Log graph transformations
Keywords
| English | Japanese | Korean | Chinese Simplified |
| Logarithmic Function | 対数関数 | 로그 함수 | 对数函数 |
| Sketch | スケッチ | 스케치 | 草图 |
| Graph Transformation | グラフの変換 | 그래프 변환 | 图形变换 |
| x-intercept | x軸との交点 | x-절편 | x轴截距 |
| Domain | 定義域 | 정의역 | 定义域 |
| Inverse Function | 逆関数 | 역함수 | 逆函数 |
| Exponential Function | 指数関数 | 지수 함수 | 指数函数 |
| Asymptotes | 漸近線 | 점근선 | 渐近线 |
| Range | 値域 | 치역 | 值域 |
| Base Change | 底の変換 | 기수 변경 | 底数变换 |
| Vertical Asymptote | 垂直漸近線 | 수직 점근선 | 垂直渐近线 |
| Different Bases | 異なる基底 | 다른 기수 | 不同基数 |
| Exponential Equations | 指数方程式 | 지수 방정식 | 指数方程 |
| Growth and Decay Models | 成長と減衰のモデル | 성장 및 감소 모델 | 增长与衰减模型 |
| Transformations | 変換 | 변환 | 变换 |
| Horizontal Shift | 水平シフト | 수평 이동 | 水平移动 |
| Vertical Shift | 垂直シフト | 수직 이동 | 垂直移动 |
| Function-inverse Relationship | 関数と逆関数の関係 | 함수-역함수 관계 | 函数与逆函数的关系 |
| Reflection over the line y=x | 直線y=xに関する反射 | 선 y=x에 대한 반사 | 关于y=x的反射 |
Factual Questions
1. What is the definition of a logarithmic function?
2. Sketch the graph of the function .
3. How do you transform the graph of f(x) = log(x) to sketch ?
4. Determine the x-intercept of the logarithmic function .
5. What is the domain of the logarithmic function ?
Conceptual Questions
1. Explain why logarithmic functions are the inverse of exponential functions.
2. Discuss the characteristics of logarithmic graphs, including their asymptotes, domain, and range.
3. How do changes in the base of a logarithmic function affect its graph?
4. Explain the significance of the vertical asymptote in the graph of a logarithmic function.
5. Compare the graphs of logarithmic functions with different bases.
Debatable Questions
1. Is understanding logarithmic functions as crucial as understanding exponential functions? Why or why not?
2. Debate the practicality of using logarithmic scales in real-world applications.
3. Can the concepts of logarithmic functions enhance one's ability to solve exponential equations?
4. Discuss the statement: "The study of logarithmic functions is essential for a deep understanding of growth and decay models."
5. Evaluate the impact of learning logarithmic functions on students' mathematical reasoning and analytical skills.
Mini-Investigation: Exploring Logarithmic Functions
Objectives:
- Understand the transformations of logarithmic functions.
- Identify the asymptotes of logarithmic functions.
- Determine the domain and range of logarithmic functions.
- Visualize the relationship between a function and its inverse.
Investigation Steps:
1. Introduction to the Logarithmic Function:
- Start with the parent function . Use your calculator to plot. Observe its shape and key characteristics.
- Identify the vertical asymptote, domain, and range.
2. Horizontal Shift:
- Transform to match .
- Use the applet to slide the horizontal shift control and observe the changes.
- Record observations on the horizontal shift's effect on the graph.
3. Vertical Shift:
- Apply a vertical shift to get
- Slide the vertical shift control to see the graph move up by 1 unit.
- Discuss the impact on domain and range.
4. Asymptote Identification:
- Identify the new vertical asymptote for the transformed function .
- Use the applet to display the equation of the asymptote and confirm if it matches .
- Explain the asymptote's significance to the function's domain.
5. Domain and Range Analysis:
- Confirm the domain and range of using the applet.
- Reflect on the reasons behind the domain restriction and the range being all real numbers.
6. Inverse Function Exploration:
- Explore the inverse function .
- Use the applet to show the inverse function and its reflection over the line .
- Sketch or capture both and on the graph with the line .
7. Conclusions:
- Summarize the effects of horizontal and vertical shifts on the logarithmic function.
- Discuss the function-inverse relationship in the context of logarithmic functions.
Reflection Questions:
1. How does changing the base of the logarithm affect the graph's shape?
2. What happens to the graph of if you subtract inside the logarithm (e.g., ?
3. How would the domain and range change if the logarithmic function were reflected over the x-axis?
3 questions selected from Christos logarithm questions. See https://www.christosnikolaidis.com/en/ for more.
Lesson Plan- Unveiling the Dynamics of Logarithmic Graph Transformations
