A model rocket is shot straight up into the air. The table shows its height, h(t), at time t. Graph the table below and determine a function, in standard form, that estimates the height of the rocket at any given time.

Question

A model rocket is shot straight up into the air. The table shows its height, h(t), at time t. Graph the table below and determine a function, in standard form, that estimates the height of the rocket at any given time.

Time(s) Height (m)
0 0.0
1 25.1
2 40.4
3 45.9
4 41.6
5 27.5
6 3.6
I graphed it and there is only one x intercept which is (0,0) and the vertex is (3, 45.9) im having trouble determining a function in standard form

Answer 1

clearly there is an x-intercept somewhere just beyond 6. See how the height is descending.

Since the given h values do not show an obvious symmetry, it is not immediately clear where the vertex is.

However, since h(0) = 0, if

h(t) = at^2 + bt + c

c = 0, so

h(t) = at^2 + bt
Plugging in t=1 and t=2, we have

a+b = 25.1
4a+2b = 40.4
a = -4.9, b=30

h(t) = 30t - 4.9t^2

So, the other t-intercept is at t = 6.12

Answer 2

So what is the vertex?

Answer 3

Huh? You said above what the vertex is.

as always, it is at t = -b/2a

It is not at t=3, but very close. Probably good enough.

Answer 4

To determine a function in standard form that estimates the height of the rocket at any given time, we need to find a quadratic equation that fits the given data points. A quadratic equation is in the form of:

h(t) = at^2 + bt + c

To find the values of a, b, and c, we can use the given data points and set up a system of equations. Let's start by plugging in the values from the table into the equation:

When t = 0, h(t) = 0:
0 = a(0)^2 + b(0) + c
0 = 0 + 0 + c
c = 0

Now we have: h(t) = at^2 + bt

Next, we use two more data points to set up two additional equations:

When t = 1, h(t) = 25.1:
25.1 = a(1)^2 + b(1) + 0
25.1 = a + b

When t = 3, h(t) = 45.9:
45.9 = a(3)^2 + b(3) + 0
45.9 = 9a + 3b

Now we have a system of equations:

25.1 = a + b
45.9 = 9a + 3b

We can solve this system of equations to find the values of a and b. Subtracting the first equation from the second equation:

20.8 = 8a + 2b

Solving for b in terms of a:

2b = 20.8 - 8a
b = 10.4 - 4a

Substituting this value of b into the first equation:

25.1 = a + (10.4 - 4a)
25.1 = 10.4 - 3a
3a = 10.4 - 25.1
3a = -14.7
a = -4.9

Now we can substitute the value of a back into the equation for b:

b = 10.4 - 4(-4.9)
b = 10.4 + 19.6
b = 30

Therefore, the function in standard form that estimates the height of the rocket at any given time is:

h(t) = -4.9t^2 + 30t

This quadratic equation represents the estimated height of the rocket as a function of time.