Transcript Angles, Degrees, and Special Triangles

Rational Functions
MATH 109 - Precalculus
S. 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
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
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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
2
x
e.g. f x    4
x2
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Vertical Asymptotes & Holes
(Example)
Ex 1: i) Find the domain ii) Identify any vertical
asymptotes:
3x 2  1
a) f x   2
x  x9
c)
x3
b) g x   2
x 1
x4
h x   2
x  16
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Horizontal Asymptotes
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
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degree of the denominator
Horizontal Asymptotes (Example)
Ex 2: Identify the horizontal asymptote if it
exists:
3x 2  1
a) f x  
9  x  x2
c)
x3
b) g x   2
x 1
x4
h x   2
x  16
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Slant Asymptotes
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
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Slant Asymptotes (Example)
Ex 3: i) State whether or not the function has a
slant asymptote and ii) if it does, find it
x2  x 1
a) f x  
x 1
x4  6x 1
b) g x   2
x  2x  9
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Graphing Rational Functions
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
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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
t
a) f t   2
t 1
x 2  3x
b) g x   2
x  x6
2
x
c) hx    4
x
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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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