The function is .

Find the intercepts:

Find the -intercept by substituting in the function.

.

Here, , and .

Here, discriminant is less than zero.

There is no real solution for .

There is no -intercept.

To find the -intercept by substituting in the function.

The -intercept is .

Find the extrema for  .

Differentiate on each side  with respect to .

Quotient rule of derivatives: .

To find the critical numbers equate to zero.

Solutions of the quadratic equation are

.

The critical numbers are .

 To find the points of inflection of the graph ,evaluate .

The first derivative of is .

Differentaite on each side with respect to .

.

.

The second derivative of is .

To find inflection points, evaluate .

.

There is no solution for the equation.

There is no possible inflection point.

Critical numbers are and .

Relative extrema points exist at critical numbers.

Perform second derivative test to identify the nature of the extrema.

Test value	Sign of	Conclusion
	\ \	Relative maximum
	\ \	Relative minimum

The function has relative maximum at .

Substitute in .

Relative maximum point is .

The function has relative minimum at .

Substitute in .

Relative minimum point is .

Find asymptote of function .

To find horizontal asymptote .

Thus, there is no horizontal asymptote.

To find vertical asymptote,equate denominator to zero.

The vertical asymptote is .

Find the slant asymptote by long division method.

Therefore, the function is reduced as .

The slant asymptote is the polynomial part of the reduced expreession.

Therefore, slant asymptote is .

Graph :

Graph the function using the above specification.

Note:The dashed lines indicates horizontal asymptote.

There is no -intercept and the -intercept is .

Relative maximum point is .

Relative minimum point is .

There is no horizontal asymptote.

The vertical asymptote is .

The slant asymptote is .

There is no -intercept and the -intercept is .

Relative maximum point is .

Relative minimum point is .

There is no horizontal asymptote.

The vertical asymptote is .

The slant asymptote is .