Find the vertex, focus, and the directrix of the parabola

In summary, for the first equation (x+1)^2 +8(y+3) = 0, the vertex is at (0,0), the focus is at (0,-3/2), and the directrix is the line y=-3/2. For the equations y^2=-6x and x+y^2=0, you can find the vertex, focus, and directrix by switching the x and y variables and using the formula (y-k)^2=4p(x-h).
  • #1
konartist
47
0
Find the vertex, focus, and the directrix of the parabola.

I get tripped up sometimes, but I know how to find all of the stuff with an equation like this :
(x+1)^2 +8(y+3) = 0

But how do I find it with equations like this
y^2=-6x

or

x+y^2=0

Can you complete the square with only two terms?
 
Physics news on Phys.org
  • #2
konartist said:
Find the vertex, focus, and the directrix of the parabola.

I get tripped up sometimes, but I know how to find all of the stuff with an equation like this :
(x+1)^2 +8(y+3) = 0

But how do I find it with equations like this
y^2=-6x

or

x+y^2=0

Can you complete the square with only two terms?

Reverse x and y! That is, change x+ y2= 0 to x2+ y= 0 which you know how to do. Once you have found the vertex, focus, directrix for that, switch back: If you found (a,b) as the vertex of x2+y= 0, then (b,a) is the vertex of x+ y2= 0. If you found y= c as the directrix of x2+ y= 0, then the directrix of
y2+ x= 0 is x= c.
 
  • #3
My work

(x-h)^2=4p(y-k)
since h and k are 0 then the vertex = (0,0)
How do I find the focus and the directrix though?
Focus maybe = 4p = -6 P= -3/2 so the focus (0,-3/2) ?
Directrix ?
 

1. What is the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction from increasing to decreasing (or vice versa). It is also the point where the axis of symmetry intersects the parabola.

2. How do you find the vertex of a parabola?

To find the vertex of a parabola, you can use the formula x = -b/2a, where a is the coefficient of the squared term and b is the coefficient of the linear term in the parabola's equation. This will give you the x-coordinate of the vertex. To find the y-coordinate, simply substitute the x-coordinate into the equation.

3. What is the focus of a parabola?

The focus of a parabola is a fixed point inside the parabola, located at a distance p from the directrix and on the axis of symmetry. The value of p is called the focus distance and is equal to 1/4a, where a is the coefficient of the squared term in the parabola's equation.

4. How do you find the focus of a parabola?

To find the focus of a parabola, you can use the formula F = (p, k + 1/4a), where p is the focus distance, k is the y-coordinate of the vertex, and a is the coefficient of the squared term in the parabola's equation.

5. What is the directrix of a parabola?

The directrix of a parabola is a fixed line outside the parabola, parallel to the axis of symmetry, and at a distance p from the focus. The equation of the directrix is x = -p, where p is the focus distance.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
Replies
44
Views
3K
Replies
7
Views
3K
  • Precalculus Mathematics Homework Help
Replies
6
Views
560
  • Precalculus Mathematics Homework Help
Replies
5
Views
2K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
21
Views
1K
  • Precalculus Mathematics Homework Help
Replies
8
Views
1K
Back
Top