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Rational functions (AA HL 2.13) (Linear over quadratic)

Topic:
Functions

Keywords

Rational Function有理関数유리 함수有理函数
Vertical Asymptotes垂直漸近線수직 점근선垂直渐近线
Domain定義域정의역定义域
Horizontal Asymptote水平漸近線수평 점근선水平渐近线
Simplify簡単化단순화简化
Degree of Numerator and Denominator分子と分母の次数분자와 분모의 차수分子和分母的次数
Holes (Removable Discontinuities)穴(取り除ける不連続性)구멍 (Removable Discontinuities)孔(可去不连续性)
Transformations変換변환变换
Intercepts切片절편截距

Inquiry questions

Factual Questions
  1. What is a rational function?
  2. How do you find the vertical asymptotes of a rational function?
  3. What is the domain of the rational function?
  4. Give an example of a rational function with a horizontal asymptote.
  5. How do you simplify the rational function?
Conceptual Questions
  1. Why do rational functions have asymptotes, and what do they represent?
  2. Explain how the degree of the numerator and denominator affects the graph of a rational function.
  3. Discuss the significance of holes in the graph of a rational function.
  4. How can transformations be used to graph more complex rational functions?
  5. Compare the behavior of a rational function near its vertical asymptote to near its horizontal or oblique asymptote.
Debatable Questions
  1. Is the concept of asymptotes more critical to understanding rational functions than intercepts?
  2. Can rational functions model real-world phenomena more effectively than polynomial functions?
  3. Debate the practicality of using rational functions in high school mathematics.
  4. Discuss the statement: "The limitations on the domain of rational functions limit their application in real-world problems."
  5. Evaluate the impact of technology on teaching and understanding rational functions.
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Mini-Investigation: Unraveling Rational Functions Objective: 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. Activity: 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?

2. How does changing the coefficient 'a' in the numerator (ax + b) impact the graph of the function? Consider both positive and negative values.

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'?

4. Examine the horizontal asymptote or the end behavior of the function. What happens as x approaches positive or negative infinity?

5. Can you find a set of coefficients where the graph crosses the horizontal asymptote?

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