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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

Mathematics
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Are you having trouble factoring?
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.
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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Part A: Says to factor to get the x-intercepts. Did you factor yet?
I think I did it wrong thats where I got a( (8, 320)
(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
Ok so how factor the function?
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.
Can you identify a,b, and c?
a would be 16, b would be 32, and 384 is c, right?
I think you had some negative numbers. I know they aren't all positive.
sorry 16 and 32 were negative
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.
You could try to make life easier and see if -16, -32,384 have a common factor (greater than one)?
ok Im trying to find that now. Ok hold on
Though you could do this with the bigger numbers.
the gcf is 16
-16
cool so lets see what happens when we factor -16 out from -16t^2-32t+384
\[-16(\frac{-16t^2}{-16}-\frac{32t}{-16}+\frac{384}{-16})\]
so -16/-16=? and -32/-16=? and 384/-16=?
-16/-16=1 -32/-16=2 384/-16=-24
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 \[t^2+2t-24\]
Ok
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.
We aren't done factoring.
I'm asking you to re-identify a,b,and c.
oh ok sorry. as you may can tell math isnt my best subject
we need to find a*c after. Then we need to find two factors of a*c that have product ac and sum b.
a= t^2 b= -2t c= -24 right
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)
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
Im so confused. wouldn't the factor be -16(t+16-24)
Where did you get -16(t+16-24) I thought we had -16(t^2+2t-24)
I'm not sure. was trying to factor I think I just put the wrong number in.
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.
a=1t^2 b=2t c=-24
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
a=1 b=2 c=-24 now what is a*c
Oh sorry -24
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?
24 and -1
so 24+(-1)=2?
i thought that was a 7. Im lost again
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
6+(-4)=2
can you tell what we were looking for?
i actually told you what we were looking for.
and where are you lost? I haven't see any 7's yet
it was the ? no 7 We were getting the b and c right
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?
won't*
6 and -4
right
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)
Ok Got it.
You may want to work on more factoring.
So what are the x-intercepts?
Yes I do. the 6 and 4 right
yes I do. the 6 and 4 right
4 is right but you should not have 6
t+6=0 when t=?
t=4
I already agreed with t=4 I don't agree with t=6 because 6+6 does not equal 0.
t+6=0 when t=?
-6
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.
the height?
which one is the height t or f(t)?
the f and the t is time?
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.
So that means the one x-intercept we found is not needed for this question. which one am I referring to?
the (6,0)
you mean the (-6,0)?
yes
Yes (-6,0)
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.
so do you know what (4,0) means?
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
so use this to say in words what (4,0) means
its the x intercept
It is 4 feet heigh at 0 secs
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
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)?
you said earlier t=4 now you are saying t=0
lol true. Yes t was first
So at 4 secs it was at 0 feet high
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).
because you know if it has 0 distance from the ground then it is got to be on the ground right?
right
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?
Do you have any questions?
I think so. But I am def looking into a tutor. lol
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.
No questions. Thats a good idea. Thank you so very much for your help
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.
:)
Now I must return to my work. Good luck.

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