Question

I have the vertex, focus, and directrix, how do I find a point on the parabola in order to draw it?

Answer 1

A parabola can have horizontal axis \(y^2=4cx\) or vertical axis \(x^2=4cy\) (see figure attached).
c is the distance between the focus & the vertex, AND the distance between the vertex and the directrix. If you have all three items, you can draw the axis of the parabola, the focus and the vertex.
Calculate the value of c (half distance between focus and directrix, or distance between focus and vertex to complete the equation of the parabola.
After that, assume different values of x (or y) to complete the plot of the conic.

Answer 2

Note: if c is negative, the directrix and parabola are mirror images about the vertex.

Answer 3

y^2 + 6y + 16x + 41 = 0 This is my equation, I have:vertex (-2,-3) focus (-6,-3) directrix (x=2). What is the step by step to finding a point to draw the parabola?

Answer 4

Write as
\((y+3)^2=4(-4)(x+2)\)
This shows a parabola with a horizontal axis, the vertex is at (-2,-3). The negative sign on the right hand side shows that c=-4, or it is a mirror image, i.e. the parabola opens to the left.
Once we have found c=-4, we know that the focus F is to the left of the vertex, and the directrix is to the right.
Assign different values of x and calculate y to plot the graph.