What is the vertex form of #y=41x^2-3x+17#?

1 Answer
Nov 26, 2015

The wording 'vertex form' is new to me but I am assuming it is completion of the square:
#color(green)(y=41(x-3/82)^2 +16 155/164)#

Explanation:

If I am wrong about the term then perhaps I am showing you something else that may prove useful.

#color(blue)(Step1)#
Write as #y=41(x^2-3/41x)+17..............................(1)#

At the moment I can use the equals because I have not changed any of the total values on the right hand side (RHS). However, the next stage does change the value on the right so at that point I must not use the equals sign.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step2)#
Consider the #-3/41x#. Suppose we change this to #-(1/2xx3/41)x#
We get #color(white)(...)color(blue)(3/82)color(red)(x).......................(2)#

Suppose I rewrite (1) and use the #color(blue)(3/82)# from (2)

#y->41(x-color(blue)(3/82))#

#color(green)(" Notice that the second "x " has gone and the first is no longer squared.")#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step3)#

Now square the bracket

#y->41(x-3/82)^(color(blue)(2))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step4)#

Add the constant ( the right still does not #= ycolor(white)(..)# yet.

#y-> 41(x-3/82)^2 +17...................................(3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step5)#
The way we have written the bracket introduces an error. Let me multiply the bracket out and you will see what I mean.

#y-> 41(x^2-6/82x+color(blue)(9/6724))+17#

#y-> (41x^2-(41xx6)/82x+color(blue)((41xx9)/(6724)))+17#

#y->41x^2-3x+color(blue)(9/164)+17#

We have introduced the extra value of #color(blue)(9/164)#

To compensate for this we remove it by subtracting the same value

#color(green)(underline("NOW IT IS = "Y))#

#y=(41x^2-3x+color(blue)(9/164))color(brown)(-9/164)+17#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step6)#

So the vertex form (completing the square) is:
Adjust equation (3) by subtracting #color(brown)(-9/164)#

#y=41(x-3/82)^2 color(brown)(-9/164) +17#

#color(green)(y=41(x-3/82)^2 +16 155/164)#

#color(red)(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~)#