Question

Can someone help me, please?

The function f(t) = t2 + 12t − 18 represents a parabola.

Part A: Rewrite the function in vertex form by completing the square. Show your work.

Answer 1

So first lets set this equation to 0. \(t^2+12t-18=0\)

Now keep the variables on the LHS but move the constants to the RHS. \(t^2 + 12t=18\)

Answer 2

Following so far?

Answer 3

I think so, yes.

Answer 4

Any questions before continuing? :o

Answer 5

I'm not too sure what the RHS and the LHS stand for- I think I have an idea but I want to be sure.

Answer 6

RHS = Right Hand Side (of the equation)

LHS = Left Hand Side (of the equation )

:)

Answer 7

Ah, okay that's what I thought. Thank you, just making sure.

Answer 8

Alright, continuing..

We're going to be expanding the LHS by completing the square, and to be consistent, whatever we do to the LHS of the equation, we must do to the RHS of the equation as well.

Now in expanding the LHS, we need to create a quadratic function such that it fits the form \(ax^2+bx+c\). Right now we only have \(ax^2+bx\) but we're missing the \(c\).

Following?

Answer 9

Yes, I believe so.

Answer 10

Alright good good.

To expand the LHS, we need to find a \(c\) value. To do that we follow the formula: \(c= \left(\dfrac{b}{2}\right)^2\)

Now what's our value of b?

Answer 11

12, right?

Answer 12

That's right, so if we substituted 12 in place of b, what would our c-value be?

Answer 13

36?

Answer 14

That's correct :)

We can now complete our quadratic by adding \(\color{red}{36}\) to both sides of our equation. We have now fit the quadratic form, \(ax^2+bx+c\)

\(t^2+12t+\color{red}{36} = 18 + \color{red}{36}\)

Answer 15

Now we simplify the LHS of our equation. \[(t+6)^2=18+36\] now we just simplify the right side, move it back over, and we're done!

Answer 16

Do you see it now?

Answer 17

Yes, and to simplify the RHS you would just need to add 18 and 36, since there are no variables, right?

Answer 18

You are correct!

Answer 19

Okay, so it would end up:
\[(t+6)^2=54\]
Right?

Answer 20

Yep, and to fit the vertex form, \(y=a(x-h)^2 +k\) , we just have to move the 54 over by subtracting it from both sides :)

Answer 21

\[y=a(x-h)^2+k \\ ~~~~~~~~~~~~~\downarrow \\ y=(t+6)^2-54\]

Answer 22

Is that the finished product or are we not finished?
Because if we aren't finished, I'm not sure what to do after that.

Answer 23

Yep, we are done. \(\boxed{y=(t+6)^2-54}\)

Answer 24

Oh okay, thank you so much! That helped a lot.

Answer 25

Your question is asking you to rewrite the function, \(y=t^2+12t-18\) in vertex form, \(y=(t+6)^2-54\) by completing the square :)

Answer 26

No problem! Glad you were able to follow through :D

Answer 27

I understand it a little bit more, too. I didn't understand it at all before cx

Answer 28

Great! That's the goal of using OpenStudy - to answer homework questions and learn more about howto solve problems similar to the ones you are stuck on :D

Answer 29

Well thank goodness I found OpenStudy cx

Answer 30

:) enjoy!

Answer 31

Thanks! c: