Answer:
Part 1) The function of the First graph is ![f(x)=(x-3)(x+1)](https://tex.z-dn.net/?f=f%28x%29%3D%28x-3%29%28x%2B1%29)
Part 2) The function of the Second graph is ![f(x)=-2(x-1)(x+3)](https://tex.z-dn.net/?f=f%28x%29%3D-2%28x-1%29%28x%2B3%29)
Part 3) The function of the Third graph is
See the attached figure
Step-by-step explanation:
we know that
The quadratic equation in factored form is equal to
![f(x)=a(x-c)(x-d)](https://tex.z-dn.net/?f=f%28x%29%3Da%28x-c%29%28x-d%29)
where
a is the leading coefficient
c and d are the roots or zeros of the function
Part 1) First graph
we know that
The solutions or zeros of the first graph are
x=-1 and x=3
The parabola open up, so the leading coefficient a is positive
The function is equal to
![f(x)=a(x-3)(x+1)](https://tex.z-dn.net/?f=f%28x%29%3Da%28x-3%29%28x%2B1%29)
Find the value of the coefficient a
The vertex is equal to the point (1,-4)
substitute and solve for a
![-4=a(1-3)(1+1)](https://tex.z-dn.net/?f=-4%3Da%281-3%29%281%2B1%29)
![-4=a(-2)(2)](https://tex.z-dn.net/?f=-4%3Da%28-2%29%282%29)
![a=1](https://tex.z-dn.net/?f=a%3D1)
therefore
The function is equal to
![f(x)=(x-3)(x+1)](https://tex.z-dn.net/?f=f%28x%29%3D%28x-3%29%28x%2B1%29)
Part 2) Second graph
we know that
The solutions or zeros of the first graph are
x=-3 and x=1
The parabola open down, so the leading coefficient a is negative
The function is equal to
![f(x)=a(x-1)(x+3)](https://tex.z-dn.net/?f=f%28x%29%3Da%28x-1%29%28x%2B3%29)
Find the value of the coefficient a
The vertex is equal to the point (-1,8)
substitute and solve for a
![8=a(-1-1)(-1+3)](https://tex.z-dn.net/?f=8%3Da%28-1-1%29%28-1%2B3%29)
![8=a(-2)(2)](https://tex.z-dn.net/?f=8%3Da%28-2%29%282%29)
![a=-2](https://tex.z-dn.net/?f=a%3D-2)
therefore
The function is equal to
![f(x)=-2(x-1)(x+3)](https://tex.z-dn.net/?f=f%28x%29%3D-2%28x-1%29%28x%2B3%29)
Part 3) Third graph
we know that
The solutions or zeros of the first graph are
x=-2 and x=6
The parabola open up, so the leading coefficient a is positive
The function is equal to
![f(x)=a(x-6)(x+2)](https://tex.z-dn.net/?f=f%28x%29%3Da%28x-6%29%28x%2B2%29)
Find the value of the coefficient a
The vertex is equal to the point (2,-8)
substitute and solve for a
![-8=a(2-6)(2+2)](https://tex.z-dn.net/?f=-8%3Da%282-6%29%282%2B2%29)
![-8=a(-4)(4)](https://tex.z-dn.net/?f=-8%3Da%28-4%29%284%29)
![a=0.5](https://tex.z-dn.net/?f=a%3D0.5)
therefore
The function is equal to
![f(x)=0.5(x-6)(x+2)](https://tex.z-dn.net/?f=f%28x%29%3D0.5%28x-6%29%28x%2B2%29)