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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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x
xxf
Vertical Asymptotes & Holes (Example)
Ex 1: i) Find the domain ii) Identify any vertical asymptotes:
a) b)
c)
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xxg
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x
xxh
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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xxxf
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xxxg
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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