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绘制对数函数图像并描述特性

绘制对数函数图像并描述特性
几何画板及绘图内容加载中，请稍等……
2014-08-06 02:00
函数解析几何函数图像
本范例演示了如何在坐标系中绘制函数f(x) = 2 • log2(x)和g(x) = 2 – log2(x)的图像，并描述其渐近线与对称性等特性。涉及的核心数学概念包括对数函数、函数图像绘制、对称性分析。作图要点使用了GeoGebra中的If、LaTeX、FractionText命令和对象，如d、c、g、a、e、p、q、r、f、stepMax、hr、layer、stepper、steps、hotspot4、hr0、prevNext、prev、next、b、prevHotspot、nextHotspot、btnInfobox、b1、pic2、b2、b3、b4、b5、b6、b7、btnsOGrandom、randomHotspot、ogHotspot、h、step1、step1_1、step1_{11}、step1_{12}、step1_2等。适合高中阶段学习对数函数图像与性质的学生使用。

GeoGebra 命令及参数

  1. If(d ≟ 1, " ", LaTeX[d]) → h
  2. LaTeX("\text{Determine function values for } f(1) \text{ and } g(1) \text{ and write the results as ordered pairs.}") → step1
  3. LaTeX("\text{Substitute } x = 1 \text{ into each function and solve.}") → step1_1
  4. LaTeX((((((("f(1) = " * a) * "\cdot \log_") * b) * "(1) = ") * a) * "\cdot 0 = 0")) → step1_{11}
  5. LaTeX(((((((((("g(1) = " * c) * " - ") * h) * " \log_") * b) * "(1) = ") * c) * " - 0 = ") * c)) → step1_{12}
  6. LaTeX("\text{Write each as an ordered pair.}") → step1_2
  7. LaTeX((((((a * "\cdot \log_2\,x = ") * c) * "-") * h) * " \log_2 x")) → step2_1
  8. LaTeX(((e * "\cdot \log_2\,x = ") * c)) → step2_2
  9. LaTeX(((((" \log_2\,x = \frac{" * p) * "}{") * q) * "}")) → step2_3
  10. LaTeX((((((a * "^{\frac{") * p) * "}{") * q) * "}}=x")) → step2_4
  11. FractionText(s) → rt
  12. LaTeX("\text{Find the function value at which the functions are equal.}") → step3
  13. LaTeX((((((((" f(2) = " * a) * "\cdot \log_") * b) * " \; (2) = ") * a) * "\cdot 1=") * f(2))) → step5_3
  14. LaTeX((((((((" g(4) = " * a) * "- \log_") * b) * " \; (4) = ") * a) * "- 2=") * g(4))) → step5_8
  15. LaTeX((((((((" g(2) = " * a) * "- \log_") * b) * " \; (2) = ") * a) * "- 1=") * g(2))) → step5_7
  16. LaTeX((((((((" g(0.5) = " * a) * "- \log_") * b) * " \; (0.5) = ") * a) * "- (-1)=") * g(0.5))) → step5_6
  17. LaTeX((((((((" g(0.25) = " * a) * "- \log_") * b) * " \; (0.25) = ") * a) * "- (-2)=") * g(0.25))) → step5_5
  18. LaTeX((((((((" f(4) = " * a) * "\cdot \log_") * b) * " \; (4) = ") * a) * "\cdot 2=") * f(4))) → step5_4
  19. LaTeX("\text{Find the value(s) of x at which f(x) = g(x). } \cr\text{This will reveal any point(s) of intersection, or solutions, of the system of equations.}") → step2
  20. LaTeX("\text{Set the functions equal to each other.}") → step2_{x1}
  21. LaTeX((("\text{Add } " * h) * " \log_2 x \text{ to both sides.}")) → step2_{x2}
  22. LaTeX((("\text{Divide both sides by }" * q) * ".")) → step2_{x3}
  23. LaTeX("\text{Simplify.}") → step2_{x4}
  24. LaTeX(((((("\text{Substitute the approximate value of x from step 2 into each function.} \cr\ \cr \cr\ \cr\ f(" * r) * ")\approx g(") * r) * ") \approx ") * s)) → step3_1
  25. LaTeX("\text{Write the approximate point at which the functions intersect.}") → step4
  26. LaTeX("\text{Write the additional points. Make sure to keep the points with the correct function.}") → step6
  27. LaTeX("\text{For f(x): As x approaches positive infinity, f(x) approaches positive infinity. } \cr\text{As x approaches 0, f(x) approaches negative infinity.}") → step8_2
  28. LaTeX("\text{For g(x): As x approaches positive infinity, g(x) approaches negative infinity. } \cr\text{As x approaches 0, g(x) approaches positive infinity.}") → step8_3
  29. LaTeX((((((((((("\text{Sketch the graphs of the functions } f(x) = " * a) * "\cdot \log_") * b) * "x \text{ and } g(x) = ") * c) * "-") * h) * "\log_") * b) * "x. \cr\text{Describe the end behavior of each function.}")) → Problem
  30. LaTeX("f(1) = 0: (1,0)") → step1_{21}
  31. LaTeX((((("g(1) = " * c) * ": (1,") * c) * ")")) → step1_{22}
  32. LaTeX((("\text{Using a calculator, x} \approx" * round((10 * a^(p / q))) / 10) * ".")) → step2_5
  33. LaTeX((((("\text{The intersection point is approximately } (" * r) * ",\, ") * s) * ").")) → step4_1
  34. LaTeX("\text{Determine additional points for sketching the graph of each function.} \cr\text{Choose values of x that can be evaluated easily. }") → step5
  35. LaTeX(" \text{Solve f(x) for x = 0.25, 0.5, 2, and 4.}") → step5_{f}
  36. LaTeX(" \text{Solve g(x) for x = 0.25, 0.5, 2, and 4.}") → step5_{g}
  37. LaTeX((((((((" f(0.25) = " * a) * "\cdot \log_") * b) * " \; (0.25) = ") * a) * "\cdot -2=") * f(0.25))) → step5_1
  38. LaTeX((((((((" f(0.5) = " * a) * "\cdot \log_") * b) * " \; (0.5) = ") * a) * "\cdot -1=") * f(0.5))) → step5_2
  39. LaTeX("\text{Plot the graphs.}") → step7
  40. LaTeX("\text{It is useful to know the limits of function values as the values go} \cr\text{beyond the finite values used for a sketch of a graph.}") → step8_1
  41. LaTeX(((((((((" \text{For f(x): A = }(0.25, " * f(0.25)) * ")\text{, B = }(0.5, ") * f(0.5)) * ")\text{, C = }(2, ") * f(2)) * ")\text{, and D = }(4,") * f(4)) * ") ")) → step6_1
  42. LaTeX(((((((((" \text{For g(x): E = }(0.25, " * g(0.25)) * ")\text{, F = }(0.5, ") * g(0.5)) * ")\text{, G = }(2, ") * g(2)) * ")\text{, and H = }(4,") * g(4)) * ") ")) → step6_2
  43. LaTeX("\text{Describe what happens to the function values as x becomes very large } \cr\text{and as x approaches 0.}") → step8