Section 5.3 – The Graph of a Rational Function

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

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

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionExample

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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)

Use numerator factors

𝑥+4=0 𝑥−3=0𝑥=−4 𝑥=3(−4 ,0) (3 ,0)

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

𝑦=1𝑎𝑛𝑑 𝑓 (𝑥 )=𝑥2+𝑥−12𝑥2−4

1= 𝑥2+𝑥−12𝑥2−4

𝑥2−4=𝑥2+𝑥−12−4=𝑥−128=𝑥(8,1)

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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)→∞

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational Function9) Sketch the graph

Section 5.3 – The Graph of a Rational FunctionExample

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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)

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational Function9) Sketch the graph

Section 5.3 – The Graph of a Rational FunctionExample

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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)

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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 xx𝑥𝑥2−𝑥−+¿4 𝑥+2

+4

4 𝑥−4−+¿0

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational FunctionGeneral Steps to Graph a Rational Function

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

Section 5.3 – The Graph of a Rational Function9) Sketch the graph