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Log graph transformations

Keywords

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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-interceptx軸との交点x-절편x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.
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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