What is the vertex form of y=(3x+1)(x-2)

Foil it out

y = 3x² - 6x + x - 2

y = 3x² - 5x - 2

Factor the coefficient of x², which is 3,
out of the first two terms only:  Not to 
take 3 out of -5x you divide -5 by 3 and get


y = 3(x² - x) - 2

Take one-half the coefficient of x:

· = 

Then square what you get:

 = 

Add that and then subtract it inside the parentheses:

y = 3(x² - x +  - ) - 2

Change the parentheses to brackets so we can factor 
and put parentheses inside:

y = 3[x² - x +  - ] - 2

Factor the first three terms inside the brackets:

y = 3[(x - )(x - ) - ] - 2 

Write (x - )(x - ) as (x - )²

y = 3[(x - )² - ] - 2

Now remove the brackets by distributing the 3 into the 
bracket, leaving the (x - )² intact.

y = 3(x - )² - 3· - 2

Simplify the last two terms:

y = 3(x - )² -  - 2 

y = 3(x - )² -  - 

y = 3(x - )² - 

Compare that to:

y = a(x - h)² + k

and equate like parts of the two equations:

a = 3,

-h = , so h = 

k = 

So the vertex is the point (, ), or

like (.8, 4.1)

or a mixed fraction is better for graphing:

The vertex is the point (, -(4)

Nasty fractions, indeed, but nasty fractions don't bother 
computers, so why should they bother us humans?  :-)
But if we can get some more points we can plot the graph.
The graph will be a parabola and we can observe if that
point really and truly is the vertex: 

We can get the y-intercept by going back to the original
equation  y=(3x+1)(x-2) and substituting x=0. We get
y = (3·0+1)(0-2) = (0+1)(-2) = (1)(-2) = -2

So we plot those two points and a bunch of others, we get
this graph.  


     

Edwin