Polynomial and Rational Functions

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Chapter 5 Polynomial and Rational Functions
5.1 Quadratic Functions and Models 5.2 Polynomial
Functions and Models 5.3 Rational Functions and
Models
A linear or exponential or logistic model either
increases or decreases but not both.
Life, on the other hand gives us many instances
in which something at first increases then
decreases or vice-versa. For situations like
these, we might turn to polynomial models.
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Higher Degree Polynomials f(x) anxn an-1xn-1
. . . a2x2 a1x a0
Graph is always a smooth curve
Leading term determines global behavior (as power
function).
To find y intercept, determine f(0) c.
To find x intercepts, solve f(x) 0 by factoring
or SOLVE command.
FACTORED FORM f(x) a(x-x1)(x-x2)(x-xk) for
x1, x2 xk zeroes of f.
possibly more turning points
Identify turning points approximately point and
click by graph.
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The speed of a car (in mph) after t seconds is
given by f(t) .005t3 0.21t2 1.31t 49
According to Maple t-intercept is -11.34 rate
of change at t 15is -1.625 rate of change at
t 26 is 0.52
(3.46, 51.27)
(24.44, 28.35)
These calculations agree with the graph, since
slope of curve is negative at t 15 and positive
at t 26.
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The cost of a days production of x pots is given
by C(x) .01x3 0.65x2 14x 20
fixed costs T/Fcost increases with of
pots(turning points) marginal cost(cost of
producing one more) e.g. C(8)-C(7)
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Rational Functions and Models A rational function
is a quotient or ratio of two polynomials.
??? A Closer Look!
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As x ? -3- , y ? 8
As x ? -3 , y ? -8
We say the graph of f has a vertical asymptote at
x -3.
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vertical asymptote at x -3.

• Vertical Asymptote at x k
• k is not in the domain of f
• the values of f increase (or decrease) without
  bound as x approaches k
• near x k, the graph of f resembles a vertical
  line

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As x ? - 8, y ? 2
As x ? 8 , y ? 2-
We say the graph of f has a horizontal asymptote
at y 2.
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horizontal asymptote at y 2.
The quotient of leading terms determines the
global behavior of a rational function.
In fact, for x large in magnitude
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y intercept f(0) -1/3
x intercept 2x 1 0 x 1/2
vertical asymptote at x -3
horizontal asymptote at y 2
two branches
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y intercept f(0) 2
x intercept -2 0 no solution
vertical asymptotes x2 1 (x-1)(x1) 0VA
POSSIBLE at x 1, -1
horizontal asymptote f(x) -2/x2 0 for x
large in magnitudeHA at y 0
three branches
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y intercept f(0) 0
x intercept 3x2 0 x 0
vertical asymptotes x2 4 (x-2)(x2) 0VA
POSSIBLE at x 2, -2
horizontal asymptote f(x) 3x2/x2 3 for x
large in magnitudeHA at y 3
three branches
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HW Page 255 33-50 TURN IN 33,36, 40( with
Maple graph), 41(with Maple graph)