1. Math IV Unit II: Rational Functions
Week 4
The Characteristics of Rational Functions

2. Today I will learn how to: To define a polynomial function
Essential Question What basic concepts must be understood before working with Rational Functions?
Today I will learn how to:
To define a polynomial function
Identify the key features of polynomial functions
Important Vocabulary
Polynomial
Range
Domain
Zeros
End Behavior

3. Warm Up Solve the equations x2 – 16 = 0 9 – x2 = 0
Essential Question What basic concepts must be understood before working with Rational Functions?
Warm Up
Solve the equations
x2 – 16 = 0
9 – x2 = 0

4. Key Vocabulary Matching Review Game (15 min)
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)

5. Match the word with it’s definition
Essential Question: What basic concepts must be understood before working with Rational Functions?
Match the word with it’s definition
Domain
Range
End Behavior
Holes
Horizontal Asymptote
Vertical Asymptote
Describes what happens when you can cancel a polynomial factor in the numerator and
denominator of a rational function.
The set of all possible x-values (independent values)
The imaginary line that the graph does not touch because using that x-value would make the
function undefined
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
The imaginary line that the function approaches at it’s ends. It usually tells us how high or low the graph will go.

6. Rational Functions Characteristics Learning Task:
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?

7. Rational Functions Characteristics Learning Task:
Essential Question: What basic concepts must be understood 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?

8. Enduring Understanding
Lesson Summary
Interpret graphs and discover characteristics of rational functions
Homework
p. 245 #1 – 4 Find the domain only.
Classwork
Cole p. 305 #7 – 31 odd
Find the domain. DO NOT sketch

9. Essential Question What are the Properties of Rational Functions?
Today I will learn how to:
Define & Identify Rational Functions
Determine the domain & range of a function
Important Vocabulary
End Behavior(Increasing, Decreasing, Approaching Infinity)
Roots & Zeros

10. Essential Question What are the Properties of Rational Functions?

11. What is a Rational Function?
Wednesday Sept. 12, 2012 Essential 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?

12. Essential Question: What are the Properties of Rational Functions?
f(x) = x + 1
g(x) = x2 + 3x – 10

13. Essential Question: What are the Properties of Rational Functions?

14. Lesson Summary Activity
Essential Question: What are some of the Characteristics of Polynomial & Rational Functions?
Practice problems Text p. 102 #14 – 19 odd
Lesson Summary Activity
Create a Rational Function which
a. Increases as x approaches infinity
b. Decreases as x approaches negative infinity
c. Has a domain of all real numbers except at x = 1
Homework p. 102 #14 – 20 even

15. Essential Question: What do Rational Functions have in common
Essential Question: What do Rational Functions have in common? Wednesday
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 #
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.

16. Essential Question: What do Rational Functions have in common
Essential Question: What do Rational Functions have in common? Wednesday
Graded warm-up p.245 Quick Review #1, 3, 6 (7 min)
Homework Review Text p. 102 #10 – 20 even (10 min)
Answers:
Domain: All Real Numbers except x = 3
Domain: All Real Numbers except x = 0 or 3
14. 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 ≤ 2
16. Domain: All Real Numbers except numbers between -4 & 0
and between 0 and 4
18. Range: [5, ∞)
20. For this one you needed a graphing calculator to get (- ∞, -1) υ
[0.75, ∞)
Homework p. 246 #
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.

17. Warm-up Complete Posters for Group work Presentations (10 min)
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.

18. What we’ve learned so far What does a rational function look like?
Essential Question: How are horizontal & vertical asymptotes found in Rational Functions? Thursday
What we’ve learned so far
What 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.

19. 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

20. Finding Horizontal & Vertical Asymptotes Task p. 13
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.