Medal!

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. (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)

Question

Medal!

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. (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)

anonymous · Answer

the equation given is a gravity equation (known because -16t^2 specifies it). it wouold also be a parabola. the negative 16 states itll open downward( making a minimum.) the axis of symmetry would be the point the minimum is.

anonymous · Answer

$$f(t) = -16t^2 - 32t + 384=-16 (t+6) (t-4) $$ is a start

anonymous · Answer

$x$ intercepts are found by setting 
$$-16 (t+6) (t-4)=0$$ and solving for $t$
you should get $t=-6,t=4$
as for the meaning, $t=-6$ has not meaning because $t$ is time which goes forward, not backward
the domain should be $t\geq 0$
as for $t=4$ that means the height is $0$ when $t=4$ i.e. it hits the ground in four seconds

anonymous · Answer

btw the person who wrote this question wasn't really paying attention 
it is $$f(t) = -16t^2 - 32t + 384=-16 (t+6) (t-4)$$
there is no $x$ in it, it has no $x$ intercepts, but rather $t$ intercepts

mckennarian · Answer

That makes sense. So the time is receding?

anonymous · Answer

i do not know what "time is receding" means
time goes forward
the bag, however, goes down from the original height of $384$ feet up to $0$ feet when it hits the ground after 4 seconds

mckennarian · Answer

That's what I meant, I worded oddly. And just to check my work, would the vertex be 400? I think I calculated it wrong.

anonymous · Answer

the first coordinate of the vertex is $-\frac{b}{2a}$which in your case is 
$$-\frac{-32}{2\times (-16)}=-1$$

anonymous · Answer

the second coordinate of the vertex is what you get when you replace $t$ by $-1$

anonymous · Answer

which is in fact 400, so you were right!

anonymous · Answer

since the parabola opens down, $400$ is the maximum if you graphed the parabola

i don't want to confuse you but it should be said that since $t$ is time, the of the function is $t\geq 0$ so if you want to be precise, $-1$ is not even in the domain of the function

i doubt the person who wrote this question had this in mind however
they clearly were not paying attention when they wrote it, which is why it has this mistake, as well as the error of "$x$" intercept

anonymous · Answer

i meant to say " the DOMAIN of the function is $t\geq 0$"

mckennarian · Answer

Alright, how do you find an axis of symmetry? I'm not sure of the process.

anonymous · Answer

you already found it
since the first coordinate of the vertex is $t=-1$ that is also the axis of symmetry, the vertical line $t=-1$

anonymous · Answer

here is a nice picture of your function http://www.wolframalpha.com/input/?i=+-16t^2+-+32t+%2B+384+

anonymous · Answer

here is what it should really look like, for domain $0\leq t\leq 4$ http://www.wolframalpha.com/input/?i=+-16t^2+-+32t+%2B+384++domain+0..4

mckennarian · Answer

Thank you so much, this really helped!