Question

A sandbag was thrown downward from a building. The function f(t) = -16t2 - 32t + 384 shows the height f(t), in feet, of the sandbag after t seconds:

Part A: Factor the function f(t) and use the factors to interpret the meaning of the x-intercept of the function.

Part B: Complete the square of the expression for f(x) to determine the vertex of the graph of f(x). Would this be a maximum or minimum on the graph?

Part C: Use your answer in part B to determine the axis of symmetry for f(x)?

Answer 1

for part A i got -16(t+6)(x-4) when factored and said that the x-intercepts of the functions is (4,0) which represents the time in seconds that the sandbag was thrown to where it lands.

Answer 2

For part b I completed the square and got (t+1)^2 = 25 but how do I find the vertex?

Answer 3

@Hero @Hero you said you would help me!

Answer 4

I said, post your next question separately. Anyone can help with these.

Answer 5

Yeah I did post it separately and you said you would help!

Answer 6

I need help on finding the vertex and I am confused on how to

Answer 7

For Part A, you are correct (almost)

Answer 8

What did I leave out? @TheSmartOne could you please explain

Answer 9

It's a simply mistake, but:

You were given \(\sf\Large f(t) = -16t^2 - 32t + 384\)

however you factored it to \(\sf\large -16(t+6)(\color{red}{x}-4)\)

Answer 10

so the t in t-4 is supposed to be x-4? @TheSmartOne

Answer 11

The variable you were given @kaite_mcgowan was \(\bf t\) so I don't understand where you brought in the \(\bf x\) from.

So you just need to change the x to a t to make it correct :)

Answer 12

oh ok thanks so much! could you please help me on b and finding the vertex with the completed square of (t+1)^2 = 25 @TheSmartOne

Answer 13

\(\sf\Large y=a(x-h)^2+k\)

The vertex is \(\sf\Large (h,k)\)

Answer 14

so how would I incorporate (t+1)^2 = 25 in to that?

Answer 15

y = a(t-1)^2 + 25 ???? @TheSmartOne

Answer 16

one sec

Answer 17

You didn't properly complete the square.

Answer 18

?? what do you mean?

Answer 19

\(\sf\Large -16t^2 - 32t + 384 \color{red}{\neq} (t-1)^2+25\)

Answer 20

so the completed square form is (t - 1)^2 + 25

Answer 21

it isn't. That's wrong.

Answer 22

oh ok i got it. How would I find it then?

Answer 23

could you show me step by step

Answer 24

\(\sf\Large -16t^2 - 32t =-384\)

What is \(\sf\Large\left( \frac{b}{2}\right)^2=\left( \frac{-32}{2}\right)^2=?\)

Answer 25

-16

Answer 26

correct; (-16)^2

Answer 27

whats next ??? @TheSmartOne

Answer 28

you have to add it to both sides:

\(\sf\Large -16t^2 - 32t+(-16^2) =-384+(-16)^2\)

Answer 29

can you factor the left hand side?

Hint:

\(\sf\Large a^2+2ab+b^2=(a+b)^2\)

Answer 30

I am so sorry i lost wifi! @TheSmartOne

Answer 31

ok so you get -16t^2 - 32t + (-256) = -384 + (-256) @TheSmartOne

Answer 32

-16t^2 - 32t + (-256) = -640 @TheSmartOne

Answer 33

hmmm

Answer 34

@TheSmartOne

Answer 35

One sec, we'll need to backtrack over here

Answer 36

ok no problem

Answer 37

do you know what went wrong?

Answer 38

we have to first factor the 16 out of the equation, so:

\(\sf\Large -16t^2 - 32t + 384 =-16(t^2+2t-24) \)

Answer 39

@Mehek14 @paki I'm not sure how to complete the square :/

Answer 40

so the factored form is -16(t+6)(t-4)

Answer 41

there are so many ways to solve math, I don't know why they limit us with this completing the square lol

Answer 42

*disappears*

Answer 43

https://mathway.com/examples/Algebra/Quadratic-Equations/Solve-by-Completing-the-Square?id=29

mhmmm

Answer 44

wait thats the answer the link above? I know that the highest point or the maximum is 400 and the symmetry of axis is -1???

Answer 45

i just dont understand completing the square?

Answer 46

the link was an example of how to do it

Answer 47

oh ok so what would I do ??

Answer 48

yes, the symmetry of axis is x=-1

Answer 49

ok I go that part and I know that the maximum is 400 i just need to show my work of how I got that

Answer 50

@zepdrix could you help us to complete the square for
\(\sf\Large f(t)=-16t^2-32t+384\)

Answer 51

@zepdrix please I really need help!

Answer 52

I can tell you how to do the first step

Answer 53

divide all the numbers by -16

Answer 54

the calculator gives you the final answer, but no steps: http://prntscr.com/7metrk

Answer 55

@mathmate could you help us complete the square?

Answer 56

t^2 + 2t +24

Answer 57

OMG EVERYONE I THINK I MAY HAVE FIGURED IT OUT! GIVE ME ONE SEC!

Answer 58

we could have just as easily found the vertex by -b/2a ¯\_(ツ)_/¯

Answer 59

a(x+d)2+e

d=−32 / 2⋅(−16)

d=1

e=384− (−32)^2 / 4⋅(−16)

e=400

−16(t+1)2+400

Answer 60

@TheSmartOne

Answer 61

that's it, I've been working it out over here as well. nicely done. :)

Answer 62

thanks so much!!!! @jtvatsim

Answer 63

well we got the completing square out of the way

Answer 64

\(\color{blue}{\text{Originally Posted by}}\) @TheSmartOne
\(\sf\Large y=a(x-h)^2+k\)

The vertex is \(\sf\Large (h,k)\)
\(\color{blue}{\text{End of Quote}}\)

Answer 65

haha yep! so to find the vertex I wold just do y = -16(x-1)^2 +400??

Answer 66

@TheSmartOne

Answer 67

or in other words the vertex is (-1,400)

Answer 68

y = -16(x+1)^2 +400

Answer 69

correct

Answer 70

and that would be a maximum or a minimum?

Answer 71

maximum!

Answer 72

correct :)

Answer 73

Thanks for all of your help @TheSmartOne and sticking with me through the whole equation unlike other people who promised before **cough cough @Hero cough cough** thanks again @TheSmartOne

Answer 74

I had to go to lunch. I would have helped otherwise.

Answer 75

yeah yeah whatever

Answer 76

I probably wouldn't have helped if it wasn't for Hero: http://prntscr.com/7mezsj

Answer 77

at least hero considered it...

Answer 78

@kaite_mcgowan
Is the question resolved?

Answer 79

yes it is thanks for asking! @mathmate

Answer 80

Sorry, I came a little late.
Here's what I would have done anyway:
Given f(t)=\(-16t^2-32t+384\)
(a) factorization
f(t)=\(-16(t^2+2t-24)\)
f(t)=-16(t+6)(t-4)
(b) Complete square
f(t)=-16(t+1 +5)(t+1 -5)
f(t)=-16[(t+1)^2-25]
maximum is at x=-1 (solve for t in t+1=0), f(-1)=400.
Maximum because leading coefficient (of t^2) is negative.
(c) Axis of symmetry
Axis of symmetry of a quadratic is location of maximum/minimum, i.e.
x=-1