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Evaluate f(x)=4x²-3 over a discrete domain and find its range

Evaluate f(x)=4x²-3 over a discrete domain and find its range
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2015-09-02 04:12
FunctionSequenceGraphing
In collections CCSS Support Math II Unit 2
This example demonstrates how to evaluate the quadratic function f(x)=4x²-3 over the discrete domain {0,1,2,3} and determine the corresponding range. The core mathematical concept is the relationship between a function's domain and range, specifically how a finite set of input values maps to output values for a quadratic expression. The construction uses GeoGebra commands such as Sequence to generate the domain points, LaTeX to display the function formula, Segment or Vector to visually connect inputs to outputs, and interactive elements like Next/Prev buttons or sliders to step through the evaluation process. This resource is suitable for middle or high school students learning basic function concepts, including function notation, tabular representation, and computing ranges for discrete domains. It also serves as an effective interactive classroom demonstration for reinforcing the idea of function mapping.

GeoGebra Commands & Parameters

  1. Segment(A_0, B_0) → Scaleline
  2. Sequence(Segment[A_0 + (i, 0), A_0 + (i, (-0.5))], i, 0, 10) → list1
  3. Sequence(Segment[A_0 + (i - 0.5, 0), A_0 + (i - 0.5, (-0.4))], i, 1, 10) → list2
  4. Sequence(Sequence[Segment[A_0 + ((0.5 * i) + (0.1 * j), 0), A_0 + ((0.5 * i) + (0.1 * j), (-0.3))], j, 1, 4], i, 0, 19) → list3
  5. LaTeX("\text{0}") → Scalevalue0
  6. LaTeX("\text{1}") → Scalevalue1
  7. LaTeX("\text{2}") → Scalevalue2
  8. LaTeX("\text{3}") → Scalevalue3
  9. LaTeX("\text{4}") → Scalevalue4
  10. LaTeX("\text{5}") → Scalevalue5
  11. LaTeX("\text{6}") → Scalevalue6
  12. LaTeX("\text{7}") → Scalevalue7
  13. LaTeX("\text{8}") → Scalevalue8
  14. LaTeX("\text{9}") → Scalevalue9
  15. LaTeX("\text{10}") → Scalevalue10
  16. Segment(D_0, (x(D_0), 0)) → j
  17. Vector(C_0, D_0) → u
  18. LaTeX("\text{Evaluate the function for each of the domain values.") → text4
  19. LaTeX("\text{Given function}") → text11
  20. LaTeX("\text{Simplify.}") → text15
  21. LaTeX((((("\text{ \textcolor{#000099}{f(" * val0) * ") = ") * sub1) * "}}")) → text14
  22. LaTeX("\text{Given function}") → text11_1
  23. LaTeX("\text{Simplify.}") → text15_1
  24. LaTeX("\text{Given function}") → text11_2
  25. LaTeX("\text{Simplify.}") → text15_2
  26. LaTeX("\text{Given function}") → text11_3
  27. LaTeX("\text{Simplify.}") → text15_3
  28. LaTeX((((("\text{ \textcolor{#000099}{f(" * val2) * ") = ") * sub3) * "}}")) → text14_2
  29. LaTeX((((("\text{ \textcolor{#000099}{f(" * val3) * ") = ") * sub4) * "}}")) → text14_3
  30. LaTeX("\text{Determine the range of the function for the given domain.}") → text17
  31. LaTeX("\text{Therefore, the range of the function over the given domain is}") → text21
  32. Segment((0, 0), (0, 1.6)) → i
  33. Segment(F_0, (x(F_0), 0)) → k
  34. Vector(E_0, F_0) → v
  35. LaTeX("\text{.Determine the range for this domain.}") → text3
  36. LaTeX((((("\text{ \textcolor{#000099}{f(" * val1) * ") = ") * sub2) * "}}")) → text14_1
  37. Factors(24) → fact
  38. LaTeX((("\text{Constant=" * valuec) * "}")) → text23
