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Chapter 5 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.

Power Functionspage 236

What happens if we multiply power functions by constants (a>0, a<0)?

Polynomial Functionspage 236

Every polynomial function is the sum of one or more power functions.

Every polynomial function can be expressed in the form:

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0

where n is a nonnegative integer

an , an-1 , . . . a2 , a1 ,a0 are constants (coefficients) with a0≠ 0

n (the highest power that appears) is called the degree

leading term is the term with the highest degree

leading coefficient is the coefficient of the leading term

examples/page 236

Polynomial Functionspage 236

Every polynomial function is the sum of one or more power functions.

Every polynomial function can be expressed in the form:

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0

Constant Functionsf(x) = kdegree 0 polynomials

Linear Functionsf(x) = mx + bfirst degree polynomials

Polynomial Functionspage 236

Every polynomial function can be expressed in the form:

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0

Section 5.1

Quadratic Functions

f(x) = ax2 + bx + c

second degree polynomials

one turning point is called the VERTEX.

BEHAVIOR is up/vertex/down OR down/vertex/up

Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet.

The model for the egg’s height t seconds after release is given by:

f(t) = -16t2 + 19t + 20

initial height

acceleration due to gravity

initial velocity

Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet.

The model for the egg’s height t seconds after release is given by:

f(t) = -16t2 + 19t + 20

When does the egg hit the ground?

How high does the egg go?

up/vertex/down

What is the velocity of the egg when it hits the ground?

parabola opening downward

What is the velocity of the egg at its maximum height?

Suppose an egg is thrown directly upward from a second story window. The initial velocity is 19 feet per second and the height of the window is 20 feet.

The model for the egg’s height t seconds after release is given by:

f(t) = -16t2 + 19t + 20

When does the egg hit the ground?

f(t) = 0factor or quadratic formula

How high does the egg go?

y coordinate of vertex

What is the velocity of the egg when it hits the ground?

rate of change

What is the velocity of the egg at its maximum height?

rate of change

Quadratic Functions

f(x) = ax2 + bx + c

Leading term determines global behavior (as power function).

To find y intercept, determine f(0) = c.

To find x intercepts of the graph of y=f(x) [or to find zeros of f or to find roots of f(x) = 0], solve f(x) = 0 by factoring or quadratic formula.

The axis of symmetry (mid-line) is given by x = -b/2a.

The coordinates of the vertex are (-b/2a, f(-b/2a)).

The number –b/2a tells where to find the greatest or least valueandf(-b/2a) is that greatest or least value.

Quadratic Functions

f(x) = ax2 + bx + c

CYU 5.1/page228

f(t) = -16t2 + 19t + 50

CYU 5.2/page233

f(t) = (5/3)t2 – 10t + 45

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

#5, #7 on page 255

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

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 by graph. (no nice formula)

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

Higher Degree Polynomials

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0

Find a formula for a polynomial whose graph is shown below.

HW

Page 255 #1-32

TURN IN: #6,8,16, 24(Maple graph), 26(Maple graph), 32