Math 139 – The Graph of a Rational Function3 examples

General Steps to Graph a Rational Function1) Factor the numerator and the denominator2) State the domain and the location of any holes in the graph

3) Simplify the function by factoring if possible.

4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0)

5) Identify any existing asymptotes (vertical, horizontal, or oblique/slant)

General Steps to Graph a Rational Function contd.

8) Analyze the behavior of the graph on each side of an asymptote

9) Sketch the graph

6) Identify any points intersecting a horizontal or oblique asymptote.

7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis

Example #1

1) Factor the numerator and the denominator

2) State the domain and the location of any holes in the graph

3) Simplify the function to lowest terms

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

𝑓 (𝑥 )=𝑥2+𝑥−12𝑥2−4

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

Domain: No holes (They occur where factors cancel)

4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)

𝑓 (0 )=(0+4)(0−3)(0+2)(0−2)

𝑓 (0 )=−12−4

=3

(0 ,3)

These are the roots of the numerator.𝑥+4=0 𝑥−3=0𝑥=−4 𝑥=3(−4 ,0) (3 ,0)

5) Identify any existing asymptotes (vertical, horizontal, or oblique

Horiz. Or Oblique Asymptotes Vertical Asymptotes

𝑦=11

𝐻𝐴 : 𝑦=1

Use denominator factors

𝑥+2=0 𝑥−2=0𝑥=−2 𝑥=2𝑉𝐴 :𝑥=−2𝑎𝑛𝑑 𝑥=2

𝑓 (𝑥 )=𝑥2+𝑥−12𝑥2−4

𝑓 (𝑥 )=(𝑥+4 )(𝑥−3)(𝑥+2)(𝑥−2)

Examine the largest exponents

Same Horiz. - use coefficients

Oblique Asymptotes occur when degree is 1 greater on top. Divide using base-x and get rid of denominator to find line.

6) Identify any points intersecting a horizontal or oblique asymptote. (this is possible because unlike a vertical asymptote, the function is defined at these types)

𝑦=1𝑎𝑛𝑑 𝑓 (𝑥 )=𝑥2+𝑥−12𝑥2−4

1= 𝑥2+𝑥−12𝑥2−4

𝑥2−4=𝑥2+𝑥−12−4=𝑥−128=𝑥(8,1)

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis

-4 -2 2 3

𝑓 (−5 )=(−5+4)(−5−3)(−5+2)(−5−2)

𝑓 (−5 )=(−)(−)(−)(−)

=+¿

𝑓 (−5 )=𝑎𝑏𝑜𝑣𝑒

𝑓 (−3 )=¿¿𝑓 (−3 )=𝑏𝑒𝑙𝑜𝑤

𝑓 (0 )=¿¿𝑓 (0 )=𝑎𝑏𝑜𝑣𝑒

𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis

-4 -2 2 3

𝑓 (2.5 )=¿¿𝑓 (2.5 )=𝑏𝑒𝑙𝑜𝑤

𝑓 (4 )=¿¿𝑓 (4 )=𝑎𝑏𝑜𝑣𝑒

𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

𝑥→−2−

𝑎𝑏𝑜𝑣𝑒

-4 -2 2 3

𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤

8) Analyze the behavior of the graph on each side of an asymptote

𝑓 (𝑥)→¿¿ 𝑓 (𝑥)→−∞

𝑥→−2+¿ ¿ 𝑓 (𝑥)→¿¿ 𝑓 (𝑥)→∞

𝑓 (𝑥 )=(𝑥+4)(𝑥−3)(𝑥+2)(𝑥−2)

𝑥→2−

8) Analyze the behavior of the graph on each side of an asymptote

𝑓 (𝑥)→¿¿ 𝑓 (𝑥)→∞

𝑥→2+¿ ¿ 𝑓 (𝑥)→¿¿ 𝑓 (𝑥 )→−∞

𝑎𝑏𝑜𝑣𝑒

-4 -2 2 3

𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤

9) Sketch the graph

Example #2

1) Factor the numerator and the denominator

2) State the domain and the location of any holes in the graph

3) Simplify the function to lowest terms

𝑓 (𝑥 )=(𝑥−2)(𝑥+3)

𝑓 (𝑥 )=𝑥2+3 𝑥−10𝑥2+8 𝑥+15

𝑓 (𝑥 )=(𝑥+5)(𝑥−2)(𝑥+5)(𝑥+3)

Domain: Hole in the graph at

4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)

𝑓 (0 )=(0−2)(0+3)

𝑓 (0 )=− 23

(0 ,−23)

Use numerator factors

𝑥−2=0𝑥=2(2 ,0)

5) Identify any existing asymptotes (vertical, horizontal, or oblique

Horiz. Or Oblique Asymptotes Vertical Asymptotes

𝑦=11

𝐻𝐴 : 𝑦=1

Use denominator factors

𝑥+3=0𝑥=−3

𝑉𝐴 :𝑥=−3

𝑓 (𝑥 )=𝑥2+3 𝑥−10𝑥2+8 𝑥+15

𝑓 (𝑥 )=(𝑥−2)(𝑥+3)

Examine the largest exponents

Same Horiz. - use coefficients

6) Identify any points intersecting a horizontal or oblique asymptote.

