how to draw y^2=(x)/(1+x) graph

Question

i really have no idea how to do this

and roughly i know how the graph looks like but i don't know how to do the workings to draw it out

i appreciate your help very much

thank you.

Answer 1

The equation is y ² = (x )/(1 + x ).

⇒ y = ± √[ (x )/(1 +x ) ].

Its, represents a rational function.

So, The rational function is y = ± √[ (x )/(1 + x ) ].

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

• Find the vertical asympotote :
  

  Vertical asypototes are found  by equating the denominator is 0.

  ± √(1 + x ) = 0

1 + x = 0

⇒ x = - 1.

The vertical asymptote is x  = - 1.

• Find the horizontal asympotote :

  To find horizontal asymptote,
  

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

  If the degree of the numerator is equals to the degree of the denominator

So, the line y = 1 is the horizontal asymptote.

• Find the x - intercept :

Find the x - intercept by equationg the numarator is 0.

± √(x ) = 0

⇒ x = 0.

The x - intercept is zero.

• Find the y - intercept :

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

y = ± √[ (0)/(1 + 0) ].

⇒ y = 0.

The y - intercept is zero.

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

x	y = ± √[ (x)/(1 + x) ]	(x, y )
- 5	y = ±√[ (- 5)/(1 + (- 5)) ] = ±√[ (- 5)/(- 4) ] = ±√(5/4)	(- 5, ±√(5/4))
- 4	y = ±√[ (- 4)/(1 + (- 4)) ] = ±√[ (- 4)/(- 3) ] = ±√(4/3)	(- 4, ±√(4/3))
- 3	y = ±√[ (- 3)/(1 + (- 3)) ] = ±√[ (- 3)/(- 2) ] = ±√(3/2)	(- 3, ±√(3/2))
- 2	y = ±√[ (- 2)/(1 + (- 2)) ] = ±√[ (- 2)/(- 1) ] = ±√2	(- 2, ±√2)
- 1	y = ±√[ (- 1)/(1 + (- 1)) ] = ±√[- 1/0] = ∞	not defined
0	y = ±√[ 0/(1 + 0) ] = 0	(0, 0)
1	y = ±√[ 1/(1 + 1) ] = ±√[1/2]	(1, ±√[1/2])
2	y = ±√[ 2/(1 + 2) ] = ±√[2/3]	(2, ±√[2/3])

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 smooth curves complete the graph.

Answer 2

Given expression is .

Graph :

Graph of the expression is

Comments

The above graph does not represent a rational function.