How to graph x - 2/x^2 + 6x

Question

How do I graph the following rational functions:

1) f(x)= x - 2/x^2 + 6x

2) f(x)= x^2/x-1

Answer 1

(1)  The rational function  f (x ) = (x - 2)/(x ² + 6x )

The graph of rational functions can be recognised by the fact two or more parts.

1) y = (x - 2) / (x ² + 6x )

To find  y intercept  x = 0 in the rational function.

y = (0-2)/(0 ² + 6(0)

Not defined.

In this case y  intercept is none.

2) to find x intercept let the numarator = 0

x - 2 = 0

x  = 2

3) to find the vertical asympotote.

To find vertical asypototes are found  bysolving denominator = 0

x ² + 6x = 0

x (x + 6) = 0

x = 0 and x + 6 = 0

Vertical asymptotes are x  = 0 and x  = -6.

4) To find horizontal asymptote.

Degree of the numarator = 1 and the degree of denominator = 2.

If the degree of the numerator is less than the degree of the denominator, the line y = 0 (x - axis)

is the horizontal asymptote.

Now i will pick few more x - values, compute the corresponding y  values and flat few more points.

x	y = (x - 2) / (x 2 + 6x)	(x , y )
-1	y = (-1 - 2) / (-1 2 + 6(-1)) = -3/-5 = 3/5	(-1,3/5)
-2	y = (-2 - 2) / (-2 2 + 6(-2)) = -4/-8 = 1/2	(-2 , 1/2)
-5	y = (-5 - 2) / (-5 2 + 6(-5)) = -7/-5 = 7/5	(-5, 7/5)
1	y  = (1 - 2) / (1 2 + 6(1)) = -1/7	(1,-1/7)
2	y  = (2 - 2) / (2 2 + 6(2)) = 0	(2,0)
5	y  = (5- 2) / (5 2 + 6(5)) = 3/55	(5,3/55)

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asympototes

3) Plot the x intercept and coordinate pairs found in the table..

4) Connect the plotted points .

When you draw your graph, use smmoth curves complete the graph.

Answer 2

2) The rational function f (x ) = x ² /(x - 1)

The graph of rational functions can be recognised by the fact two or more parts.

•  y = x ² /(x - 1)

To find  y intercept  x = 0 in the rational function.

y = 0 ² /(0 - 1)

y = 0

In this case y  intercept is 0.

•  to find x intercept let the numarator = 0

x ² = 0

x  = 0

In this case x  intercept is 0.

• Vertical asymptotes are found  by solving denominator = 0

x - 1 = 0

x  = 1

Vertical asymptote is x  = 1.

• To find slant asymptote.

Degree of the numarator = 2 and the degree of denominator = 1.

If the degree of the numerator is one grater than the degree of the denominator, then the function will have slant

asymptote.

Slant asymptote is foung by long division.

x -1 ) x ²      ( x + 1

        x ² - x

   (-)_______

          x

          x  - 1

   (-)_______

            1

x ² = (x - 1) (x + 1) + 1

Quotient is slant asymptote.

In this case slant asymptote is y  = x  + 1.

Now i will pick few more x - values, compute the corresponding y  values and flat few more points.

x	y = x 2 /(x - 1)	(x , y )
-2	y = (-2) 2 /(-2 - 1) = - 4/3	(-2, -4/3)
-4	y = (-4) 2 /(-4 - 1) = - 16/5	(-4 , -16/5)
6	y = (6) 2 /(6 - 1) = 36/5	(6, 36/5)
1	y = (1) 2 /(1 - 1) = 1/0	not defined
2	y = (2) 2 /(2 - 1) = 4	(2, 4)
4	y = (4) 2 /(4 - 1) = 16/3	(4,16/3)

Graph

1) Draw the coordinate plane.

2) Next dash the slant and vertical asymptotes.

3) Plot the x intercept and coordinate pairs found in the table..

4) Connect the plotted points .

When you draw your graph, use smooth curves complete the graph.