SOLUTION: What is the vertex form of y=(3x+1)(x-2)and y=3(x-2)? Is it y=3(x-5/6)^2-4.03?

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Question 84031: What is the vertex form of y=(3x+1)(x-2)and y=3(x-2)?
Is it y=3(x-5/6)^2-4.03?
Found 2 solutions by vertciel, Edwin McCravy:
Answer by vertciel(183) About Me  (Show Source):
You can put this solution on YOUR website!
You're off by a little.
The answer should be y = 3(x - (5/6))^2 - 49/12.
I expressed 4.08333333.... as 49/12 because you want be consistent.

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!

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
-5%2F3

y = 3(x² - 5%2F3x) - 2

Take one-half the coefficient of x:

1%2F2·5%2F3 = 5%2F6

Then square what you get:

%285%2F6%29%5E2 = 25%2F36

Add that and then subtract it inside the parentheses:

y = 3(x² - 5%2F3x + 25%2F36 - 25%2F36) - 2

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

y = 3[x² - 5%2F3x + 25%2F36 - 25%2F36] - 2

Factor the first three terms inside the brackets:

y = 3[(x - 5%2F6)(x - 5%2F6) - 25%2F36] - 2 

Write (x - 5%2F6)(x - 5%2F6) as (x - 5%2F6)²

y = 3[(x - 5%2F6)² - 25%2F36] - 2

Now remove the brackets by distributing the 3 into the 
bracket, leaving the (x - 5%2F6)² intact.

y = 3(x - 5%2F6)² - 3·25%2F36 - 2

Simplify the last two terms:

y = 3(x - 5%2F6)² - 25%2F12 - 2 

y = 3(x - 5%2F6)² - 25%2F12 - 24%2F12

y = 3(x - 5%2F6)² - 49%2F12

Compare that to:

y = a(x - h)² + k

and equate like parts of the two equations:

a = 3,

-h = -5%2F6, so h = 5%2F6

k = -49%2F12

So the vertex is the point (5%2F6, -49%2F12), or

like (.8, 4.1)

or a mixed fraction is better for graphing:

The vertex is the point (5%2F6, -(41%2F12)

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