SOLUTION: find (a)the directrix, (b)the focus, and (c)the roots of the parabola y = x^2 - 5x + 4

We have to get it in the form 






Swap sides:



Add -4 to both sides the get the x-terms alone on the left.



Multiply the coefficient of x, which is  by 
This gives .  Now we square  and get 
We add  to both sides:



Factor the left side, and write the  as 



Write the left side as the square of a binomial,
combine the fractions on the right:



Now so that equation will look like this:



we put the right side in parentheses and put a 1
coefficient before the parentheses, like this:



Now we can compare it to the equation:



and get , so 

, so 

So the vertex is (,) or (,)

And , so 

Now let's begin by plotting the vertex, which is (,),

But for plotting purposes, mixed numbers are better
than improper fractions, so for plotting vertex (,),
we rewrite it as (,)




Now we will find the x-intercepts, by settng 
in the original equation, and finding the "roots":




, so 
, so 

So the x-intecepts are (1,0) and (4,0) 

So we plot those:



and sketch in the parabola:



Now the focus is p units from the vertex INSIDE
the parabola, so since the parabola opens upward,
we add P or  to the y-coordinate of the
vertex. Since the vertx is (,),
the focus = (,) = (,) 
= (,)

So we draw that point:



Now the directrix is a line OUTSIDE the parabola which is
also p-units, or  from the vertex.

Since the y-coordinate of the vertex is  we want
the directrix to be  unit below the vertex, so we
subtract 

So the directrix is the horizontal line whose equation is 

.  I'll draw it in in green:



So the focus is the POINT (,) and the 

directrix is the LINE 

The "roots" are really the y-coordinates of the 
x-intercepts or 1 and 4.

Edwin