1. I'm assumig that the paths are perfect parabolas
this means that their general forms can be written in ![y=ax^2+bx+c](https://tex.z-dn.net/?f=y%3Dax%5E2%2Bbx%2Bc)
it's easier to find vertex form first then expand to get general form
vertex form is
where the vertex is (h,k) and a is a constant
firework #1
vertex is (10,50), so (h,k)=(10,50) and h=10, k=50
![h_1=a(t-10)^2+50](https://tex.z-dn.net/?f=h_1%3Da%28t-10%29%5E2%2B50)
to find the value of
, subsitute another point
(0,0)
![0=a(0-10)^2+50)](https://tex.z-dn.net/?f=0%3Da%280-10%29%5E2%2B50%29)
![0=100a+50](https://tex.z-dn.net/?f=0%3D100a%2B50)
![a=\frac{-1}{2}](https://tex.z-dn.net/?f=a%3D%5Cfrac%7B-1%7D%7B2%7D)
so the equation in vertex form is ![h_1=\frac{-1}{2}(t-10)^2+50](https://tex.z-dn.net/?f=h_1%3D%5Cfrac%7B-1%7D%7B2%7D%28t-10%29%5E2%2B50)
expand to get general form
![h_1=\frac{-1}{2}(t^2-20t+100)+50](https://tex.z-dn.net/?f=h_1%3D%5Cfrac%7B-1%7D%7B2%7D%28t%5E2-20t%2B100%29%2B50)
![h_1=\frac{-1}{2}t^2+10t-50+50](https://tex.z-dn.net/?f=h_1%3D%5Cfrac%7B-1%7D%7B2%7Dt%5E2%2B10t-50%2B50)
![h_1=\frac{-1}{2}t^2+10t](https://tex.z-dn.net/?f=h_1%3D%5Cfrac%7B-1%7D%7B2%7Dt%5E2%2B10t)
2.
same as last time
vertex is (10,72) so (h,k)=(10,72) so h=10 and k=72
equation is ![h_2=a(t-10)^2+72](https://tex.z-dn.net/?f=h_2%3Da%28t-10%29%5E2%2B72)
find ![a](https://tex.z-dn.net/?f=a)
use another point
(0,22)
![22=a(0-10)^2+72](https://tex.z-dn.net/?f=22%3Da%280-10%29%5E2%2B72)
![22=100a+72](https://tex.z-dn.net/?f=22%3D100a%2B72)
![-50=100a](https://tex.z-dn.net/?f=-50%3D100a)
![a=\frac{-1}{2}](https://tex.z-dn.net/?f=a%3D%5Cfrac%7B-1%7D%7B2%7D)
so the equation in vertex form is ![h_2=\frac{-1}{2}(t-10)^2+72](https://tex.z-dn.net/?f=h_2%3D%5Cfrac%7B-1%7D%7B2%7D%28t-10%29%5E2%2B72)
3.
range is the numbers that h is allowed to be
think about what h represents. it represents the height of the rocket
from the graph, we can see that the lowest possible height is 0yd and the highest height is 50yd
so range is 0 to 50 or 0≤h≤50
domain is the numbers that t is allowed to be
think about what t represents. it represents how long the rocket has been flying
it will stop flying when it hits the ground or at t=20
it starts flying at t=0
so domain is from 0 to 20 or 0≤t≤20