Question

Sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result.

y = x/ 1 - x

Answer 1

Step 1 :

Thu function is

Domain :

Thu function

The function continuous for all the points except at .

Because at the function is undefined.

Thus, the domain of the function is .

Intercepts :

y - intercept is :

Thus, y - intercept is .

x - intercept :

Consider and solve for x.

Thus, x - intercept is .

Step 2 :

Symmetry :

If , then the function is even and it is symmetric about x-axis.

If , then the function is odd and it is symmetric about origin.

Here

Thus, the function is neither even nor odd.

Step 3 :

Asymptotes :

Vertical asymptote exist when denominator is zero.

Equate denominator to zero.

Vertical asymptote is

Horizontal asymptote:

The line is called a horizontal asymptote of the curve if either

  or

Thus, the horizontal asymptote is

Answer 2

Contd.........

Step 4 :

Intervals of increase or decrease :

Differentiate with respect to x:

is never zero on it's domain.

f  is increasing on its domain because

Step 5 :

Determination of extrema :

f  is an increasing function, hence there is no chance of local minimum or maximum.

Step 6 :

Determination of inflection point:

Differentiate with respect to x:

is never zero.

Hence, there is no inflection points.

But at the function is undefined.

Consider the test intervals as and 

Interval	Test Value	Sign of	Concavity
			Up
			Down

Thus, the graph is concave up on the interval .

The graph is concave down on the interval .

Step 7 :

Graph of the function  :

Solution :

Graph of the function  :