Polynomial and Rational Functions

1
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.
2
Quadratic Functions f(x) ax2 bx c
CYU 5.2/page233 f(t) (5/3)t2 10t 45
3
Quadratic Functions f(x) ax2 bx c
FACTORED FORM f(x) a(x-x1)(x-x2)
for x1 and x2 zeroes of f.
VERTEX FORM f(x) a(x-h)2 k for
vertex (h,k).
All quadratic functions are tranformations of
f(x) x2
CYU 5.3/page 234
4
Optimization(finding maximum/minimum values in
context)
(page232) Suppose Jack has 188 feet of fencing to
make a rectangular enclosure for his cow. Find
the dimensions of the enclosure with maximum area.
Build an area function and find maximum value.
More Practice 15/257
5
Higher Degree Polynomials (5.2) 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 by graph.
(no nice formula)
6
Higher Degree Polynomials f(x) anxn an-1xn-1
. . . a2x2 a1x a0
The speed of a car after t seconds is given
by f(t) .005t3 0.21t2 1.31t 49
(3.46, 51.27)
extended view
global behavior matches leading term
(24.44, 28.35)
local maximum and local minimum
CYU 5.5/240
7
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.
8
Higher Degree Polynomials f(x) anxn an-1xn-1
. . . a2x2 a1x a0
Find a formula for a polynomial whose graph is
shown below.
9
HW Page 255 1-32 TURN IN 16, 24(Maple graph),
26(Maple graph), 32