How to find the focus and directrix?

The equation of a parabola is of the form

y = A(x - h)^2 + k

where (h, k) is the vertex of the parabola, and A is a constant that determines the shape of the parabola. In your equation, A = 1/4, so the parabola is symmetric about the line x = h.

To find the vertex, we complete the square. First, we move the constant term to the left side of the equation:

y - 2 = 1/4x^2 + x

Then, we can factor out a 1/4 from the right side:

y - 2 = 1/4(x^2 + 4x)

To complete the square, we need to add (4/2)2=4 to both sides of the equation. This give us:

y - 2 + 4 = 1/4(x^2 + 4x + 4)

We can now rewrite the expression in parentheses as a squared term:

y + 2 = 1/4(x + 2)^2

The vertex of the parabola is therefore the point (h, k) = (-2, 2).

The focus of a parabola is always located on its axis of symmetry, and is the same distance away from the vertex and the directrix. In this case, the axis of symmetry is the line x = -2, so the focus is also at the point (-2, 2 + p), where p is the distance between the focus and the vertex.

We can find the value of p using the formula

p = 1/(4A)

In this case, A = 1/4, so p = 1/(4 * 1/4) = 1.

Therefore, the focus of the parabola is at the point (-2, 3).

The directrix of a parabola is always parallel to the axis of symmetry, and is located the same distance below (or above) the vertex as the focus is above (or below) the vertex. In this case, the directrix is the line y = 2 - 1 = 1.

Therefore, the focus and directrix of the parabola y = 1/4x^2 + x + 2 are (-2, 3) and y = 1, respectively.

I hope this helps! Let me know if you have any other questions.

Arrange the given parabola equation to vertex (h,k)    form:  y =  a ( x- h) ^2 + k   

          by completing the square for x :

                 1/4 ( x^2 + 4x) +2

                 1/4 ( x^2 + 4x + 4)   -1 + 2

                 1/4 ( x+2)^2 +1      Shows vertex = -2,1   

 the equation of a parabola in vertex form y=a(x−h)^2+k, ======> the focus is (h,k+1/4a)

               focus is then  -2 , 1 + 1/(4 * 1/4) =   (  -2,2 )  

    and directrix is equal distance from the vertex as the focus and is    y = 0