Find the x-intercepts of the parabola with the vertex (4,-1) and y-intercept (0,15)

1. A parabola is the graph of the function 



(0, 15 ) is a point of this parabola, so 

So c=15, which means we decrease the number of unknowns and write again: 


2. Now, since (4, -1) is another point in the parabola:


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: 





solution 1: x-4=1, x=5
solution 2: x-4=-1, x=3

The x-intercepts are (3, 0) and (5, 0)
 

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