SOLUTION: A parabola has its vertex at (-2,1) and its focus at (-2,-1). Write the equations of the parabola, the directrix, and the axis of symmetry. Graph the parabola.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A parabola has its vertex at (-2,1) and its focus at (-2,-1). Write the equations of the parabola, the directrix, and the axis of symmetry. Graph the parabola.       Log On


   

Question 1067892: A parabola has its vertex at (-2,1) and its focus at (-2,-1). Write the equations of the parabola, the directrix, and the axis of symmetry. Graph the parabola.
Found 2 solutions by josgarithmetic, math_helper:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Directrix is on the other side of the vertex than the focus, by 1-(-1)=2 units, at y=3. You can also call this the general point, (x,3). This parabola will have its vertex as the maximum point. Coefficient of the x%5E2 term will be negative.

The given vertex and focus tell you axis of symmetry is x=-2.

Definition of a parabola uses the directrix, the focus, and the distance formula.
Distance from a point on parabola to focus is equal to distance of same point on parabola to directrix:

Simplify this equation and put into whatever form you want or need.

highlight%28-8%28y-1%29=%28x%2B2%29%5E2%29
OR
highlight%28y=-%281%2F8%29%28x%2B2%29%5E2%2B1%29, in standard form

Derivation of Equation for Parabola given directrix and focus - video

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
With vertex at (-2,1) and focus at (-2,-1):
Since the vertex and focus are on the line x=-2, the axis of symmetry is +highlight%28x=-2%29
Also, for this case, the directrix will be y = b, where b = 1+(1-(-1)) = 1+2 = 3 (the y component of the vertex + the distance between vertex and focus in vertical direction).
Directrix: +highlight%28y+=+3%29
Equation of parabola with vertex (h,k): +%28x-h%29%5E2+=+4p%28y-k%29+
p = -b = -3
h = -2
k = 1
+%28x-%28-2%29%29%5E2+=+4%28-3%29%28y-1%29+
solving for y:
Eq of Parabola: ++highlight%28y+=+-%281%2F12%29%28x%2B2%29%5E2+%2B+1%29+


Green line is directrix.

—
Sorry to say, I made a mistake with p, it is only -2, not -3 (negative of distance from directrix to vertex, not just -b like I originally posted), thus the proper equation for the parabola is:
+highlight%28y+=+-%281%2F8%29%28x%2B2%29%5E2+%2B+1%29+