Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

3.3 Power Functions and End Behavior, Study Guides, Projects, Research of Mathematics

Montclair State University (MSU)Mathematics

The end behavior is the behavior of the graph of a function as the input decreases without bound and increases without bound. A power function is of the form: ...

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

loche
loche ๐Ÿ‡บ๐Ÿ‡ธ

4.3

(16)

241 documents

1 / 4

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
3.3 Power Functions and End Behavior
โ€ขThe end behavior is the behavior of the graph of a function as the input decreases without
bound and increases without bound.
โ€ขApower function is of the form: f(x) = kxpwhere kand pare constant. pdetermines the
degree of the power function and both kand pdetermine the end behavior.
x
y
c
Power function, p: odd, k > 0
End behavior:
yโ†’ โˆž as xโ†’ โˆž
yโ†’ โˆ’โˆž as xโ†’ โˆ’โˆž
x
y
c
Power function, p: odd, k < 0
End behaviour:
yโ†’ โˆ’โˆž as xโ†’ โˆž
yโ†’ โˆž as xโ†’ โˆ’โˆž
x
y
c
Power function, p: even, k > 0
End behaviour:
yโ†’ โˆž as xโ†’ โˆž
yโ†’ โˆž as xโ†’ โˆ’โˆž
x
y
c
Power function, p: even, k < 0
End behaviour:
yโ†’ โˆ’โˆž as xโ†’ โˆž
yโ†’ โˆ’โˆž as xโ†’ โˆ’โˆž
Comparing Power Functions:
x
y
y=x3
y=x5
y=x7
x
y
y=x2
y=x4
y=x6
pf3
pf4

Partial preview of the text

Download 3.3 Power Functions and End Behavior and more Study Guides, Projects, Research Mathematics in PDF only on Docsity!

3.3 Power Functions and End Behavior

  • The end behavior is the behavior of the graph of a function as the input decreases without bound and increases without bound.
  • A power function is of the form: f (x) = kxp^ where k and p are constant. p determines the degree of the power function and both k and p determine the end behavior.

x

y

c

Power function, p: odd, k > 0 End behavior: y โ†’ โˆž as x โ†’ โˆž y โ†’ โˆ’โˆž as x โ†’ โˆ’โˆž

x

y

c

Power function, p: odd, k < 0 End behaviour: y โ†’ โˆ’โˆž as x โ†’ โˆž y โ†’ โˆž as x โ†’ โˆ’โˆž

x

y

c

Power function, p: even, k > 0 End behaviour: y โ†’ โˆž as x โ†’ โˆž y โ†’ โˆž as x โ†’ โˆ’โˆž

x

y

c

Power function, p: even, k < 0 End behaviour: y โ†’ โˆ’โˆž as x โ†’ โˆž y โ†’ โˆ’โˆž as x โ†’ โˆ’โˆž

Comparing Power Functions:

x

y y = x^3

y = x^5

y = x^7

x

y

y = x^2

y = x^4

y = x^6

Polynomials

  • A polynomial function is a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Leading coefficient coefficient coefficient Constant coefficient

โ†“ degree

โ†“

f (x) = anxn

leading term

+anโˆ’ 1 xnโˆ’^1 ... +a 1 x + a 0

constant term

  • The leading term is the term containing the highest power of the variable. The leading coeffi- cient is the coefficient of the leading term.
  • A continuous function is a function whose graph has no break or jump. A smooth function is a function whose graph has no sharp corners.
  • A polynomial is smooth and continuous. The end behavior of a polynomial is the same as its leading term. (Note that the leading term is a power function.)
  • The degree of a polynomial the highest power of the variable that occurs in a polynomial.
  • Turning point is the location at which the graph of a function changes direction. (a.k.a a local Extremum) The number of turning points for a polynomial of degree n is โ‰ค n โˆ’ 1. The number of real roots for a polynomial of degree n is โ‰ค n.
  • Other Forms: Often we refer to a polynomial in its factored form:

P (x) = an(x โˆ’ x 1 )(x โˆ’ x 2 )...(x โˆ’ xn). To find the standard form of the polynomial, foil all factors.

  1. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of d, the number of days elapsed.
  2. By cutting away identical squares of side length x in. from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 16 in. long and 12 in. wide, find a function that models the volume of the box as a function of x.

l = 16 โˆ’ 2 x

w = 12 โˆ’ 2 x 12

x

x