slide1
Download
Skip this Video
Download Presentation
Chapter 5 Polynomial and Rational Functions

Loading in 2 Seconds...

play fullscreen
1 / 16

Chapter 5 Polynomial and Rational Functions - PowerPoint PPT Presentation


  • 110 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Chapter 5 Polynomial and Rational Functions' - makelina-justin


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

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.

slide2

Power Functionspage 236

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

slide3

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

slide4

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

slide5

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

slide6

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

slide7

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?

slide8

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

slide9

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.

slide10

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

slide11

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

slide12

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

slide13

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)

slide14

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

slide15

Higher Degree Polynomials

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

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

slide16

HW

Page 255 #1-32

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

ad