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)

Answer 1

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

Answer 2

Well, the first part tells you to factor the polinomy,
\[-16t^2-64t+80 \] You can do it because
\[-16t^2-64t+80=0 \Rightarrow-t^2-4t+5=0\] has solutions,
\[t=\frac{4\pm\sqrt{{4^2+4\cdot80}}}{-2}\] You will obtain two solutions, the x-intercepts.

Answer 3

As t means time only the positive solution has physical sense.

Answer 4

So the factored function is -16^2-64t+80, then what about the meaning of the x-intercept. It means time only the positive solution has a physical sense?

Answer 5

The correct one it is,
\[t=\frac{4\pm\sqrt{4^2+4\cdot5}}{-2}\] The two solutions are t=-5 and t=1. The factored polinomy is,
f(t)=-16(t+5)(t-1)

Answer 6

To interpret the meaning of the x intercept you should know that only positive time has physical sense. So the positive time indicates you the time when the sandbag hits the ground.

Answer 7

Oh! Got it, so my answer "The factored function of -16^2-64t+80 is f(t)=-16(t+5)(t-1). The x-intercepts are when the sandbag hits the ground."

Answer 8

Exactly. But I would add "the positive x-intercept"

Answer 9

Okay thanks, can we work on part B?

Answer 10

To complete the square, you should put it in the form
\[(a+b)^2=a^2+b^2+2ab\]

Answer 11

It should be something like this,
\[f(t)=-16(t^2+4t-5)=-16(t^2+4t+4-4-5)=\\=-16((t+2)^2-9)=-16(t+2)^2+144\]

Answer 12

You will find the vertex then in the point (-2,144).

Answer 13

Okay, so then to find the vertex I have to put a number in for all the "x's" but what number? Wait, (-2,144) is the vertex?

Answer 14

Yes, the vertex has x coordinate -2, and y coordinate 144. It is a maximum.

Answer 15

Okay so "The completed square of the function is -16(t+2)^2+144, and the vertex is (-2, 144). This would be a maximum."

Answer 16

Yes.

Answer 17

Yay! Then part c ask to find the axis of symmetry.. The formula is x=-b over 2a I think.

Answer 18

You don't need the formula now, but you can use it. You don't need the formula because you have found the axis in the part B. x=-2.

Answer 19

So the axis of symmetry is just x=-2?