Rational functions AASL
Keywords
[br][table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Reciprocal Function[/td][br][td]逆関数[/td][br][td]역함수[/td][br][td]倒数函数[/td][br][/tr][br][tr][br][td]Simplify[/td][br][td]簡素化する[/td][br][td]단순화하다[/td][br][td]简化[/td][br][/tr][br][tr][br][td]Graph[/td][br][td]グラフ[/td][br][td]그래프[/td][br][td]图形[/td][br][/tr][br][tr][br][td]Characteristics[/td][br][td]特性[/td][br][td]특성[/td][br][td]特性[/td][br][/tr][br][tr][br][td]x-intercepts[/td][br][td]x軸との交点[/td][br][td]x-절편[/td][br][td]x轴截距[/td][br][/tr][br][tr][br][td]Vertical Asymptotes[/td][br][td]垂直漸近線[/td][br][td]수직 점근선[/td][br][td]垂直渐近线[/td][br][/tr][br][tr][br][td]Horizontal Asymptotes[/td][br][td]水平漸近線[/td][br][td]수평 점근선[/td][br][td]水平渐近线[/td][br][/tr][br][tr][br][td]Polynomial Functions[/td][br][td]多項式関数[/td][br][td]다항 함수[/td][br][td]多项式函数[/td][br][/tr][br][tr][br][td]Asymptotes[/td][br][td]漸近線[/td][br][td]점근선[/td][br][td]渐近线[/td][br][/tr][br][tr][br][td]Discontinuities[/td][br][td]不連続[/td][br][td]불연속[/td][br][td]不连续[/td][br][/tr][br][tr][br][td]Rational Functions[/td][br][td]有理関数[/td][br][td]유리 함수[/td][br][td]有理函数[/td][br][/tr][br][tr][br][td]Coordinates[/td][br][td]座標[/td][br][td]좌표[/td][br][td]坐标[/td][br][/tr][br][tr][br][td]Intercepts[/td][br][td]切片[/td][br][td]절편[/td][br][td]截距[/td][br][/tr][br][tr][br][td]Curve[/td][br][td]曲線[/td][br][td]곡선[/td][br][td]曲线[/td][br][/tr][br][/table][br]
[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]1. What is the definition of a reciprocal function?[/td][br][td]1. Explain why the graph of a reciprocal function has vertical asymptotes at the zeros of the original function.[/td][br][td]1. Is the study of reciprocal functions of linear functions as important as studying polynomial functions? Why or why not?[/td][br][/tr][br][tr][br][td]2. How do you graph the reciprocal function of [math]f(x)=x+1[/math]?[/td][br][td]2. Discuss the behavior of a reciprocal function as [math]x[/math] approaches the zeros of the original linear function.[/td][br][td]2. Can reciprocal functions of linear functions be effectively used to model real-world situations?[/td][br][/tr][br][tr][br][td]3. What are the characteristics of the graph of a reciprocal function of a linear function?[/td][br][td]3. How does the graph of a reciprocal function relate to the graph of its original linear function?[/td][br][td]3. Debate the difficulty of understanding the concepts of asymptotes and discontinuities in reciprocal functions.[/td][br][/tr][br][tr][br][td]4. Determine the x-intercepts of the reciprocal function for [math]f(x)=3-x.[/math][/td][br][td]4. Explain the conditions under which a reciprocal linear function will have horizontal asymptotes.[/td][br][td]4. Discuss the statement: "The graphical analysis of reciprocal functions is more complex than that of their original functions."[/td][br][/tr][br][tr][br][td]5. Explain how to verify that two functions are inverses of each other.[/td][br][td]5. Compare and contrast the graphs of reciprocal functions derived from different linear functions.[/td][br][td]5. Evaluate the impact of learning reciprocal functions on students' overall mathematical development.[/td][br][/tr][br][/table][br]
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.[br][br]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![br]
Task 1: Asymptote Archipelago**[br]- Chart a course for [math]y=\frac{ax+b}{cx+d}[/math] and discover the coordinates of the Vertical and Horizontal Asymptotes. Use the sliders to change a, b, c, and d and record your findings.[br]- Can you predict the location of the asymptotes before they appear? Test your hypothesis using the applet.
Task 2: Captain's Challenge[br]- 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?[br]- 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[br]- Legends speak of a strange phenomenon where the sea turns upside down. Can you create a function that flips over its horizontal asymptote?[br]- 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 [br]Q10-17 Section A style short answer[br]Q18-19 Long answer[br]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:[br]Consider the rational function[math]f(x)=(2x+3)/(x-4)[/math]. What is the domain of [math]f(x)[/math]?[br][br][br]
Question 2:[br]What is the horizontal asymptote of the function g(x) = (5x - 1)/(3x + 2)?[br][br][br][br]
Question 3:[br]For the function h(x) = (3x + 4)/(2x - 5), what is the vertical asymptote?[br][br][br]
Question 4:[br]The function [math]p(x)=(4x-7)/(x+3)[/math] has a y-intercept at:[br][br][br]
Question 5:[br]Which of the following represents a rational function with a hole at [math]x=-2[/math]?[br]
[MAA 2.7] ASYMPTOTES
[MAA 2.7] ASYMPTOTES_solutions
Lesson Plan- Navigating the World of Rational Functions and Asymptotes
Rational functions - Intuition pump (thought experiments and analogies)
