Problems with detailed solutions on the graphing of the hyperbola equation are presented in this tutorial.

A hyperbola with center at the origin (0,0), is the graph of

The graph of the equation on the __left__ has the
following properties: x intercepts at ± a , no y intercepts, foci at (-c , 0)
and (c , 0), asymptotes with equations y = ± x (b/a). The graph of
the equation on the __right__ has the following properties: y intercepts at ± a , no x intercepts, foci at (0 , -c)
and (0 , c), asymptotes with equations y = ± x (a/b).

a, b and c are related by c^{2} = a^{2}
+ b^{2}.

The length of the transverse axis is 2a, and the length of the conjugate axis is 2b.

Applying the symmetry tests for graphs of equations in two variables, the hyperbola is symmetric with respect to x axis, y axis and the origin.

A tutorial on the definition and properties of hyperbolas can be found in this site.

9 x^{2} - 16 y^{2} = 144

a) Find the x and y intercepts, if possible, of the graph of the equation.

b) Find the coordinates of the foci.

c) Sketch the graph of the equation.

__Solution to Problem 1__

a) We first write the given equation in standard form by dividing both sides of the equation by 144

9x^{2} / 144 - 16y^{2} / 144 = 1

x^{2} / 16 - y^{2} / 9 = 1

x^{2} / 4^{2} - y^{2} / 3^{2} =
1

We now compare the equation obtained with the
standard equation (left) in the review above and we can say that the given equation is
that of an hyperbola with

** a = 4** and

Set y = 0 in the equation obtained and find the x intercepts.

x^{2} / 4^{2} = 1

Solve for x.

x^{2} = 4^{2}

x^{ } = ± 4

Set x = 0 in the equation obtained and find the y intercepts.

y^{2} / 3^{2} = 1

NO y intercepts since the above equation does not have real solutions.

b) We need to find c first (see formula above).

c^{2} = a^{2} + b^{2}

a and b were found in part a).

c^{2} = 4^{2} + 3^{2}

c^{2} = 25

Solve for c .

c = ± 5

The foci are F_{1}
(5 , 0)
and F_{2} (-5 , 0)

c)

1 - Find the asymptotes y = - (b/a) x and y = (b/a) x and plot them.

y = -(3/4) x and y = (3/4) x

2 - plot the x intercepts

3 - Find extra points (if necessary)

set x = 6 and find y

9(6)^{2} - 16y^{2} = 144

- 16y^{2} = 144 - 324

y^{2} = 45 / 4

Solve for y

y = 3(5)^{1/2} / 2

y = - 3(5)^{1/2} / 2

so the points (6 , 3(5)^{1/2} / 2)
and (6 , -3(5)^{1/2} / 2) are on the graph of the
hyperbola.

Also because of the symmetry of the graph of the
hyperbola, the points (-6 , 3(5)^{1/2} / 2)
and

(-6 , -3(5)^{1/2} / 2) are also on the graph
of the hyperbola.

** Matched 2:** Given the following equation

x^{2} - y^{2} = 9

a) Find the x and y intercepts, if possible, of the graph of the equation.

b) Find the coordinates of the foci.

c) Sketch the graph of the equation.

Find the Points of Intersection of Two Hyperbolas

Points of Intersection of a Hyperbola and a Line