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Bellwork
168178)( 234 xxxxxf
2. Find all zeros of the function, write the polynomial as a product of linear factors.
1. Find a polynomial function with integer coefficients that has the given zeros.
4, 3i
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Last Nights Homework
3637.45
343310.43
2525.41
24
23
23
xx
xxx
xxx
53. -3/2, ±5i55. -3, 5, ±2i65. No, Setting h = 64 and solving the resulting equation yields imaginary roots,
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2.6 Rational Functions and Asymptotes-How to find domains of rational functions?-How to find horizontal and vertical asymptotes of graphs of rational functions?
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Rational Functions and Asymptotes• A rational function can be written in the form
)(
)()(
xD
xNxf
The most basic rational function
Where N(x) and D(x) are polynomials
Domain: (-∞,∞)
Horizontal Asymptote: x = 0
Vertical Asymptote: y = 0
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• The line x = a is a vertical asymptote of the graph of f if f(x)→∞ or f(x)→-∞ as x→a, either from the right or the left.
• The line y = b is a horizontal asymptote of the graph of f if f(x)→b as x→∞ or x→-∞.
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Vertical Asymptotes
• Let f be the rational function
)(
)()(
xD
xNxf
Where N(x) and D(x) have no common factors.
The graph of f has vertical asymptotes at the zeros of D(x).
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Example 1: Find the Vertical Asymptotes.
2
4)()
2
x
xxfa
xx
xxgb
3
3)()
2
209
82)()
2
xx
xxwc
Hole atx = -2No VA
VA at x = 0, Hole at x = 3
VA at x = 5
Hole at x = 4
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Horizontal Asymptotes
• The graph f has at most one horizontal asymptote determined by looking at the exponents of the numerator and the denominator. • If n < m, then y = 0 is the H.A.• If n = m, then y = a/b is the H.A.• If n > m, then there is no H.A.
...
...)(
m
n
bx
axxf
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Example 2: Find the H.A. of the following functions.
13
2)()
2 x
xxfa
13
2)()
2
2
x
xxgb 13
2)()
2
3
x
xxhc
Bigger exponent in D(x).H.A. y = 0
Same exponent in N(x)and D(x). H.A. y = 2/3
Bigger exponent inN(x). No H.A.
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Example 3: Find a functions vertical asymptotes,
and horizontal asymptotes.
54
273)()
3
23
x
xxxfa
3
4
5.. xAV
4
3. yAH
12
42)()
2
23
x
xxxgb
4
1,4
1 xxVA
0yHA
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Tonight’s Homework• Pg 195 #7-19. #40, 41