𝑦=1𝑎𝑛𝑑 𝑓 (𝑥 )=𝑥−2𝑥+3

1=𝑥−2𝑥+3

𝑥+3=𝑥−23=−2𝑙𝑜𝑠𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑛𝑜𝑝𝑜𝑖𝑛𝑡𝑠𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑜𝑛 h𝑡 𝑒𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒

𝑓 (𝑥 )=(𝑥−2)(𝑥+3)

7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis

𝑓 (−4 )=(−4−2)(−4+3)

𝑓 (−4 )=(−)(−)

=+¿

𝑓 (−4 )=𝑎𝑏𝑜𝑣𝑒

𝑓 (0 )=(−)¿¿

𝑓 (0 )=𝑏𝑒𝑙𝑜𝑤

𝑓 (3 )=¿¿𝑓 (3 )=𝑎𝑏𝑜𝑣𝑒

𝑎𝑏𝑜𝑣𝑒 𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒

-3 2

𝑓 (𝑥 )=(𝑥−2)(𝑥+3)

𝑥→−3−

8) Analyze the behavior of the graph on each side of an asymptote

𝑓 (𝑥)→(−)(0−) 𝑓 (𝑥)→∞

𝑥→−3+¿¿𝑓 (𝑥)→

(−)¿¿

𝑓 (𝑥 )→−∞

-3 2

9) Sketch the graph

Example #3

1) Factor the numerator and the denominator

2) State the domain and the location of any holes in the graph

3) Simplify the function to lowest terms

𝑓 (𝑥 )=(𝑥+2)(𝑥+1)(𝑥−1)

𝑓 (𝑥 )=𝑥2+3 𝑥+2𝑥−1

𝑓 (𝑥 )=(𝑥+2)(𝑥+1)𝑥−1

Domain: No holes

4) Find the y-intercept (x = 0) and the x-intercept(s) (y = 0) y-intercept (x = 0) x-intercept(s) (y = 0)

𝑓 (0 )=(0+2)(0+1)(0−1)

𝑓 (0 )= 2−1

=−2

(0 ,−2)

Use numerator factors

𝑥+2=0𝑥=−2(−2 ,0)

𝑥+1=0𝑥=−1(−1 ,0)

5) Identify any existing asymptotes (vertical, horizontal, or oblique

Horiz. or Oblique Asymptotes Vertical Asymptotes

𝑥−1

O 𝐴 : 𝑦=𝑥+4

Use denominator factors

𝑥−1=0𝑥=1

𝑉𝐴 :𝑥=1

𝑓 (𝑥 )=𝑥2+3 𝑥+2𝑥−1

𝑓 (𝑥 )=(𝑥+2)(𝑥+1)

(𝑥−1)

Examine the largest exponents Oblique: Use long division

232 xx𝑥

𝑥2−𝑥−+¿4 𝑥+2

+4

4 𝑥−4−+¿0

6) Identify any points intersecting a horizontal or oblique asymptote.

𝑦=𝑥+4 𝑎𝑛𝑑 𝑓 (𝑥 )=(𝑥+2)(𝑥+1)𝑥−1

𝑥+4=(𝑥+2)(𝑥+1)

𝑥−1

(𝑥+4 )(𝑥−1)=(𝑥+2)(𝑥+1)𝑥2+3𝑥−4=𝑥2+3 𝑥+2𝑙𝑜𝑠𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑛𝑜𝑝𝑜𝑖𝑛𝑡𝑠𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑜𝑛 h𝑡 𝑒𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒

𝑓 (𝑥 )=(𝑥+2)(𝑥+1)(𝑥−1)

7) Use test points between the zeros and vertical asymptotes to locate the graph above or below the x-axis

𝑓 (−4 )=(−)(−)(−)

=−

𝑓 (−1.5 )=¿¿

𝑓 (−4 )=𝑏𝑒𝑙𝑜𝑤 𝑓 (0 )=¿¿𝑓 (0 )=𝑏𝑒𝑙𝑜𝑤

𝑓 (3 )=¿¿𝑓 (3 )=𝑎𝑏𝑜𝑣𝑒

𝑎𝑏𝑜𝑣𝑒𝑏𝑒𝑙𝑜𝑤 𝑎𝑏𝑜𝑣𝑒

-2 1-1

𝑓 (−1.5 )=𝑎𝑏𝑜𝑣𝑒

𝑏𝑒𝑙𝑜𝑤

𝑓 (𝑥 )=(𝑥+2)(𝑥+1)(𝑥−1)

𝑥→1−

8) Analyze the behavior of the graph on each side of an asymptote

𝑓 (𝑥)→¿¿ 𝑓 (𝑥 )→−∞

𝑥→1+¿¿ 𝑓 (𝑥)→¿¿ 𝑓 (𝑥 )→∞

1

9) Sketch the graph