Recall the standard form of an ellipse centered at the origin, (x^2) / (a^2) + (y^2) / (b^2) = 1
where a and b are the radii.
Define the vertices. For an ellipse centered at the origin, the vertices are (a,0) (-a,0) (0,b) (0,-b)
Identify a and b from the standard form letting "a" be the larger number and "b" the smaller.
For example, if given (x^2) / 81 + (y^2) / 16 = 1,
a = 81^ (1/2) = 9
b = 16^(1/2) = 4.
Write the vertices. In the example these would be, (9,0) (-9,0) (0,4) (0,-4).
Plot the vertices on the graph.
Connect the dots of the vertices to complete the ellipse. Remember that conic sections are parabolic and therefore "circular" in nature.
Recall the equation of a hyperbola, (x^2) / (a^2) - (y^2) / (b^2) = 1 and identify the a and b terms. For example, if given the hyperbola (x^2) / 4 - (y^2) / 16 = 1, a = 2 since 4 = 2^2 and b = 4 since 16 = 4^2.
Find the foci c from the relationship, c = (a^2 + b^2)^(1/2). Using the example:
c = (2^2 + 4^2)^(1/2)
c =(4+16)^(1/2)
c = 20^(1/2)
c = 4.47
The foci are then (4.47, 0) and (-4.47,0)
Check for intercepts. For example, if asked to graph the hyperbola (x^2) / 4 - (y^2) / 16 = 1, set x equal to zero to find any y intercepts. Here it would yield:
0 - (y^2) /16 = 1
(-y^2) = 16
so there is no real solution. Now check for x intercepts. Set y equal to zero and solve for x:
(x^2) / 4 = 1
x^2 =4
x = 2, x = -2
Plot the intercepts, (a,0) (-a,0), which in the example are: (2,0) (-2, 0).
Plot the points (0,b) (0,-b), which in the example these are (0,4) (0, -4).
Plot the foci in the example, which are (4.47,0) and (-4.47,0).
Draw a rectangle containing the four points: (a,0) (-a,0) (0,b) (0,-b). These four points in the example are:
(2,0) (-2,0) (0,4) (0,-4).
Draw the diagonal lines of the constructed rectangle. These lines are the asymptotes. By definition the asymptotes are defined as y = b/a, and -b/a.
Construct the hyperbola going through the vertices (a,0) and (-a,0) and approaching the asymptotes but not crossing them.