Math IVUnit II: Rational Functions

Week 4The Characteristics of Rational

Functions

Essential Question

What basic concepts must be understood before working with Rational Functions?

Today I will learn how to:Today I will learn how to:•To define a polynomial function•Identify the key features of polynomial functions

Important VocabularyImportant Vocabulary•Polynomial•Range•Domain•Zeros•End Behavior

Essential Question

What basic concepts must be understood before working with

Rational Functions?Warm UpWarm Up

Solve the equations1.x2 – 16 = 02.9 – x2 = 0

Essential Question:What basic concepts must be understood before working with Rational Functions?

Agenda•Key Vocabulary Matching Review Game (15 min)•Basic Concepts Review (40 min)•Collect Graded Assignments (5 min)•Answer Essential Question (5 min)•Review Test #1 (10 min)

Essential Question:What basic concepts must be understood before working with Rational Functions?

Match the word with it’s definitionDomain Range

Horizontal Asymptote Vertical Asymptote

HolesEnd Behavior

The set of all possible x-values (independent values)

The set of all possible y-values (dependent values)

Describes what happens to the graph as x gets really, really big or really, really small

Describes what happens when you can cancel a polynomial factor in the numerator anddenominator of a rational function.

The imaginary line that the graph does not touch because using that x-value would make thefunction undefined

The imaginary line that the function approaches at it’s ends. It usually tells us how high or low the graph will go.

Essential Question:What basic concepts must be understood before working with Rational Functions?

Rational Functions Characteristics Learning Task: •What do you know about the polynomial

g(x) = x2 + 3x – 10? •What is the Domain? How do you determine the Domain? •What is the Range? How do you determine the Range? •Where are the Roots or Zeros found? What are some different ways you know to find them?

Essential Question:Essential Question:What basic concepts must be understood What basic concepts must be understood before working with Rational Functions?before working with Rational Functions?

Rational Functions Characteristics Learning Task: •What is the End Behavior? How do you know? •What do you know about the polynomial

f(x) = x + 1? •What is the Domain? •What is the Range? •What are the Roots or Zeros? •What is the End Behavior? How do you know?

Enduring UnderstandingEnduring UnderstandingLesson Summary

Interpret graphs and discover characteristics of rational functions

Homeworkp. 245 #1 – 4 Find the domain only.

ClassworkCole p. 305 #7 – 31 odd

Find the domain. DO NOT sketch

Essential Question

What are the Properties of Rational Functions?

Today I will learn how to:Today I will learn how to:•Define & Identify Rational FunctionsDefine & Identify Rational Functions•Determine the domain & range of a functionDetermine the domain & range of a function

Important VocabularyImportant Vocabulary•End Behavior(Increasing, Decreasing, Approaching Infinity)•Roots & Zeros

Essential Question

What are the Properties of Rational Functions?

Wednesday Sept. 12, 2012Essential Question

What are the Properties of Rational Functions?

What is a Rational Function?Rational Functions Functions That Are Not Rational

What is a Rational Function?

Essential Question:What are the Properties of Rational Functions?

g(x) = x2 + 3x – 10f(x) = x + 1

Essential Question:What are the Properties of Rational

Functions?

Essential Question:What are some of the Characteristics of Polynomial &

Rational Functions?

Practice problems Text p. 102 #14 – 19 odd

Lesson Summary ActivityLesson Summary ActivityCreate a Rational Function which a. Increases as x approaches infinityb. Decreases as x approaches negative infinityc. Has a domain of all real numbers except at x = 1

Homework p. 102 #14 – 20 even

Essential Question:What do Rational Functions have in common?

Wednesday

MM4A1. Students will explore rational functions.a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.

Graded warm-up p.245 Quick Review #1, 3, 6 (7 min)Homework Review Text p. 102 #10 – 20 even (10 min)

Task Completion in Groups pp 7 – 9 (20 min)(Each group will be assigned one of three problems)Group Presentations (35 min)Practice Problems Text p. 245 #1 – 4 (8 min)

Lesson Summary ( 5min)Answer Essential Question (5 min)Homework p. 246 #63 - 68

Essential Question:

What do Rational Functions have in common? Wednesday

MM4A1. Students will explore rational functions.a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.

Graded warm-up p.245 Quick Review #1, 3, 6 (7 min)Homework Review Text p. 102 #10 – 20 even (10 min)Answers:10.Domain: All Real Numbers except x = 312.Domain: All Real Numbers except x = 0 or 314. Domain: All Real Numbers except x = 3 and since 4 – x2 ≥ 0 this means we can only use numbers between -2 and 2. Written mathematically the domain is -2 ≤ x ≤ 216. Domain: All Real Numbers except numbers between -4 & 0 and between 0 and 418. Range: [5, ∞)20. For this one you needed a graphing calculator to get (- ∞, -1) υ [0.75, ∞)

Homework p. 246 #63 - 68

Essential Question:How are horizontal & vertical asymptotes found in Rational

Functions? Thursday

Warm-up Complete Posters for Group work Presentations (10 min)Homework Review (10 min)

Finding Horizontal & Vertical Asymptotes Task p. 13 (Rotating Groups) (30 min)Practice Problems p. 103 #55 – 61 odd (10 min)

Group Presentations (20 min)Lesson Summary (5 min)Answer Essential Question (5 min)

Tonight’s Homework p. 103 #56 ,60, 62 MM4A1. Students will explore rational functions.a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.

Essential Question:How are horizontal & vertical asymptotes found in Rational

Functions?Thursday

What we’ve learned so farWhat does a rational function look like?How do you find the domain of a rational function?How do you find the range of a rational function?How do you find the zeros of a rational function?How can you tell what’s happening at the ends of a rational function?

Tonight’s Homework p. 103 #56 ,60, 62

MM4A1. Students will explore rational functions.a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.

Essential Question:How are horizontal & vertical asymptotes found in Rational

Functions?Thursday

Warm-up Complete Posters for Group work Presentations (10 min)Homework Review p. 246 #63 – 68 (10 min)#63 False. If the denominator is never zero, there will be no vertical asymptote. For example, f(x) = 1/(x2 + 1) is a rational function and has no vertical asymptotes.#64 False. A rational function is the quotient of two polynomials, and √(x2 +4) is not a polynomial.#65 The answer is E. Factoring the denominator yields x(x+3).#66 The answer is A. Could have been skipped. #67 The answer is D. Since x + 5 = 0 when x = -5, there is a vertical asymptote. We did not cover slant asymptotes, so it is understandable if you did not get that part.#68 The answer is E. The quotient of the leading terms is x4.Tonight’s Homework p. 103 #56 – 62 even

Essential Question:How are horizontal & vertical asymptotes found in Rational

Functions?Thursday

Finding Horizontal & Vertical Asymptotes Task p. 13 (Rotating Groups) (30 min)

Practice Problems p. 103 #55 – 61 odd (10 min)

Group Presentations (20 min)Lesson Summary (5 min)Answer Essential Question (5 min)

Tonight’s Homework p. 103 #56 – 62 even

MM4A1. Students will explore rational functions.a. Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior.