GeoGebra Classroom

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