Question

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

Answer 1

Are you having trouble factoring?

Answer 2

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.

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

(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

Answer 7

Ok so how factor the function?

Answer 8

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.

Answer 9

Can you identify a,b, and c?

Answer 10

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

Answer 11

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

Answer 12

sorry 16 and 32 were negative

Answer 13

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.

Answer 14

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

Answer 15

ok Im trying to find that now. Ok hold on

Answer 16

Though you could do this with the bigger numbers.

Answer 17

the gcf is 16

Answer 18

-16

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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 \[t^2+2t-24\]

Answer 24

Ok

Answer 25

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.

Answer 26

We aren't done factoring.

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

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)

Answer 32

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

Answer 33

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

Answer 34

Where did you get -16(t+16-24)

I thought we had -16(t^2+2t-24)

Answer 35

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

Answer 36

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.

Answer 37

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

Answer 38

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

Answer 39

a=1
b=2
c=-24

now what is a*c

Answer 40

Oh sorry -24

Answer 41

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?

Answer 42

24 and -1

Answer 43

so 24+(-1)=2?

Answer 44

i thought that was a 7. Im lost again

Answer 45

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

Answer 46

6+(-4)=2

Answer 47

can you tell what we were looking for?

Answer 48

i actually told you what we were looking for.

Answer 49

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

Answer 50

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

Answer 51

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?

Answer 52

won't*

Answer 53

6 and -4

Answer 54

right

Answer 55

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)

Answer 56

Ok Got it.

Answer 57

You may want to work on more factoring.

Answer 58

So what are the x-intercepts?

Answer 59

Yes I do. the 6 and 4 right

Answer 60

yes I do. the 6 and 4 right

Answer 61

4 is right
but you should not have 6

Answer 62

t+6=0 when t=?

Answer 63

t=4

Answer 64

I already agreed with t=4

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

Answer 65

t+6=0 when t=?

Answer 66

-6

Answer 67

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.

Answer 68

the height?

Answer 69

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

Answer 70

the f and the t is time?

Answer 71

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.

Answer 72

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

Answer 73

the (6,0)

Answer 74

you mean the (-6,0)?

Answer 75

yes

Answer 76

Yes (-6,0)

Answer 77

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.

Answer 78

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

Answer 79

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

Answer 80

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

Answer 81

its the x intercept

Answer 82

It is 4 feet heigh at 0 secs

Answer 83

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

Answer 84

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)?

Answer 85

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

Answer 86

lol true. Yes t was first

Answer 87

So at 4 secs it was at 0 feet high

Answer 88

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).

Answer 89

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

Answer 90

right

Answer 91

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?

Answer 92

Do you have any questions?

Answer 93

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

Answer 94

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.

Answer 95

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

Answer 96

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.

Answer 97

:)

Answer 98

Now I must return to my work.
Good luck.