Graphing Logarithmic Functions by Walch Education

GeoGebra Commands & Parameters

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