Rational functions (AA HL 2.13) (Linear over quadratic)
Keywords
[table][br][tr][br][td]Rational Function[/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]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]Domain[/td][br][td]定義域[/td][br][td]정의역[/td][br][td]定义域[/td][br][/tr][br][tr][br][td]Horizontal Asymptote[/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]Simplify[/td][br][td]簡単化[/td][br][td]단순화[/td][br][td]简化[/td][br][/tr][br][tr][br][td]Degree of Numerator and Denominator[/td][br][td]分子と分母の次数[/td][br][td]분자와 분모의 차수[/td][br][td]分子和分母的次数[/td][br][/tr][br][tr][br][td]Holes (Removable Discontinuities)[/td][br][td]穴（取り除ける不連続性）[/td][br][td]구멍 (Removable Discontinuities)[/td][br][td]孔（可去不连续性）[/td][br][/tr][br][tr][br][td]Transformations[/td][br][td]変換[/td][br][td]변환[/td][br][td]变换[/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][/table][br]
Inquiry questions
[br][table][br][tr][br] [td][b]Factual Questions[/b][br] [list=1][br] [*]What is a rational function?[br] [*]How do you find the vertical asymptotes of a rational function?[br] [*]What is the domain of the rational function?[br] [*]Give an example of a rational function with a horizontal asymptote.[br] [*]How do you simplify the rational function?[br] [/list][br] [/td][br] [td][b]Conceptual Questions[/b][br] [list=1][br] [*]Why do rational functions have asymptotes, and what do they represent?[br] [*]Explain how the degree of the numerator and denominator affects the graph of a rational function.[br] [*]Discuss the significance of holes in the graph of a rational function.[br] [*]How can transformations be used to graph more complex rational functions?[br] [*]Compare the behavior of a rational function near its vertical asymptote to near its horizontal or oblique asymptote.[br] [/list][br] [/td][br] [td][b]Debatable Questions[/b][br] [list=1][br] [*]Is the concept of asymptotes more critical to understanding rational functions than intercepts?[br] [*]Can rational functions model real-world phenomena more effectively than polynomial functions?[br] [*]Debate the practicality of using rational functions in high school mathematics.[br] [*]Discuss the statement: "The limitations on the domain of rational functions limit their application in real-world problems."[br] [*]Evaluate the impact of technology on teaching and understanding rational functions.[br] [/list][br] [/td][br][/tr][br][/table][br]
Mini-Investigation: Unraveling Rational Functions[br][br]Objective:[br]To delve into the behavior of rational functions where the numerator is linear and the denominator is quadratic, and to understand how the parameters of the function affect its graph.[br][br]Activity:[br]Using the applet, manipulate the coefficients to model a real-world situation where a ratio decreases rapidly at first and then levels off, such as the concentration of a drug in the bloodstream over time after it is administered.
1. What patterns do you notice in the graph when the coefficients of the quadratic in the denominator are altered?[br]
2. How does changing the coefficient 'a' in the numerator (ax + b) impact the graph of the function? Consider both positive and negative values.[br]
3. Identify the vertical asymptotes of the function and relate them to the denominator's factors. How do they shift when you tweak 'c', 'd', and 'e'?[br]
4. Examine the horizontal asymptote or the end behavior of the function. What happens as x approaches positive or negative infinity?[br]
5. Can you find a set of coefficients where the graph crosses the horizontal asymptote?[br]
6. Challenge: Create a scenario where the function has a "hole" (a removable discontinuity). What condition on the coefficients would lead to this situation?
Lesson Plan- Rational Functions - Linear over Quadratic
Rational functions - Intuition pump (thought experiments and analogies)
