Answer:
Part 1) ![-\frac{1}{2}x^2+10x=0](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B10x%3D0)
Part 2)![y=-\frac{1}{2}(x-10)^{2}+72](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7D%28x-10%29%5E%7B2%7D%2B72)
Part 3) The domain of firework #1 is the interval (-∞,∞) and the range is the interval (-∞,50]
Step-by-step explanation:
1.) What is the equation, in standard form, of the path of firework #1?
we know that
The equation of a vertical parabola in vertex form is equal to
![y=a(x-h)^{2}+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E%7B2%7D%2Bk)
where
a is a coefficient
(h,k) is the vertex
In this problem we have
The vertex is the point (10,50)
substitute
![y=a(x-10)^{2}+50](https://tex.z-dn.net/?f=y%3Da%28x-10%29%5E%7B2%7D%2B50)
we have the roots
(0,0) and (20,0)
For x=0,y=0
substitute in the equation and solve for a
![0=a(0-10)^{2}+50](https://tex.z-dn.net/?f=0%3Da%280-10%29%5E%7B2%7D%2B50)
![100a=-50](https://tex.z-dn.net/?f=100a%3D-50)
![a=-\frac{1}{2}](https://tex.z-dn.net/?f=a%3D-%5Cfrac%7B1%7D%7B2%7D)
substitute
![y=-\frac{1}{2}(x-10)^{2}+50](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7D%28x-10%29%5E%7B2%7D%2B50)
Convert to standard form
A quadratic equation in standard form is equal to
so
![0=-\frac{1}{2}(x^2-20x+100}+50](https://tex.z-dn.net/?f=0%3D-%5Cfrac%7B1%7D%7B2%7D%28x%5E2-20x%2B100%7D%2B50)
![0=-\frac{1}{2}x^2+10x-50+50](https://tex.z-dn.net/?f=0%3D-%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B10x-50%2B50)
![-\frac{1}{2}x^2+10x=0](https://tex.z-dn.net/?f=-%5Cfrac%7B1%7D%7B2%7Dx%5E2%2B10x%3D0)
2.) What is the equation, in vertex form, of the path of firework #2?
we know that
The equation of a vertical parabola in vertex form is equal to
![y=a(x-h)^{2}+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E%7B2%7D%2Bk)
where
a is a coefficient
(h,k) is the vertex
In this problem we have
The vertex is the point (10,72)
substitute
![y=a(x-10)^{2}+72](https://tex.z-dn.net/?f=y%3Da%28x-10%29%5E%7B2%7D%2B72)
we have the root
(22,0)
For x=22,y=0
substitute in the equation and solve for a
![0=a(22-10)^{2}+72](https://tex.z-dn.net/?f=0%3Da%2822-10%29%5E%7B2%7D%2B72)
![144a=-72](https://tex.z-dn.net/?f=144a%3D-72)
![a=-\frac{1}{2}](https://tex.z-dn.net/?f=a%3D-%5Cfrac%7B1%7D%7B2%7D)
substitute
![y=-\frac{1}{2}(x-10)^{2}+72](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7D%28x-10%29%5E%7B2%7D%2B72)
3.) What is the domain and range of firework #1?
we know that
The domain of firework #1 is the interval (-∞,∞)
All real numbers
The range of firework #1 is the interval (-∞,50]
![y\leq 50](https://tex.z-dn.net/?f=y%5Cleq%2050)
All real numbers less than or equal to 50