Question

A sandbag was thrown downward from a building. The function f(t) = -16t2 - 64t + 80 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. (4 points)

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? (4 points)

Part C: Use your answer in part B to determine the axis of symmetry for f(x)? (2 points)

Answer 1

Well x-intercept means y=0, so
\[f(t) = -16t^2 - 64t + 80 \\ \\ 0 = -16t^2 - 64t + 80 \]
Can you factor this?

Answer 2

yes I am factoring it right now

Answer 3

Is it -16(t-1) (t+5) ?

Answer 4

are the factors 1 and 5?

Answer 5

I mean 1 and -5

Answer 6

correct!

Answer 7

what does that mean about the x intercept of the funtion

Answer 8

the \(x-\)intercept, is the value of \(x\) such that \(f(x)=0\)
in this case it is better to speak about \(t-\)intercept, such the function is \(f(t)\)
and the \(t-\)intercepts are \(t=1,t=-5\)

Answer 9

it is easy, using the factorization above, to check, these conditions:
\(f(1)=0,\;f(-5)=0\)

Answer 10

f(1) = 0 and f(-5) = 0?

Answer 11

yes! For example if I substitute \(t=1\), I can write this:
\[f\left( 1 \right) = - 16\left( {1 - 1} \right) \cdot \left( {1 + 5} \right) = - 16 \cdot 0 \cdot 6 = 0\]
similarly, if I substitute \(t=-5\)

Answer 12

also,
the height f(t), in feet, of the sandbag

if f(t) is zero , that means the height of the sandbag above the ground is 0
i.e. that is the bag hits the ground at that time.

Answer 13

does it hit the ground at 5 or 1?

Answer 14

I think at \(t=1\), since, time has to be positive

Answer 15

the f(-5) is 5 seconds before the sandbag was dropped.
it is the "theoretical" time when you would have "launched" the bag up into the air to follow the parabola. Of course we don't do that... we start near the top by standing on a building

Answer 16

for part B
hint:
I add and subtract \(64\) so I get this:
\[\begin{gathered}
f\left( t \right) = - 16{t^2} - 64t - 64 + 64 + 80 = \hfill \\
\hfill \\
= \left( { - 16{t^2} - 64t - 64} \right) + 144 \hfill \\
\end{gathered} \]
please continue

Answer 17

are you completing the square?

Answer 18

yes, that is showing how to complete the square

Answer 19

I think i understand the question now. Thank you so much for your help