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.

Answer 1

so... .what did you get for the function factors?

Answer 2

I dont understand what to do

Answer 3

well, have you covered quadratic factoring yet?

as in factoring quadratic equations, or equations of 2nd degree

Answer 4

Yeah

Answer 5

so... the x-intercepts or "zeros" or solutions of the quadratic
are when y = 0, or f(x) is 0
so

\(\bf f(t)=-16t^2-64t+80\implies 0=-16t^2-64t+80
\\ \quad \\
16t^2+64t-80=0\implies 16(t^2+4t-5)=0\)

so... any idea on the factors?

Answer 6

Im not exactly sure, im not exactly good at this. Can you explain what to do? I dont understand how to factor out 16(t^2+4t-5)=0.

Answer 7

t=1 and -5?

Answer 8

\(\bf f(t)=-16t^2-64t+80\implies 0=-16t^2-64t+80
\\ \quad \\
16t^2+64t-80=0\implies 16(t^2+4t-5)=0
\\ \quad \\
t^2+4t-5=\cfrac{0}{16}\implies t^2+4t-5=0\)

how about now?

Answer 9

yeap.... 1 and -5 are the x-intercepts

Answer 10

can you help with me part b?

Answer 11

what's part b?

Answer 12

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

Answer 13

do you know what a perfect square trinomial is?

Answer 14

Isnt the middle number supposed to be 2 times more then ones on the side>?

Answer 15

@jdoe0001

Answer 16

yes site is a bit sluggish :/

Answer 17

yeap
so... let us do some grouping first

\(\bf f(t)=-16t^2-64t+80\implies 16t^2+64t-80=f(t)
\\ \quad \\
(16t^2+64t)-80=f(t)\implies (16t^2+64t+{\color{red}{ \square }}^2)-80=f(t)\)

any ideas on what our missing number is there, to get a "perfect square trinomial" from the parenthesized group?

Answer 18

8? OR 10?
I dont know. im confused again.

Answer 19

well... lemme simplify it some

\(\bf f(t)=-16t^2-64t+80\implies 16t^2+64t-80=f(t)
\\ \quad \\
(16t^2+64t)-80=f(t)\implies 16(t^2+4t)-80=f(t)
\\ \quad \\
16(t^2+4t+{\color{red}{ \square }}^2)-80=f(t)\)

Answer 20

hmm actually, should be negative f(t) btw =)

so \(\bf f(t)=-16t^2-64t+80\implies 16t^2+64t-80=-f(t)
\\ \quad \\
(16t^2+64t)-80=-f(t)\implies 16(t^2+4t)-80=-f(t)
\\ \quad \\
16(t^2+4t+{\color{red}{ \square }}^2)-80=-f(t)\)

Answer 21

\[ax^2+bx+c \\ =ax^2+\frac{a}{a}bx+c \\ \text{ I multiplied the second term by } \frac{a}{a} \\ \text{ I did this because I thought it would be easier } \\ \text{ for you to understand how I factor out } a \\ \text{ from the first two terms in the following step } \\ =a(x^2+\frac{1}{a}bx)+c \\ \\ =a(x^2+\frac{b}{a}x)+c \\ \text{ I'm going to leave a space } \\ \text{ this space will be meant for the number we need to add in } \\ \text{ so that we can complete the square } \\ \text{ but remember whatever I add in } \\ \text{ I have to subtract it out } \]
\[=\color{red}{a}(x^2+\frac{b}{a}x+\color{red}{(\frac{b}{2a})^2})+c \color{red}{-a(\frac{b}{2a})^2 }\\ =a(x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2\]

Answer 22

why is the f(t) negative? and is it 2?

Answer 23

ohh, because I moved over to the right-hand side =)

Answer 24

So is it 2?

Answer 25

yes....but, notice, there's a 16 outside the group
so, if we expand it, it'd be 16 * 2, or 32
so hold the mayo

Answer 26

keep in mind, as freckles said, all we're doing is "borrowing" from our good friend Mr Zero, 0

so if we ADD \(16*2\), we also have to SUBTRACT \(16*2^2\)

one sec

Answer 27

how do you vind the vertex?
or the maximum or minimum? Im getting lost

Answer 28

\(\bf 16(t^2+4t+{\color{red}{ 2 }}^2-{\color{red}{ 2}}^2)-80=-f(t)
\\ \quad \\
16(t^2+4t+{\color{red}{ 2}}^2)-(16\cdot {\color{red}{ 2}}^2)-80=-f(t)
\\ \quad \\
16(t+2)^2-64-80=-f(t)
\\ \quad \\
f(t)=-16(t+2)^2+64+80
\\ \quad \\
f(t)=-16(t+2)^2+144\implies f(t)=-16(t-{\color{brown}{ (-2)}})^2+{\color{blue}{ 144}}

\\ \quad \\ \quad \\
y=(x-{\color{brown}{ h}})^2+{\color{blue}{ k}}\\
x=(y-{\color{blue}{ k}})^2+{\color{brown}{ h}}\qquad\qquad vertex\ ({\color{brown}{ h}},{\color{blue}{ k}})\)


see the vertex now?

Answer 29

What is a vertex?

Answer 30

a vertex is where the graph makes a U-turn

Answer 31

is it like parabola

Answer 32

also, notice, the number in front of the parentheses, is negative, -16
that means, the parabola opens downards, or it is going down