Graphing Logarithmic and Linear Functions

GeoGebra Commands & Parameters

1. If(d ≟ 1, " ", LaTeX[(d * "\cdot")]) → 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("\text{Write each as an ordered pair.}") → step1_2
5. FractionText(s) → rt
6. LaTeX("\text{Find the function value at which the functions are equal.}") → Xstep3X
7. 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)) → Xstep3_1X
8. LaTeX(((((("f(1) = " * d) * "+ \log (1) = ") * d) * " + 0 =") * d)) → step1_{11}
9. LaTeX("g(1) = \log (1) = 0 ") → step1_{12}
10. LaTeX("f(x) = g(x)") → step2_1
11. LaTeX(" x=0") → step2_3
12. LaTeX((" g(0.1) = \log \, (0.1) = " * g(0.1))) → step3_{g1}
13. LaTeX((" g(1) = \log \, (1) = " * g(1))) → step3_{g2}
14. LaTeX((" g(2) = \log \, (2) \approx " * g(2))) → step3_{g3}
15. LaTeX((" g(3) = \log \, (3) \approx " * g(3))) → step3_{g4}
16. LaTeX((" g(4) = \log \, (4) \approx " * g(4))) → step3_{g5}
17. If(d ≟ 1, " ", LaTeX[d]) → h_2
18. LaTeX((h_2 * "x + \log \, x = \log \, x")) → step2_2
19. LaTeX("\text{Write f(x) in terms of g(x).}") → step5
20. LaTeX("\text{Often functions are related, which can simplify graph sketching.}") → step5_1
21. LaTeX("\text{Describe what happens to the function values for domain values less than 1.}") → step7
22. 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
23. LaTeX("\text{The value x=0 is not in the domain of these functions.} \cr\text{Therefore, there is no point of intersection or solution for f(x)= g(x).} ") → step2_4
24. LaTeX("\text{Determine additional points for sketching the graph of each function.} ") → step3
25. LaTeX((((((((("f(x) = " * h_2) * "x + \log \, x = ") * h_2) * " x + g(x) \text{ or } f(x) - g(x) = ") * h_2) * "x, \cr\text{which means the difference between the function values } \cr\text{at any value of x is } ") * h_2) * "x.")) → step5_2
26. LaTeX("\text{Plot the graphs.}") → step6
27. LaTeX("\text{Plot the graphs.}") → FormulaText["\text{Plot of the graphs.}"]
28. LaTeX("\text{The graph indicates the function diverges, but the pattern is unclear.}") → step7_1
29. LaTeX("\text{For g(x): The function values approach negative infinity as x approaches 0.} \cr\text{As x approaches 1, the function values converge on }0.") → step7_3
30. LaTeX("\text{Describe what happens to the function values for domain values greater than 1.}") → step8
31. LaTeX("\text{The graph indicates the function values increase, but the pattern is unclear.}") → step8_1
32. LaTeX("\text{For g(x): The function values approach positive infinity as x increases } \cr\text{from x=1 to positive infinity. As x approaches 1 from the right, } \cr\text{the function values converge on }0.") → step8_3
33. LaTeX((("\text{Sketch the graphs of the functions } f(x) = " * h_2) * " x + \log \,x \text{ and } g(x) = \log \,x. \cr\text{Describe the end behavior of each function. Then, write f(x) in terms of g(x).}")) → Problem
34. LaTeX((((((((" f(0.1) = " * h) * " (0.1)+ \log \, (0.1) = ") * h) * " 0.1 + (") * g(0.1)) * ") = ") * f(0.1))) → step3_{f1}
35. LaTeX((((((((" f(1) = " * h) * " (1)+ \log \, (1) = ") * h) * " 1 + (") * g(1)) * ") = ") * f(1))) → step3_{f2}
36. LaTeX((((((((" f(2) = " * h) * " (2)+ \log \, (2) = ") * h) * " 2 + (") * g(2)) * ") \approx ") * f(2))) → step3_{f3}
37. LaTeX((((((((" f(3) = " * h) * " (3)+ \log \, (3) = ") * h) * " 3 + (") * g(3)) * ") \approx ") * f(3))) → step3_{f4}
38. LaTeX((((((((" f(4) = " * h) * " (4)+ \log \, (4) = ") * h) * " 4 + (") * g(4)) * ") \approx ") * f(4))) → step3_{f5}
39. LaTeX(" \text{Solve f(x) for x = 0.1, 1, 2, 3, and 4.}") → step3_{f}
40. LaTeX(" \text{Solve g(x) for x = 0.1, 1, 2, 3, and 4.}") → step3_{g}
41. LaTeX(((((((((((" \text{For } f(x): \cr\ A=(0.1, " * f(0.1)) * ") \cr\ B=(1, ") * f(1)) * ") \cr\ C=(2, ") * f(2)) * ") \cr\ D=(3,") * f(3)) * ") \cr\ E=(4,") * f(4)) * ") ")) → step4_1
42. LaTeX(((((((((((" \text{For } g(x): \cr\ F=(0.1, " * g(0.1)) * ") \cr\ G=(1, ") * g(1)) * ") \cr\ H=(2, ") * g(2)) * ") \cr\ I=(3,") * g(3)) * ") \cr\ J=(4,") * g(4)) * ") ")) → step4_2
43. LaTeX((("\text{For f(x): The function values approach negative infinity as x approaches 0.} \cr\text{As x approaches 1, the function values converge on }" * d) * ".")) → step7_2
44. LaTeX((("\text{For f(x): The function values approach positive infinity as x increases } \cr\text{from x=1 to positive infinity. As x approaches 1 from the right, } \cr\text{the function values converge on }" * d) * ".")) → step8_2
45. LaTeX((((("g(1) = " * g(1)) * ": (1, ") * g(1)) * ")")) → step1_{22}
46. LaTeX((((("f(1) = " * d) * " : (1, ") * d) * ")")) → step1_{21}
47. LaTeX("\text{Write and plot the additional points. Make sure to keep the points with the correct function.}") → step4