Rational Functions

MATH 109 - PrecalculusS. Rook

Overview

• Section 2.6 in the textbook:– Vertical asymptotes & holes– Horizontal asymptotes– Slant asymptotes– Graphing rational functions

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Vertical Asymptotes & Holes

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Definition of a Rational Function

• Recall f(x) = N(x) / D(x) is a rational function for polynomials N(x) and D(x)– Domain is where D(x) ≠ 0

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Undefined and No Common Factors – Vertical Asymptote

• Vertical Asymptote: a vertical line x = k where the value of f(x) either dives down to -oo or soars to +oo

f(x) gets “extremely close” but can never touch the line x = k

• Factor D(x) if possible and look for values such that D(x) = 0

• If the rational function f(x) = N(x) / D(x) does NOT contain x – k as a common factor [in both N(x) and D(x)]:

f(x) has a vertical asymptote at x = k

Undefined, but with a Common Factor

– If the rational function f(x) = N(x) / D(x) DOES contain x – k as a common factor [in both N(x) and D(x)]:

f(x) will contain a hole at x = k

e.g.

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Vertical Asymptotes & Holes (Example)

Ex 1: i) Find the domain ii) Identify any vertical asymptotes:

a) b)

c)

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Horizontal Asymptotes

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Horizontal Asymptotes• Horizontal Asymptote: a horizontal line y = k where the

value of f(x) is EVENTUALLY bounded by k as x approaches -oo or +oo

• Unlike a vertical asymptote, f(x) IS ALLOWED TO CROSS a horizontal asymptote– Just so long as x becomes

infinitely large or as x becomes infinitely small, f(x) is bounded by y = k

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Horizontal Asymptotes (Continued)

• Given the rational function f(x) = N(x) / D(x):– Let anxn and bmxm be the leading terms of N(x) and D(x) respectively

• N(x) and D(x) MUST be in descending degree!– Then the horizontal asymptote of f(x) is:

• y = 0 (the x-axis) if n < m– i.e. Degree of the numerator is less than the degree of the

denominator• y = an / bm if n = m

– i.e. Degree of the numerator equals the degree of the denominator

• Nonexistent if m > n – i.e. Degree of the numerator is greater than the degree of the

denominator

Horizontal Asymptotes (Example)

Ex 2: Identify the horizontal asymptote if it exists:

a) b)

c)

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Slant Asymptotes

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Slant Asymptotes• Some rational functions have neither vertical nor horizontal

asymptotes, but asymptotes of the form y = mx + b• Given f(x) = N(x) / D(x), let anxn and bmxm be the leading terms

of N(x) and D(x) (in degree order) respectively f(x) has a slant asymptote if m = n + 1• i.e. the degree of the

numerator is ONE greater than the denominator

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Slant Asymptotes (Continued)

• To find the slant asymptote of the rational function f(x) = N(x) / D(x):– Ensure that f(x) meets the criteria for having a

slant asymptote– Perform polynomial long division of D(x) into N(x)• The quotient is the equation of the slant asymptote in y = mx + b format

Slant Asymptotes (Example)

Ex 3: i) State whether or not the function has a slant asymptote and ii) if it does, find it

a) b)

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Graphing Rational Functions

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Graphing Rational Functions

• To graph a rational function f(x) = N(x) / D(x):– Simplify f(x) by factoring and dividing out common

factors if they exist– Sketch the vertical asymptotes for x – k that are not

common factors and holes for x – k that are common factors

– Sketch the horizontal asymptote or slant asymptote if it exists

– Plot the y-intercept if it exists– Find the x-intercepts• Those values of x such that N(x) = 0 and D(x) ≠ 0

Graphing Rational Functions (Continued)

– Use the zeros and asymptotes to divide (-oo, +oo) into subintervals

– Pick additional points in each subinterval, especially near any vertical asymptotes• Recall that the value of the function has the same sign

for EVERY value in a particular interval

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Graphing Rational Functions (Example)

Ex 4: i) State the domain ii) Identify all intercepts iii) Find any vertical, horizontal, or slant asymptotes iv) Plot additional points in each subinterval to sketch the function

a) b)

c)

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Summary

• After studying these slides, you should be able to:– Identify the domain of a rational function– State where the vertical asymptotes and/or holes lie on a

rational function– Find the horizontal asymptote if it exists– Find the slant asymptote if it exists– Graph a rational function

• Additional Practice– See the list of suggested problems for 2.6

• Next lesson– Nonlinear Inequalities (Section 2.7)

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