Log graph transformations
Keywords
[br][table][br][tr][br][td]English[/td][br][td]Japanese[/td][br][td]Korean[/td][br][td]Chinese Simplified[/td][br][/tr][br][tr][br][td]Logarithmic Function[/td][br][td]対数関数[/td][br][td]로그 함수[/td][br][td]对数函数[/td][br][/tr][br][tr][br][td]Sketch[/td][br][td]スケッチ[/td][br][td]스케치[/td][br][td]草图[/td][br][/tr][br][tr][br][td]Graph Transformation[/td][br][td]グラフの変換[/td][br][td]그래프 변환[/td][br][td]图形变换[/td][br][/tr][br][tr][br][td]x-intercept[/td][br][td]x軸との交点[/td][br][td]x-절편[/td][br][td]x轴截距[/td][br][/tr][br][tr][br][td]Domain[/td][br][td]定義域[/td][br][td]정의역[/td][br][td]定义域[/td][br][/tr][br][tr][br][td]Inverse Function[/td][br][td]逆関数[/td][br][td]역함수[/td][br][td]逆函数[/td][br][/tr][br][tr][br][td]Exponential Function[/td][br][td]指数関数[/td][br][td]지수 함수[/td][br][td]指数函数[/td][br][/tr][br][tr][br][td]Asymptotes[/td][br][td]漸近線[/td][br][td]점근선[/td][br][td]渐近线[/td][br][/tr][br][tr][br][td]Range[/td][br][td]値域[/td][br][td]치역[/td][br][td]值域[/td][br][/tr][br][tr][br][td]Base Change[/td][br][td]底の変換[/td][br][td]기수 변경[/td][br][td]底数变换[/td][br][/tr][br][tr][br][td]Vertical Asymptote[/td][br][td]垂直漸近線[/td][br][td]수직 점근선[/td][br][td]垂直渐近线[/td][br][/tr][br][tr][br][td]Different Bases[/td][br][td]異なる基底[/td][br][td]다른 기수[/td][br][td]不同基数[/td][br][/tr][br][tr][br][td]Exponential Equations[/td][br][td]指数方程式[/td][br][td]지수 방정식[/td][br][td]指数方程[/td][br][/tr][br][tr][br][td]Growth and Decay Models[/td][br][td]成長と減衰のモデル[/td][br][td]성장 및 감소 모델[/td][br][td]增长与衰减模型[/td][br][/tr][br][tr][br][td]Transformations[/td][br][td]変換[/td][br][td]변환[/td][br][td]变换[/td][br][/tr][br][tr][br][td]Horizontal Shift[/td][br][td]水平シフト[/td][br][td]수평 이동[/td][br][td]水平移动[/td][br][/tr][br][tr][br][td]Vertical Shift[/td][br][td]垂直シフト[/td][br][td]수직 이동[/td][br][td]垂直移动[/td][br][/tr][br][tr][br][td]Function-inverse Relationship[/td][br][td]関数と逆関数の関係[/td][br][td]함수-역함수 관계[/td][br][td]函数与逆函数的关系[/td][br][/tr][br][tr][br][td]Reflection over the line y=x[/td][br][td]直線y=xに関する反射[/td][br][td]선 y=x에 대한 반사[/td][br][td]关于y=x的反射[/td][br][/tr][br][/table][br]
Factual Questions[br]1. What is the definition of a logarithmic function?[br]2. Sketch the graph of the function [math]f(x)=log(x)[/math].[br]3. How do you transform the graph of f(x) = log(x) to sketch [math]f(x)=log(x-2)[/math]?[br]4. Determine the x-intercept of the logarithmic function [math]f(x)=log(x)+3[/math].[br]5. What is the domain of the logarithmic function [math]f(x)=log(2x-1)[/math]?[br][br]Conceptual Questions[br]1. Explain why logarithmic functions are the inverse of exponential functions.[br]2. Discuss the characteristics of logarithmic graphs, including their asymptotes, domain, and range.[br]3. How do changes in the base of a logarithmic function affect its graph?[br]4. Explain the significance of the vertical asymptote in the graph of a logarithmic function.[br]5. Compare the graphs of logarithmic functions with different bases.[br][br]Debatable Questions[br]1. Is understanding logarithmic functions as crucial as understanding exponential functions? Why or why not?[br]2. Debate the practicality of using logarithmic scales in real-world applications.[br]3. Can the concepts of logarithmic functions enhance one's ability to solve exponential equations?[br]4. Discuss the statement: "The study of logarithmic functions is essential for a deep understanding of growth and decay models."[br]5. Evaluate the impact of learning logarithmic functions on students' mathematical reasoning and analytical skills.[br]
Mini-Investigation: Exploring Logarithmic Functions[br][br]Objectives:[br]- Understand the transformations of logarithmic functions.[br]- Identify the asymptotes of logarithmic functions.[br]- Determine the domain and range of logarithmic functions.[br]- Visualize the relationship between a function and its inverse.[br][br]Investigation Steps:[br][br]1. Introduction to the Logarithmic Function:[br] - Start with the parent function [math]g(x)=log_{10}(x)[/math]. Use your calculator to plot. Observe its shape and key characteristics.[br] - Identify the vertical asymptote, domain, and range.[br][br]
2. Horizontal Shift:[br] - Transform [math]g(x)[/math] to match [math]f(x)=log_{10}(x-2)[/math].[br] - Use the applet to slide the horizontal shift control and observe the changes.[br] - Record observations on the horizontal shift's effect on the graph.
3. Vertical Shift:[br] - Apply a vertical shift to get [math]f(x)=log_{10}(x-2)+1[/math][br] - Slide the vertical shift control to see the graph move up by 1 unit.[br] - Discuss the impact on domain and range.
4. Asymptote Identification:[br] - Identify the new vertical asymptote for the transformed function [math]f(x)[/math].[br] - Use the applet to display the equation of the asymptote and confirm if it matches [math]x=2[/math].[br] - Explain the asymptote's significance to the function's domain.[br]
5. Domain and Range Analysis:[br] - Confirm the domain and range of [math]f(x)[/math] using the applet.[br] - Reflect on the reasons behind the domain restriction and the range being all real numbers.
6. Inverse Function Exploration:[br] - Explore the inverse function [math]f^{-1}(x)[/math].[br] - Use the applet to show the inverse function and its reflection over the line [math]y=x[/math].[br] - Sketch or capture both [math]f(x)[/math]and [math]f^{-1}(x)[/math]on the graph with the line [math]y=x[/math].
7. Conclusions:[br] - Summarize the effects of horizontal and vertical shifts on the logarithmic function.[br] - Discuss the function-inverse relationship in the context of logarithmic functions.[br][br]Reflection Questions:[br]1. How does changing the base of the logarithm affect the graph's shape?[br]2. What happens to the graph of [math]f(x)[/math] if you subtract inside the logarithm (e.g., [math]f(x)=log_{10}(x+2)[/math]?[br]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
