Question

Find the center, vertices, foci, and the equations  of  the  asymptotes  of  the  hyperbola,  and  sketch  its graph using the asymptotes as an aid.

Answer 1

Step 1:

The equation is .

Compare it to the standard form of the hyperbola .

Here transverse axis is parallel to the axis.

is the center of the hyperbola.

is the distance between center and focus.

.

Step 2:

The center of the hyperbola is .

The vertices of the hyperbola are .

The foci of the hyperbola are .

Find the points to form a rectangle :

.

.

The extensions of the diagonals of the rectangle are the asymptotes of the hyperbola.

Asymptotes will provide the end behavior of the hyperbola.

Asymptotes of the hyperbola are .

Substitute the values of in .

Asymptotes are .

Step 3:

Graph :

(1) Draw the coordinate plane.

(2) Draw the equation of the hyperbola.

(3) Plot the foci and vertices.

(4) Form a rectangle containing the points , .

(5) Draw the asymptotes of the hyperbola.

 

Solution :

The center of the hyperbola is .

The vertices of the hyperbola are .

The foci of the hyperbola are .

Asymptotes of the hyperbola are .

Graph of the hyperbola and asymptotes :