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Rational functions AASL

Author:Cliff Packman

Keywords

EnglishJapaneseKoreanChinese Simplified
Reciprocal Function逆関数역함수倒数函数
Simplify簡素化する단순화하다简化
Graphグラフ그래프图形
Characteristics特性특성特性
x-interceptsx軸との交点x-절편x轴截距
Vertical Asymptotes垂直漸近線수직 점근선垂直渐近线
Horizontal Asymptotes水平漸近線수평 점근선水平渐近线
Polynomial Functions多項式関数다항 함수多项式函数
Asymptotes漸近線점근선渐近线
Discontinuities不連続불연속不连续
Rational Functions有理関数유리 함수有理函数
Coordinates座標좌표坐标
Intercepts切片절편截距
Curve曲線곡선曲线
Image
Factual QuestionsConceptual QuestionsDebatable Questions
1. What is the definition of a reciprocal function?1. Explain why the graph of a reciprocal function has vertical asymptotes at the zeros of the original function.1. Is the study of reciprocal functions of linear functions as important as studying polynomial functions? Why or why not?
2. How do you graph the reciprocal function of ?2. Discuss the behavior of a reciprocal function as approaches the zeros of the original linear function.2. Can reciprocal functions of linear functions be effectively used to model real-world situations?
3. What are the characteristics of the graph of a reciprocal function of a linear function?3. How does the graph of a reciprocal function relate to the graph of its original linear function?3. Debate the difficulty of understanding the concepts of asymptotes and discontinuities in reciprocal functions.
4. Determine the x-intercepts of the reciprocal function for 4. Explain the conditions under which a reciprocal linear function will have horizontal asymptotes.4. Discuss the statement: "The graphical analysis of reciprocal functions is more complex than that of their original functions."
5. Explain how to verify that two functions are inverses of each other.5. Compare and contrast the graphs of reciprocal functions derived from different linear functions.5. Evaluate the impact of learning reciprocal functions on students' overall mathematical development.

The Asymptotic Adventure

Ahoy, intrepid explorers! Today, we set sail on the high seas of mathematics to navigate the mysterious curves of rational functions and hunt for the elusive asymptotes. Record your findings in your captain's log (notebook) and share the tales of your adventure with your fellow mathematicians. May the winds of curiosity always fill your sails!

Task 1: Asymptote Archipelago** - Chart a course for and discover the coordinates of the Vertical and Horizontal Asymptotes. Use the sliders to change a, b, c, and d and record your findings. - Can you predict the location of the asymptotes before they appear? Test your hypothesis using the applet.

Task 2: Captain's Challenge - As captain, create the most complex function you can, with both asymptotes and intercepts. Can your crew (classmates) determine the function's equation just by looking at the graph? - Try to write the function's equation in a different form. Does it still lead to the same graph? Use the applet to check.

Bonus Task: Legend of the Inverted Sea - Legends speak of a strange phenomenon where the sea turns upside down. Can you create a function that flips over its horizontal asymptote? - Discuss with your crew what mathematical transformations lead to this 'inverted sea'. How do the asymptotes and intercepts help guide you to this phenomenon?

See the PDF for questions and solutions

Q1-9 Basic practice Q10-17 Section A style short answer Q18-19 Long answer Q20 Extension Towards HL

Part 2 - Checking your understanding

Watch the following video to see how the asymptotes can be calculated, after you have made your conjectures.

Question 1: Consider the rational function. What is the domain of ?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 2: What is the horizontal asymptote of the function g(x) = (5x - 1)/(3x + 2)?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 3: For the function h(x) = (3x + 4)/(2x - 5), what is the vertical asymptote?

Select all that apply
  • A

Question 4: The function has a y-intercept at:

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 5: Which of the following represents a rational function with a hole at ?

Select all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

[MAA 2.7] ASYMPTOTES

[MAA 2.7] ASYMPTOTES.pdf

[MAA 2.7] ASYMPTOTES.pdf

[MAA 2.7] ASYMPTOTES_solutions

[MAA 2.7] ASYMPTOTES_solutions.pdf

[MAA 2.7] ASYMPTOTES_solutions.pdf

Lesson Plan- Navigating the World of Rational Functions and Asymptotes

Lesson Plan- Navigating the World of Rational Functions and Asymptotes.pdf

Lesson Plan- Navigating the World of Rational Functions and Asymptotes.pdf

Rational functions - Intuition pump (thought experiments and analogies)

Rational functions AASLLinlin.pdf

Rational functions AASLLinlin.pdf

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