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

Question

This a ratioal function in which I need to sketch a graph for. This is for college algebra.

Answer 1

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

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

1) y = (x ² - x)/(x + 2 )

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

y =(0 ² - 0)/(0 + 2 )

y = 0

In this case y  intercept is 0.

2)To find x intercept let the numarator = 0

x ² - x = 0

x ( x - 1) = 0

x  = 0 and x  = 1

x intercepts are x  = 0 and x = 1

3) To find the vertical asymptote.

Vertical asymptotes are found  by solving denominator = 0

x + 2 = 0

x  = - 2

Vertical asymptotes are x  = -2.

4) 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 found by long division.

x + 2 ) x ² - x ( x - 3

            x ² + 2x

         (-)    (-)

      ______________

               - 3x

              - 3x - 6

            (+)    (+)

      ________________

                  6

Quotient is slant asymptote.

In this case slant asymptote is y  = x  - 3.

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

x	y = (x 2 - x)/(x + 2 )	(x , y )
-10	y = (-10)2 - (-10)/(-10 + 2 )= -13.75	(-10, -13.75)
-5	y = (-5)2 - (-5)/(-5 + 2 )= -10	(-5 , -10)
5	y = (5)2 - (5)/(5 + 2 )= 2.85	(5, 2.85)
10	y = (10)2 - (10)/(10 + 2 )= 7.5	(10, 7.5)

Graph

1) Draw the coordinate plane.

2) Next dash the slant and vertical asymptotes.

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

4) Connect the plotted points .

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