A sandbag was thrown downward from a building. The function f(t)=-16t^2-32t+384 shows the height f(t), in feet, of the sandbag after t sec;
Part A: factor the function f(t) and use the factors to interpret the meaning of the x intercept of the function

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- myininaya

Are you having trouble factoring?

- myininaya

f(t)=at^2+bt+c
if f is factorable, you should be able to find two integer numbers that when multiplied give you a product of a*c and a when added give you a product of b.

- anonymous

I don't think I understand it. I came up with (8, 320) for first part but not sure if right and I don't understand second part

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- myininaya

Part A: Says to factor to get the x-intercepts.
Did you factor yet?

- anonymous

I think I did it wrong thats where I got a( (8, 320)

- myininaya

(8,320) doesn't lay on the x-axis
points that lay on the axis are points in the form (a,0) where a is any number
notice the y is 0 that means we don't go up and or down from the x-axis and that we are actually on the x-axis
The part of the graph that crosses the x-axis is called the x-intercept. (You can have more than one)
(8,320) does not have the y being 0 so it is definitely not on the x-axis

- anonymous

Ok so how factor the function?

- myininaya

I will copy what I said before:
f(t)=at^2+bt+c
if f is factorable, you should be able to find two integer numbers that when multiplied give you a product of a*c and a when added give you a product of b.

- myininaya

Can you identify a,b, and c?

- anonymous

a would be 16, b would be 32, and 384 is c, right?

- myininaya

I think you had some negative numbers.
I know they aren't all positive.

- anonymous

sorry 16 and 32 were negative

- myininaya

so a=-16, b=-32, and c=384
now recalling my quote from earlier:
"f(t)=at^2+bt+c
if f is factorable, you should be able to find two integer numbers that when multiplied give you a product of a*c and a when added give you a product of b. "
We need to find what a*c is.
Then we need to find two integer numbers that have product a*c and also have sum b.

- myininaya

You could try to make life easier and see if -16, -32,384 have a common factor (greater than one)?

- anonymous

ok Im trying to find that now. Ok hold on

- myininaya

Though you could do this with the bigger numbers.

- anonymous

the gcf is 16

- anonymous

-16

- myininaya

cool so lets see what happens when we factor -16 out from -16t^2-32t+384

- myininaya

\[-16(\frac{-16t^2}{-16}-\frac{32t}{-16}+\frac{384}{-16})\]

- myininaya

so -16/-16=?
and -32/-16=?
and 384/-16=?

- anonymous

-16/-16=1
-32/-16=2
384/-16=-24

- myininaya

so therefore you are saying we have
\[-16(t^2+2t-24)\]
Which I totally agree with
So instead of looking at those really huge numbers
let's re-identify what is a, b, and c
We are now just looking at \[t^2+2t-24\]

- anonymous

Ok

- anonymous

So that is the answer for A, but for the second part of the question how do I use that to interpret the x-intercept.

- myininaya

We aren't done factoring.

- myininaya

I'm asking you to re-identify a,b,and c.

- anonymous

oh ok sorry. as you may can tell math isnt my best subject

- myininaya

we need to find a*c after.
Then we need to find two factors of a*c that have product ac and sum b.

- anonymous

a= t^2
b= -2t
c= -24
right

- myininaya

a,b, and c are the constant values that are in front of the variable squared, in front of the variable, and then lastly the constant not attach to a variable respectively
in other words
a is the coefficient of t^2
b is the coefficient of t
c is the left over term (the constant term)

- myininaya

we have at^2+bt+c and we are comparing this to t^2+2t-24
now since 1*k=k for all k
then 1*t^2=t^2
so you could say we are comparing at^2+bt+c to 1t^2+2t-24
if you think that makes it easier

- anonymous

Im so confused. wouldn't the factor be -16(t+16-24)

- myininaya

Where did you get -16(t+16-24)
I thought we had -16(t^2+2t-24)

- anonymous

I'm not sure. was trying to factor I think I just put the wrong number in.

- myininaya

ok so we are looking at factoring t^2+2t-24
which is the samething as saying 1t^2+2t-24
what is a,b, and c?
remember you are comparing 1t^2+2t-24 to at^2+bt+c
to determine what a,b, and c are.

- anonymous

a=1t^2
b=2t
c=-24

- myininaya

again a,b, and c are suppose to be constants
c is right but a and b for some reason you are inputting the variable part

- myininaya

a=1
b=2
c=-24
now what is a*c

- anonymous

Oh sorry -24

- myininaya

ok so since a=1 we can use a shortcut to the factoring style mentioned earlier
what are two numbers that multiplied to be -24 and add up to be 2?

