Rational functions (AAHL 2.13) quadratic over linear
Keywords
[table][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]Quadratic over linear[/td][br][td]線形分母の二次関数[/td][br][td]1차식 위의 이차식[/td][br][td]一次式除以二次式[/td][br][/tr][br][tr][br][td]Domain and range[/td][br][td]定義域と値域[/td][br][td]정의역과 치역[/td][br][td]定义域和值域[/td][br][/tr][br][tr][br][td]Asymptotes: vertical, horizontal, oblique[/td][br][td]漸近線: 縦, 横, 斜め[/td][br][td]비대칭선: 수직, 수평, 경사선[/td][br][td]渐近线：垂直，水平，斜线[/td][br][/tr][br][tr][br][td]Graph analysis[/td][br][td]グラフの分析[/td][br][td]그래프 분석[/td][br][td]图形分析[/td][br][/tr][br][tr][br][td]Coefficients effect[/td][br][td]係数の影響[/td][br][td]계수의 영향[/td][br][td]系数影响[/td][br][/tr][br][tr][br][td]Vertical asymptote[/td][br][td]垂直漸近線[/td][br][td]수직 비대칭선[/td][br][td]垂直渐近线[/td][br][/tr][br][tr][br][td]x-intercepts[/td][br][td]x切片[/td][br][td]x절편[/td][br][td]x截距[/td][br][/tr][br][tr][br][td]Removable discontinuity[/td][br][td]可除不連続点[/td][br][td]제거 가능한 불연속점[/td][br][td]可去间断点[/td][br][/tr][br][/table]
Inquiry questions
[br][table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What is a rational function, specifically one that is quadratic over linear?[br] [*]How can the domain and range of a rational function with a quadratic numerator and linear denominator be determined?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]Why does the presence of a quadratic expression in the numerator and a linear expression in the denominator affect the function's behavior, such as asymptotes and intercepts?[br] [*]How can the concepts of vertical, horizontal, or oblique asymptotes be applied to analyze the graph of a rational function that is quadratic over linear?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Can understanding rational functions with a quadratic numerator over a linear denominator provide deeper insights into other areas of mathematics or applied fields? Provide examples.[br] [*]How do advancements in graphing technology influence the way students understand and interact with complex functions such as rational functions that are quadratic over linear?[br] [/list][br] [/td][br][/tr][br][/table][br]
Mini-Investigation: Rational Functions – Quadratic over Linear[br][br]Objective:[br]To delve into the characteristics of rational functions where the numerator is a quadratic expression and the denominator is linear.[br]
1. How does changing the coefficients of the quadratic numerator affect the shape of the rational function graph? Experiment with different values for a, c, and e.
2. What happens to the graph when the linear denominator has coefficients other than 1? Modify the values of a and d and observe the changes.
3. Identify the vertical asymptote of the function and discuss how it relates to the denominator's coefficients.
5. Explore the behavior of the function around the vertical asymptote. How does the function behave as x approaches the asymptote from the left and from the right?
6. Can you determine the x-intercepts of the function by setting the numerator equal to zero?
7. If the numerator has a real factor (a real root), how does that affect the graph of the rational function?
8. Challenge: Adjust the coefficients to create a "hole" in the graph (a removable discontinuity). What conditions must be met for a hole to occur?
Lesson Plan- Exploring Rational Functions - Quadratic over Linear
Rational functions - Intuition pump (thought experiments and analogies)
