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.

Rational Functions and Models

A rational function is a quotient or ratio of two polynomials.

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

The quotient of leading terms determines the asymptotic (global) behavior of a rational function.

Horizontal Asymptote at y = m

• global behavior tends toward a constant value m

• graph resembles a horizontal line for x large in magnitude.

43/259 Write a formula for the function that could represent the graph. Give your reasoning.

horizontal asymptote: at y = 1.

f(x) ≈ ??/(x+1) ≈ 1

vertical asymptotes: at x = -1.

denominator contains (x+1)

y intercept: f(0) = 0

x intercept: 0 = f(x) only when x = 0.numerator = 0 only when x = 0.

xf(x) =

x + 1

37/259 After the engine of a moving motorboat is cut off, the boat’s velocity decreases according to the model, where t is elapsed time in

seconds and v is the velocity in feet per second.

300v(t) =

15 + t

a) Sketch a graph of the abstract function v(t).

b) Is the domain continuous or discrete?

c) How fast was the boat moving when the engine was cut off?

d) After how many seconds did the velocity reach 10 ft/ sec ?

e) Find the acceleration [rate of change of velocity] of the boat after 5 seconds.

33&50/259 The cost C (dollars) of operating a studio on a day in which x pots are produced is given by the function C(x) = 0.01x3 – 0.65x2 + 14x + 20.

Let A(x) be the average cost of producing each ceramic pot on a day when x pots are made.

a) Use the formula for C(x) to find a formula for A(x).

c) Is the domain continuous or discrete?

d) Find the coordinates of the local minimum.

e) To minimize the average cost per pot, how many should the studio make in a given day and

what would be the average cost of each?

b) Sketch the graph of the abstract function A(x).

(33.3, 4)

33&50/259 The cost C (dollars) of operating a studio on a day in which x pots are produced is given by the function C(x) = 0.01x3 – 0.65x2 + 14x + 20.

Let A(x) be the average cost of producing each ceramic pot on a day when x pots are made.

Describe global behavior for A(x).

HW

Page 255 #33-50

PROJECT

Lab 5A The Doormats LabDUE: Wednesday April 23, 2008

Report should start with a well-written summary of each of the three models as outlined on the bottom of page 283 with graphs and

asymptotic analysis for each model as necessary for support of your summary.