Chapter 2 section 6 and 7
Rational Functions and Asymptotes, Graphing Rational Functions
Rational Functions
 N(x) and D(x) are poly.
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 Domain: all real numbers except x-values that make the denominator zero
Look at all examples in section 6 and 7
Vertical Asymptotes and Holes
 For
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where N(x) and D(x) have no common factors (reduce first)
 The graph has vertical asymptotes at the zeros of D(x)
 The graph has a hole at any reduced factor
 Try this: Example: Find the vertical asymptotes and holes:
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answer: vertical asymptote at x = 1, hole (discontinued point) at (-1, -1/2)
work: factor and reduce: ( )( )
factor in the denominator (x – 1) is the vertical asymptote: x = 1
factor that reduced out (x + 1) is the hole at x = -1
substitute -1 into reduced equation,
, to find the exact point of the hole
, therefore the hole is at (-1, -1/2)
Horizontal Asymptotes (memorize these rule)
 For
( )
( )
 1. If n < m, then y = 0
(Ex.
)
 2. If n = m, then
(Ex.
, then y = )
 3. If n > m, then no horizontal asymptote (Ex.
)
 Try this: Example
 Find the domain and asymptotes.
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answer: Domain:
Vertical Asy:
; Horizontal Asy:
; No holes
work: Domain comes from 4x – 6 ≠ 0
Horizontal Asy: , x both to 1st degree therefore coef. is HA
Vertical Asy: 4x – 6 = 0
Holes: nothing reduced out
Slant Asymptotes (also called, Oblique) (See page 202)
 If the degree of N(x) is 1 more than D(x), then slant asymptote
 Use long division to find.
 This happens when there is no horizontal asy.
Steps to graph rational functions (See page 199 box)
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 N(x) and D(x) have no common factors
1. Find y-intercepts, f(0)
2. Find zeros, N(x) = 0
3. Find vertical asymptotes and holes D(x) = 0
4. Find horizontal asymptotes
5. Find slant (oblique) asymptotes
6. Graph points on each side of vertical asy.
Example – Find domain, all asymptotes, holes and intercepts and
graph with additional points
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answer: Domain:
(bottom)
HA: y = 1 (leading terms)
VA: x = 3, x = -3 (bottom)
Hole: none (bottom)
Slant asy: none
x –intercept: (0, 0) (numerator = 0, x2 = 0)
y –intercept: (0, 0)
(let x = 0)
additional points: (-4, 16/7), (-5, 25/16), (-1, -1/8), (1, -1/8)
(4, 16/7), (5, 25/16)
Example Find Domain, all asymptotes, holes and intercepts and graph with additional points
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answer: Reduce:
Domain: x ≠ -3, x ≠ 2
HA: y = 0
VA: x = 2
Hole: (-3, -1/5)
Slant Asy: none
x-intercept: none
y-intercept (0, -1/2)
additional points: (3, 1) (4, ½) (1, -1)
Example
 Graph
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answer: Domain: x ≠ 1
HA: none
VA: x = 1
Hole: none
Slant Asy.: y = x (long division, disregard remainder)
x-int. (2, 0) ( -1, 0)
y-int (0, 2)
additional points: (3, 2) (3/2, -5/2) (-2, -4/3)