Suppose a person on the surface of an asteroid kicks a ball. The table shows the height, h, in meters, of the ball over time, t, in seconds, after it is kicked into the air.
t h
0
0 0
3 16.2
6 28.8
9 37.8
12 43.2
15 45
18 43.2
21 37.8
24 28.8
27 16.2
30 0
Write the quadratic relation in vertex form that models this situation.
since h(0)=0 and h(30)=0, the vertex must be at t=15
h(t) = 45-a(t-15)^2
45-a*15^2 = 0
225a = -45
a = -1/5
h(t) = -1/5 (t-15)^2 + 45
peak (vertex) at t = 15, h =45
note symmetry about t = 15
(t-15)^2 = 4a(h-45)
when t = 0, h = 0
225 = 4a (-45)
a = -5/4
(t-15)^2 = -5(h-45)
check if h = 0 at t = 30
(15)^2 =-5(h-45)
225 = -5 h +225 sure enough h = 0
Thank you so much guys I appreciate it!
To write the quadratic relation in vertex form, we need to find the equation that represents the height of the ball as a function of time. The vertex form of a quadratic equation is given as:
y = a(x-h)^2 + k
Where (h, k) represents the vertex of the parabola.
In this scenario, the time (t) is represented by the variable x, and the height (h) is represented by the variable y.
To find the vertex form equation, we need to determine the values of a, h, and k.
First, let's look at the table of values. From the table, we can see that when t is 15 seconds, the height of the ball is at its maximum, which is 45 meters. So the vertex occurs at (15, 45).
Using the vertex coordinates, we can substitute them into the vertex form equation:
y = a(x-h)^2 + k
y = a(x-15)^2 + 45
Now we need to find the value of a. Since we know the height of the ball at t=0, we can substitute those coordinates into the equation:
0 = a(0-15)^2 + 45
0 = 225a + 45
-45 = 225a
a = -45/225
a = -1/5
Now we have all the values needed to write the quadratic equation in vertex form:
y = (-1/5)(x-15)^2 + 45
So the quadratic relation in vertex form that models this situation is y = (-1/5)(x-15)^2 + 45.