Question

Suppose the function g(x) is used to model the height y, of a ball x seconds after the ball is thrown up in the air. Suppose further that a ball reaches its maximum height of 7.5 feet in 5 seconds, and then hits the ground 3 seconds later. What would be an appropriate domain for g(x)?

Answer 1

I thinking its is 0 ≤ x ≤ 8

because domain is x-values and range is the y-values.

Not 100% sure, my domain n range skills are rusty.

Answer 2

thanks!

Answer 3

The max height occurs at the vertex

Answer 4

The vertex lies on the axis of symmetry

Answer 5

So this means that there is a vertical line that cuts the parabola (which opens downward) in 2 exact mirroring halves at x = 5

Answer 6

So if it takes 3 seconds (from x = 5) to hit the ground, which is the x-axis, then it should take 3 seconds for the ball to launch into the air and hit the peak

So the domain is actually 2 <= x <= 8 because 2 and 8 are both 3 seconds away from x = 5 seconds

Answer 7

im sorry i had to step away from the computer

Answer 8

thanks your answer makes alot more sense and i get it thanks alot!

Answer 9

that's great, glad you do

Answer 10

wait i have multiple choice and thats not a choice

Answer 11

what are your choices?

Answer 12

there is no 2 <= x <= 8

Answer 13

they are 3< or equal to x<or equal to 5
-5<or equal to x<or equal to 5
0<or equal to x<or equal to 8
0<or equal to x< or equal to 7.5

Answer 14

well it can't be 0 to 8 because the midpoint of this interval is 4 (because 8+0 = 8 and you divide this in half to get 4)

Answer 15

this is an odd problem

Answer 16

I know im really stuck

Answer 17

however, if you use x = 0 to x = 8 and you discount x = 0 to x = 2, then I guess it could work (in an odd way)

Answer 18

but there is no physical way to have a ball reach its peak in 5 seconds, come back down in 3 seconds and have it start on the ground at 0 seconds...it's just not possible

Answer 19

so your guess would be 0<=x<=8

Answer 20

yeah that's if you ignore the fact that x = 0 to x = 2 (excluding x = 2 itself) don't work

Answer 21

which is why this is an odd problem

Answer 22

could you help with one more!

Answer 23

sure

Answer 24

So I can get that one right!

Answer 25

go for it

Answer 26

A biologist took count of the nuber of migrating waterfowl at a particular lake and recounted the lakes population of waterfowl on each of the next six weeks.

Answer 27

week: 0 1 2 3 4 5 6
Population: 635 644 719 860 1,067 1,340 1,679

Answer 28

this is the table

Answer 29

Find a quadratic function that models the data as a function of x, the number of weeks. Use the model to estimate the number of waterfowl at the lake on week 8.

Answer 30

are you familiar with polynomial interpolation?

Answer 31

no whats that!

Answer 32

it's what is needed to solve this problem

polynomial interpolation is the method to find the polynomial (in this case, the quadratic) that goes through every point

Answer 33

no i dont

Answer 34

hmm seems odd how you need to use something you haven't learned yet

Answer 35

could you explain it

Answer 36

sure I can do that

Answer 37

these are my answers p(x)=33x^2-24x+635:2,555 waterfowl
p(x)=38x^2+24x+585: 2615
p(x)=33x^2-24x+635:2,084
p(x)=38x^2+24x+585: 3,209

Answer 38

well since you have the answer choices, we can do it a much simpler way if you want

Answer 39

yeah

Answer 40

ok, for each answer choice, the number at the very end is going to be ignored for now, so erase those numbers for each answer choice to get


p(x)=33x^2-24x+635
p(x)=38x^2+24x+585
p(x)=33x^2-24x+635
p(x)=38x^2+24x+585

Answer 41

ok

Answer 42

Now look at the last numbers of each choice

They are: 635, 585, 635, 585

Answer 43

These are the y - intercepts of each answer choice

Answer 44

so how do i know which one to pick

Answer 45

remember that the y-intercepts are the points (0, b) where b is some number

Answer 46

because we have the point (0, 635), this means that we can rule out choices B and D because they have the y-intercept of (0, 585)

Answer 47

So we now have these two answer choices


A) p(x)=33x^2-24x+635

C) p(x)=33x^2-24x+635

Answer 48

they are the same polynomial luckily, so this means that the answer to the first part is p(x)=33x^2-24x+635

Answer 49

to answer the second part, you plug in x = 8 and evaluate/simplify

Answer 50

so its A because it equals 2555

Answer 51

you got it

Answer 52

you make it seem so easy :)

Answer 53

thanks lol

Answer 54

the other way would have done the trick too, but it was much longer/harder, so i figured i'd go this way

Answer 55

Can you check my answer for this one. there is a table
x -2 0 4
f(x) 0 -6 78

Answer 56

i have to write equation indard form of parabola that models the values of the table
I got y=5x^2+4x-6

Answer 57

in standard form sorry!

Answer 58

is that right

Answer 59

unfortunately no, here's why

plug in x = -2 to get...

5x^2+4x-6

5(-2)^2+4(-2)-6

5(4)+4(-2)-6

20 - 8 - 6

12 - 6

6

So 5x^2+4x-6 = 6 when x = -2

BUT we want 5x^2+4x-6 to equal 0 when x = -2

So it can't be the answer because of this reason.

Answer 60

ohh so how could i do it

Answer 61

my other answers choices are y=6x^2+5x-4
y=-4x^2-5x+6
Y=4x^2+5x-6

Answer 62

is there a way i can work backwards or something

Answer 63

well you know that one point is (0, -6)

Answer 64

so the y-intercept is (0, -6) which means that the last number must be a -6

Answer 65

so in this case it has to be the last one

Answer 66

so that leaves the last choice y=4x^2+5x-6

sure enough, plugging in x = -2 gives you

y=4x^2+5x-6

y=4(-2)^2+5(-2)-6

y=0

which is what we want (so far so good)

and plugging in x = 4 gives you

y=4x^2+5x-6

y=4(4)^2+5(4)-6

y=78

which is also what we want

So all 3 points lie on this parabola

Answer 67

I cant explain how grateful i am! thanks so much!!!

Answer 68

You really were a big help!!

Answer 69

yw

Answer 70

I'm glad that I was

Answer 71

hmm interesting, thx for letting me know

Answer 72

@lala2 to answer the confusion of the first question you asked and @jim_thompson5910 the reason the solution is 0<x<8 is because the x axis is the second and y is the feet high. So he would have thrown the ball at 0 seconds not 2 seconds.