Vertex form for quadratic function

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frctl

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Find the vertex form for each quadratic function.
Then find each of the following:
(A)Intercepts
(B)Vertex
(C)Maximum or minimum
(D)Range

Given:
g(x) = 0.25x2 - 1.5x - 7

I can't take the LCF of the first two terms.

Is this correct? g(x) = 0.25(x2 - 6 - 7)
 
frctl said:
Given:
g(x) = 0.25x2 - 1.5x - 7

I can't take the LCF of the first two terms.

Is this correct? g(x) = 0.25(x2 - 6 - 7)
Click to expand...
Almost.

Since you only factored 0.25 out of the first two terms, the last term should be outside the parentheses.

And you also omitted an x.

Try this instead: g(x) = 0.25(x2 - 6x) - 7

One more thing: you aren't working with an "LCF" (or a GCF, which you probably meant); you're just factoring out the coefficient of the squared term.
 
In order to obtain vertex form now,

g(x) = 0.25(x2 - 6x) - 7
g(x) = 0.25(x - 3)2 - 7 + 9
g(x) = 0.25(x - 3)2 + 2
 
frctl said:
In order to obtain vertex form now,

g(x) = 0.25(x2 - 6x) - 7
g(x) = 0.25(x - 3)2 - 7 + 9
g(x) = 0.25(x - 3)2 + 2
Click to expand...
Check each step!

First, look for anything that might make one line have a different value from the previous one.

Then, if necessary, plug in some specific number (maybe x=2) and see if all lines give the same value.

Then let us know what you found. This is an important skill to develop.
 
g(x) = 0.25(x - 3)2 is correct, I'm not sure what happens outside the perfect square to be equivalent to the beginning function.

How do I check each step?
 
You multiply out, combine like terms and see if you get the same result as the line before.

(x-3)2 = x2 -6x + 9 so .25((x-3)2)= .25(x2 -6x + 9) = .25x2- 1.5x + 2.25.

Now is .25x2- 1.5x + 2.25 what you started with? Remember you started with 0.25x2 - 1.5x - 7. If not can you fix it?
 
frctl said:
0.25(x2 -6x - 28)
Click to expand...
Check it and see!

But there are several slightly different ways I've seen the procedure taught; you should check how your textbook teaches it.

The way I think of it is like this: if we add 9 inside the parentheses, that's equivalent to adding 0.25*9 on the outside (since the 9 inside is being multiplied by 0.25. So to avoid changing the value, you have to subtract 0.25*9 on the outside.
 
1st line is fine.
2nd line. You subtracted 0.25*28 and also subtracted 7 so in the end you subtracted 14 from the right hand side which will not yield g(x). You need to fix that.

Here are the first few steps.

g(x) = 0.25x2 - 1.5x - 7
= 0.25(x2-6x) - 7
= 0.25(x2-6x + 9) - 7 -??
= 0.25(x-3)2 - 7 -??

Just find out what ?? should be and report back.
 
now there is no 28..
g(x) = 0.25(x2-6x) - 7
g(x) = 0.25(x2-6x + 9) - 7 - 9
g(x) = 0.25(x-3)2 - 7 - 9(0.25)
g(x) = 0.25(x-3)2 - 9.25
 
There is no 28 since (x^2 - 6x -28) is not a perfect square. I also think that you meant to say that there is no -28. Note that 28 and -28 are not the same number just as 28 and 4 are not the same number.
The third line from the bottom is wrong. You clearly subtracted 9 (at the end) but you never added 9. So it is not balanced. Can you try to fix that?
 
g(x) = 0.25(x2-6x) - 7
g(x) = 0.25(x2-6x + 9) - 7 - 9
g(x) = 0.25(x-3)2 + 9 - 7 - 9(0.25)
g(x) = 0.25(x-3)2 - 9.25
 
The 2nd line is wrong. It looks like the same equation that I said was wrong before.
The 3rd line does not equal the 2nd line
The 4th line does not equal the 3rd line.

Do you really believe that if have $9 and then spend (ie subtract) $7 and the spend 9-quarters you are at -$9.25. That is if you start of with $9, then spend $7, then spend 9-quarters you had to borrow $9.25
 
g(x) = 0.25x2 - 1.5x - 7
g(x) = 0.25(x2-6x) - 7
g(x) = 0.25(x2-6x + 9) - 7 -9(0.25)
g(x) = 0.25(x-3)2 - 7 -9.25
 
Ah, I think I see what you are doing. Yes, 9 + .25 = 9.25 and yes, -9 - 0.25 = -9.25. The problem is that we are not adding in 9(0.25), rather we are multiplying.

The only errors I see are in your last line. One being that you never finished by computing -7-9.25 and my 2nd concern is that 9(0.25) is not 9.25. What is 9(0.25)? This means to add 0.25 to itself 9 times. 0.25 is a quarter. So what is the value of 9 quarters.
 
Last edited:
g(x) = 0.25x2 - 1.5x - 7
g(x) = 0.25(x2-6x) - 7
g(x) = 0.25(x2-6x + 9) - 7 -9(0.25)
g(x) = 0.25(x-3)2 - 7 - 2.25
g(x) = 0.25(x-3)2 - 9.25

Thank you
 
(4, -9.25)

For the y intercept though,
I obtain the wrong answer

g(0) = 0.25(0 - 3)2 - 9.25
g(0) = -11.5
 
(0-3)2=(-3)2 = 9 NOT -9!

Rather -32 = -(32) = -9

Are you sure about your vertex (4, -9.25)??
 
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