SOLUTION: Find the directrix, the focus and the roots of the parabola y=x^2 - 5x + 4 Please help!!!! I'm so confused. If you could help with this one problem I think I could get the oth

Algebra ->  Finance -> SOLUTION: Find the directrix, the focus and the roots of the parabola y=x^2 - 5x + 4 Please help!!!! I'm so confused. If you could help with this one problem I think I could get the oth      Log On


   

Question 162445: Find the directrix, the focus and the roots of the parabola
y=x^2 - 5x + 4
Please help!!!! I'm so confused. If you could help with this one problem I think I could get the others like it.
Thanks!!
Answer by KnightOwlTutor(293) About Me  (Show Source):
You can put this solution on YOUR website!
Use the quadratic equation to get the roots

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-5x%2B4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-5%29%5E2-4%2A1%2A4=9.

Discriminant d=9 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+9+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+9+%29%29%2F2%5C1+=+4
x%5B2%5D+=+%28-%28-5%29-sqrt%28+9+%29%29%2F2%5C1+=+1

Quadratic expression 1x%5E2%2B-5x%2B4 can be factored:
1x%5E2%2B-5x%2B4+=+1%28x-4%29%2A%28x-1%29
Again, the answer is: 4, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-5%2Ax%2B4+%29


In order to find the vertex point you need to complete the square. You take the middle term half it and then square it.
y=x^2 - 5x + 4
Subtract 4 from each side
y-4=x^2 - 5x take -5 and divide by 2 -5/2 and then square this term 25/4
y-4+25/4=(x-5/2)^2
y-16/4+25/4=(x-5/2)^2
y+9/4=(x-5/2)^2
Subtract 9/4 from both sides
y=(x-5/2)^2 -9/4
The vertex o=is (5/2,-9/4)
We know that since a is positive the parabola is facing upward
the directrix is below the vertex at distance -c on the x axis
the focus is a distance +c above the vertex
a=coefficient on x^2 term
It is 1 The equation for determining the c value is a=1=1/4c


4c=1
c=1/4
Y value for vertex+1/4= Focus, x value for vertex
the directrix is a line y=8/4=2