I used a TI-83 calculator and am still stumped on 
 
how exactly the values affect the graph.

Thanks a lot.

Date: 05/04/2001 at 23:29:23
From: Doctor Peterson
Subject: Re: The effects of changing the values of a, b, and c in a 
quadratic function

Hi, Trey.

It's easier to see the effects of changing the parameters in a 
quadratic written in completed-square form. If we had been given it in 
this form,

    y = A(x - B)^2 + C

we would know that A makes the graph wider or narrower, B moves it 
left and right, and C moves it vertically. 

Let's put it in that form:

    y = ax^2 + bx + c

      = a[x^2 + b/a x + (b/2a)^2] + (c - a(b/2a)^2)

      = a[x + b/(2a)]^2 + (c - b^2/(4a))

Think about how a, b, and c affect A, B, and C, when

    A = a
    B = -b/(2a)
    C = c - b^2/(4a)

The constant c will only affect C, so it will just move the graph up 
or down; but b will affect both B and C, and a will affect A, B, and 
C, so their effects on the graph will be more complicated. Changing b 
will move the vertex - you'll have to decide how - and changing a will 
both widen it and change the vertex.

I hope that gives you some ideas to work with. There are some things 
you could probably figure out just by playing with the equation on the 
calculator and paying attention to the position of the vertex; but 
knowing what's going on behind the scenes makes it a lot easier to 
see.

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