- anonymous

24 and -1

- myininaya

so 24+(-1)=2?

- anonymous

i thought that was a 7. Im lost again

- myininaya

which one of these is true:
24+(-1)=2
-24+1=2
12+(-2)=2
-12+2=2
-6+4=2
6+(-4)=2
8+(-3)=2
-8+3=2

- anonymous

6+(-4)=2

- myininaya

can you tell what we were looking for?

- myininaya

i actually told you what we were looking for.

- myininaya

and where are you lost?
I haven't see any 7's yet

- anonymous

it was the ? no 7 We were getting the b and c right

- myininaya

My thing want copy and paste.
But I asked you to find two numbers that multiplied to be -24 and add up to be 2?
Can you tell me what two integer numbers do this?

- myininaya

won't*

- anonymous

6 and -4

- myininaya

right

- myininaya

so again since a=1
we don't have to go through the longer process which requires factor by grouping
so since a=1, t^2+2t-24=(t+6)(t-4)
so
we started with f(t)=-16t^2-32t+384
and then we factor f
and got f(t)=-16(t+6)(t-4)

- anonymous

Ok Got it.

- myininaya

You may want to work on more factoring.

- myininaya

So what are the x-intercepts?

- anonymous

Yes I do. the 6 and 4 right

- anonymous

yes I do. the 6 and 4 right

- myininaya

4 is right
but you should not have 6

- myininaya

t+6=0 when t=?

- anonymous

t=4

- myininaya

I already agreed with t=4
I don't agree with t=6 because 6+6 does not equal 0.

- myininaya

t+6=0 when t=?

- anonymous

-6

- myininaya

right
so the x-intercepts of f(t)=-16(t-4)(t+6) are t=4 or t=-6.
Recall that anything that lays on the x-axis has the y-value being 0.
So the x-intercepts in order-pair form are (4,0) or (-6,0).
Now what does t represent and what does y represent (or what does f(t) represent)?
Refer to your problem for this.
We are using this to interpret the x-intercepts.

- anonymous

the height?

- myininaya

which one is the height
t or f(t)?

- anonymous

the f and the t is time?

- myininaya

right
y=f(t) is the height
t is the seconds
Now I don't think we invented time travel yet and I think time travel would actually cause multiple dimensions (multiple possibilities)
So I'm assuming time has to be 0 or greater that 0.

- myininaya

So that means the one x-intercept we found is not needed for this question.
which one am I referring to?

- anonymous

the (6,0)

- myininaya

you mean the (-6,0)?

- anonymous

yes

- anonymous

Yes (-6,0)

- myininaya

Ok so we only really have a graph for t>=0
or interval notation [0,infinity)
So the only x-intercept we have to determine is (4,0) since this point actually exists for t>=0.

- myininaya

so do you know what (4,0) means?

- myininaya

I will give you a hint.
You already said t represented seconds
And f(t) or y whichever one you want to call it represents height

- myininaya

so use this to say in words what (4,0) means

- anonymous

its the x intercept

- anonymous

It is 4 feet heigh at 0 secs

- myininaya

why do you say that?
do you think the point should be read (4,0) or (0,4)
you are getting confused which is which

- myininaya

we should really say t-intercept when we say x-intercept since there is no x's
lol
but anyways
isn't it (t,y) not (y,t)?

- myininaya

you said earlier t=4
now you are saying t=0

- anonymous

lol true. Yes t was first

- anonymous

So at 4 secs it was at 0 feet high

- myininaya

So we are definitely not interpreting (0,4) I'm not even sure this point is on the graph unless I check it but that is unnecessary work.
We are looking at (4,0).
which says yes after 4 seconds, the sandbag is 0 ft from the ground (or on the ground).

- myininaya

because you know if it has 0 distance from the ground then it is got to be on the ground right?

- anonymous

right

- myininaya

ok Do you feel like you understand this problem better?
Do you feel like if your teacher asked you to explain you would have the confidence to do so?

- myininaya

Do you have any questions?

- anonymous

I think so. But I am def looking into a tutor. lol

- myininaya

You might want to try to work this problem without looking at what we did from the very beginning and see if you arrive at the same conclusions as we did.

- anonymous

No questions. Thats a good idea. Thank you so very much for your help

- myininaya

np
You deserve a medal for putting in work and not giving up on the problem.
I see so many people give up. It really is awesome to come across people like you who care to understand it.
Thanks for trying super hard.

- anonymous

:)

- myininaya

Now I must return to my work.
Good luck.

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