Question

What are the vertex, focus, and directrix of the parabola with equation y = x^2 – 10x + 33?

Answer 1

opens up so
(x-h)^2=4P(y-k)
complete the square with the x term
take 1/2 of -10 and square it and add that to both sides

y+25=x^2-10x+25+33
factor perfect square
y+25=(x-5)^2+33
minus 33 both sides
y-8=(x-5)^2
force factor out a 4
4(1/4)(y-8)=(x-5)^2
(x-5)^2=4(1/4)(y-8)
vertex is (5,8)

distance from directix is P or 1/4 or 0.25
since opens up, directix is ycoordinate-0.25 aka y=7.75 is directix
focus=(5,8+0.25)=(5,8.25)


vertex=(5,8)
directix: y=7.75
focus: (5,8.25)

Answer 2

Vertex:(5,8)

Focus:(5,8.25)

Directrix: y=7.75

Explanation:

For the equation of the parabola in the form:

(x - h)² = 4p(y - k), where p≠ 0

The vertex of this parabola is at (h, k).

The focus is at (h, k + p).

The directrix is the line y = k - p.

Converting the equation y=x²-10x+33 in the above form

y=x²-10x+25+8

y=(x-5)²+8

(x-5)²=(y-8)

On comparing this equation with the above equation, we see that

h=5 k=8 and 4p=1

or p=1/4

Vertex=(5,8)

Focus=(5,8+1/4)

=(5,8.25)

Directrix: y=8-1/4

y= 7.75