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

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) (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 (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=9 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 4, 1. Here's your graph:

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

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