1. A parabola is the graph of the function
![y=f(x)=ax^2+bx+c](https://tex.z-dn.net/?f=y%3Df%28x%29%3Dax%5E2%2Bbx%2Bc)
(0, 15 ) is a point of this parabola, so
![15=f(0)=a(0)^2+b(0)+c=c](https://tex.z-dn.net/?f=15%3Df%280%29%3Da%280%29%5E2%2Bb%280%29%2Bc%3Dc)
So c=15, which means we decrease the number of unknowns and write again:
![y=f(x)=ax^2+bx+15](https://tex.z-dn.net/?f=y%3Df%28x%29%3Dax%5E2%2Bbx%2B15)
2. Now, since (4, -1) is another point in the parabola:
![-1=f(4)=a(4)^2+b(4)+15](https://tex.z-dn.net/?f=-1%3Df%284%29%3Da%284%29%5E2%2Bb%284%29%2B15)
16a+4b=-16
dividing by 4:
4a+b=-4
We also know that -b/2a gives the x-coordinate of the vertex:
-b/2a=4
-b=8a
b=-8a
Substitute b=-8a in 4a+b=-4,
we get 4a-8a=-4
-4a=-4
a=1, then b=-8
So y=f(x)=x^2-8x+15
The roots of the expression, which are the x-intercepts can be found by solving the equation:
![x^2-8x+15=0](https://tex.z-dn.net/?f=x%5E2-8x%2B15%3D0)
![x^2-8x+16-1=0](https://tex.z-dn.net/?f=x%5E2-8x%2B16-1%3D0)
![(x-4)^2=1](https://tex.z-dn.net/?f=%28x-4%29%5E2%3D1)
solution 1: x-4=1, x=5
solution 2: x-4=-1, x=3
The x-intercepts are (3, 0) and (5, 0)