Question 1035268
{{{y=x^2+6x+5}}}
<pre><b>
Since the coefficient of x<sup>2</sup> is positive,
we know that the parabola opens upward.

We must get it into the form:

{{{(x-h)^2=4p(y-k)}}}

where (h,k) is the vertex and p is the number of units
the vertex is from both the focus (point) and the 
directrix (line).

{{{y=x^2+6x+5}}}

Get the x terms on the left and other terms on the right

{{{-x^2-6x=-y+5}}}

Multiply through by -1

{{{x^2+6x=y-5}}}

Complete the square on the left:
1. Multiply the coefficient of x by 1/2, getting 6(1/2) = 3
2. Square the result of step 1, getting +9
3. Add +9 to both sides 

{{{x^2+6x+9=y-5+9}}}

Factor the left side, combine like terms on the right:

{{{(x+3)(x+3)=y+4}}}

Write (x+3)(x+3) as (x+3)<sup>2</sup>

{{{(x+3)^2=y+4}}}

Factor out 1 on the right to show the value of 4p

{{{(x+3)^2=1(y+4)}}}

Compare to

{{{(x-h)^2=4p(y-k)}}}

-h=+3 so h=-3
4p=1 so p=1/4
-k=+4 so k=-4

Vertex = (h,k) = (-3,-4)

We can easily get the intercepts from
the original equation.

{{{y=x^2+6x+5}}}
{{{x^2+6x+5=0}}}
{{{(x+5)(x+1)=0}}}
So the x-intercepts are (-5,0) and (-1,0)

To get the y-intercept:

{{{y=x^2+6x+5}}}
{{{y=0^2+6*0+5}}}
{{{y=5}}}

So the y-intercept is (0,5), so we draw
the graph:

{{{drawing(400,400,-8,3,-5,6,

graph(400,400,-8,3,-5,6,x^2+6x+5),

circle(-5,0,0.09),circle(-5,0,0.07),circle(-5,0,0.05),circle(-5,0,0.03),circle(-5,0,0.01),

circle(-1,0,0.09),circle(-1,0,0.07),circle(-1,0,0.05),circle(-1,0,0.03),circle(-1,0,0.01),

circle(0,5,0.09),circle(0,5,0.07),circle(0,5,0.05),circle(0,5,0.03),circle(0,5,0.01),

circle(-3,-4,0.09),circle(-3,-4,0.07),circle(-3,-4,0.05),circle(-3,-4,0.03),circle(-3,-4,0.01)

)}}}

The focus is the point which is {{{p=1/4}}} of a unit above the
vertex, and the directrix is a line {{{p=1/4}}} of a unit below
the vertex. I'll make them green:

{{{drawing(400,400,-8,3,-5,6,
green(line(-10,-4.25,10,-4.25)),
graph(400,400,-8,3,-5,6,x^2+6x+5),


green(circle(-3,-3.75,0.09),circle(-3,-3.75,0.07),circle(-3,-3.75,0.05),circle(-3,-3.75,0.03),circle(-3,-3.75,0.01)),




circle(-5,0,0.09),circle(-5,0,0.07),circle(-5,0,0.05),circle(-5,0,0.03),circle(-5,0,0.01),

circle(-1,0,0.09),circle(-1,0,0.07),circle(-1,0,0.05),circle(-1,0,0.03),circle(-1,0,0.01),

circle(0,5,0.09),circle(0,5,0.07),circle(0,5,0.05),circle(0,5,0.03),circle(0,5,0.01),

circle(-3,-4,0.09),circle(-3,-4,0.07),circle(-3,-4,0.05),circle(-3,-4,0.03),circle(-3,-4,0.01)

)}}}

Since the focus (the green point) is p=1/4 of a unit above 
the vertex (-3,-4), the focus has the same x-coordinate as 
the vertex, and the y-coordinate is 1/4 of a unit above the 
the y-coordinate of the vertex. So we add 1/4 to the y-coordinate 
{{{-4+1/4=-16/4+1/4=-15/4}}}

So the focus (the green point) is the point {{{(matrix(1,3,-3,",",-15/4))}}}

The directrix (the green line) is a horizontal line which is
1/4 of a unit below the y-coordinate of the vertex, so we subtract
1/4 from the y-coordinate of the vertex to find out how far below
the x-axis the directrix is.
{{{-4-1/4=-16/4-1/4=-17/4}}}

So the directrix is the horizontal line which has the equation:

{{{y = -17/4}}}
   
Edwin</pre></b>