Figuring out how the rational function transforms and approaches

Question

Having trouble understanding how graphs transforms and approaches from the sides. I can figure out how this works for example how these rational functions transform and are graphed:
 

f(x) = (x+5)/(x^2-2x-3)

f(x) = -x^4 + 13x^2 - 36

f(x) = (x^2+x-20)/(x^2-3x-18)

 

please explain,

Thank you

Answer 1

(1).

The function is f(x) = (x + 5)/(x^2 - 2x  - 3).

First find y-intercept by substituting x = 0 in the original function.

f(x) = (0 + 5)/[0^2 - 2(0)  - 3]

f(x) = - 5/3.

The y-intercept is - 5/3 and the point (0, - 5/3).
 

Next  find x-intercept by substituting f(x) = 0 in the original function.

0 = (x + 5)/(x^2 - 2x  - 3).

x = - 5

The x-intercept is - 5 and the point is (-5, 0).

 

Find the vertical asymptote :

To find the vertical asymptote, find the zeros of the denominator.

x^2 - 2x  - 3 = 0

x^2 - 3x + x - 3 = 0

x(x - 3) + 1(x - 3) = 0

(x + 1)(x - 3) = 0

x = - 1 and x = 3.

The vertical asymptote x = - 1 and x = 3.

 

Find the horizontal asymptote :

Since degree of numerator < degree of denominator, the horizontal asymptote y = 0.

 

Make a table and find the additional points to graph the function.

x	f(x) = (x + 5)/(x^2 - 2x  - 3)	y	(x, y)
-2	(-2+ 5)/[(-2)^2 – 2(-2) – 3] = 0.167	0.6	(-2, 0.6)
-1	(-1+ 5)/[(-1)^2 – 2(-1) – 3] = 0.167	undefined
0	(0+ 5)/[(0)^2 – 2(0) – 3] = - 1.167	-1.67	(0, -1.67)
1	(1+ 5)/[(1)^2 – 2(1) – 3] = 0.167	-1.5	(1, -1.5)
2	(2+ 5)/[(2)^2 – 2(2) – 3] = 0.167	-2.3	(2, -2.3)
3	(3+ 5)/[(3)^2 – 2(3) – 3] = 0.167	undefined
4	(4+ 5)/[(4)^2 – 2(4) – 3] = 1.8	1.5	(4, 1.8)
5	(5+ 5)/[(5)^2 – 2(5) – 3] = 0.83	0.83	(5, 0.83)

Draw a smooth curve through these points

 

Answer 2

2) The function f(x) = - x⁴ + 13x² - 36

Real zeros are x intercepts of the graph.

Find the real zeros

f(x) = - x⁴ + 13x² - 36

= - x⁴ + 9x²  + 4x² - 36

= - x² (x² - 9) + 4(x² - 9)

= (x² - 9)( 4 -  x²)

y = (x - 3)(x + 3)( 2 - x)(2 + x)

Real zeros are 3, -3, 2 and - 2.

Test points

Make the table of values to for the polynomial.

Choose random values for x and find the corresponding values for y.

x	y = - x4 + 13x2 - 36	(x, y )
- 1	y = - (-1)4 + 13(-1)2 - 36 = -24	(- 1, - 24)
- 2.5	y = - (-2.5)4 + 13(-2.5)2 - 36 = 6.1875	(-2.5, 6.18)
0	y = - (0)4 + 13(0)2 - 36 = -36	(0, - 36)
1	y = - (1)4 + 13(1)2 - 36 = -24	(1, - 24)
2.5	y = - (2.5)4 + 13(2.5)2 - 36 = 6.1875	(2.5, 6.18)

End behavior y = - x⁴ + 13x² - 36

Degree of polynomial is 4 and leading coefficient -1.

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

All even degree polynomials behave on their ends like quadratics.

All even degree polynomials are either up on both ends and or down on both ends.depending on whether the polynomial has, respectively, a positive or negative leading coefficient.

The above polynomial even degree  polynomial with a negative leading coefficient .

So the graph down on both ends.

Graph

1.Draw a coordinate plane.

2.Plot the coordinate points found in the table.

3.Then sketch the graph, connecting the points with a smooth curve.

Answer 3

1) The rational function f(x) = (x² + x - 20)/(x² - 3x - 18)

y = (x² + x - 20)/(x² - 3x - 18)

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

1) y = (x² + x - 20)/(x² - 3x - 18)

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

y = (0² + 0 - 20)/(0² - 3(0) - 18)

y = 20/18 = 1.11

 y  intercept is 1.11.

2) To find x intercepts let the numerator = 0

x² + x - 20 = 0

(x - 4)(x + 5)  = 0

x intercepts are x = 4 and - 5.

Vertical asymptote can be found by making denominator = 0.

x² - 3x - 18 = 0

(x + 3)(x - 6) = 0

x = - 3 and x = 6

Vertical asymptote are x = - 3 and x = 6.

To find horizontal asymptote, first find the degree of the numerator and  the degree of denominator.

Degree of the numerator = 2 and the degree of denominator = 2.

Since the degree of the numerator is equal to the degree of the denominator,horizontal asymptote is the ratio of the leading coefficient of numerator and denominator.

Leading coefficient of numerator =  1, leading coefficient of denominator = 1

y = 1 is the horizontal asymptote.

We need some more points to more accurate graph.

Choose random values for x and find the corresponding values for y.

x	y = (x2 + x - 20)/(x2 - 3x - 18)	(x, y)
-1	y = [(-1)2 - 1 - 20]/[(-1)2 - 3(-1) - 18]	(-1, 1.42)
-2	y = [(-2)2 - 2 - 20]/[(-2)2 - 3(-2) - 18]	(-2, 2.25)
2	y = [22 + 2 - 20]/[22 - 3(2) - 18]	(2, 0.7)
5	y = [52 + 5 - 20]/[52 - 3(5) - 18]	(5,-1.25)

 

Comments

Contd..

Graph

1) Draw the coordinate plane.

2) Next dash the horizontal and vertical asymptotes

3) Plot the x,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.