  39. LaTeX((((("\text{Evaluate \textcolor{#000099}{f(x) = " * valuex) * "x² \;– \;") * valuec) * "} over the domain}")) → text1
  40. LaTeX((("\text{Evaluate \textcolor{#000099}{f(" * val0) * ")}.}")) → text9
  41. LaTeX((("\text{Evaluate \textcolor{#000099}{f(" * val1) * ")}.}")) → text9_1
  42. LaTeX((("\text{Evaluate \textcolor{#000099}{f(" * val2) * ")}.}")) → text9_2
  43. LaTeX((("\text{Substitute \textcolor{#000099}{" * val2) * "} for \textcolor{#000099}{x}.}")) → text13_2
  44. LaTeX((("\text{Evaluate \textcolor{#000099}{f(" * val3) * ")}.}")) → text9_3
  45. LaTeX((("\text{Substitute \textcolor{#000099}{" * val3) * "} for \textcolor{#000099}{x}.}")) → text13_3
  46. LaTeX("\text{values we substituted for \textcolor{#000099}{x}.}") → text19
  47. LaTeX((("\text{Substitute \textcolor{#000099}{" * val1) * "} for \textcolor{#000099}{x}.}")) → text13_1
  48. LaTeX("\text{\mathbf{Quadratic \;function} a function that can be written in the form \textcolor{#000099}{f(x) = ax² + bx + c}, }") → text24
  49. LaTeX("\text{where \textcolor{#000099}{a ≠ 0}. The graph of any quadratic function is a parabola.}") → text25
  50. LaTeX("\text{ function.}") → text27
  51. LaTeX("\text{ function.}") → text29
  52. LaTeX((("\text{Substitute \textcolor{#000099}{" * val0) * "} for \textcolor{#000099}{x}.}")) → text13
  53. LaTeX((("\text{Coefficient of x=" * valuex) * "}")) → text2
  54. LaTeX("\text{ \mathbf{Domain} the set of all inputs of a function; the set of \textcolor{#000099}{x}-values that are valid for the}") → text26
  55. LaTeX("\text{\mathbf{Range} the set of all outputs of a function; the set of \textcolor{#000099}{y}-values that are valid for the }") → text28
  56. LaTeX((("\text{\textcolor{#000099}{" * sub1) * "}.}")) → text30
  57. LaTeX("\text{The outputs from the previous step were}") → text20
  58. LaTeX((((("\text{To evaluate the function \textcolor{#000099}{f(x) = " * valuex) * "x² -") * valuec) * "} over the domain")) → text5
  59. LaTeX((((("\text{into \textcolor{#000099}{f(x) = " * valuex) * "x²-") * valuec) * "}.")) → text8
  60. LaTeX((((("\text{ \textcolor{#000099}{f(x) = " * valuex) * "x²-") * valuec) * "}}")) → text10
  61. LaTeX((((((((("\text{ \textcolor{#000099}{f(" * val0) * ") = ") * valuex) * "(") * val0) * ")²-") * valuec) * "}}")) → text12
  62. LaTeX((((("\text{ \textcolor{#000099}{f(x) = " * valuex) * "x²-") * valuec) * "}}")) → text10_1
  63. LaTeX((((((((("\text{ \textcolor{#000099}{f(" * val1) * ") = ") * valuex) * "(") * val1) * ")² -") * valuec) * "}}")) → text12_1
  64. LaTeX((((("\text{ \textcolor{#000099}{f(x) = " * valuex) * "x²-") * valuec) * "}}")) → text10_2
  65. LaTeX((((((((("\text{ \textcolor{#000099}{f(" * val2) * ") = ") * valuex) * "(") * val2) * ")²-") * valuec) * "}}")) → text12_2
  66. LaTeX((((("\text{ \textcolor{#000099}{f(x) = " * valuex) * "x²-") * valuec) * "}}")) → text10_3
  67. LaTeX((((((((("\text{ \textcolor{#000099}{f(" * val3) * ") = ") * valuex) * "(") * val3) * ")²-") * valuec) * "}}")) → text12_3
  68. LaTeX("\text{, substitute the values from the domain") → text7
  69. LaTeX((((("\text{When \textcolor{#000099}{" * val0) * "} is substituted for \textcolor{#000099}{x},the value of \textcolor{#000099}{f(") * val0) * ")} is}")) → text16
  70. LaTeX((((((("\text{When \textcolor{#000099}{" * val1) * "} is substituted for \textcolor{#000099}{x},the value of \textcolor{#000099}{f(") * val1) * ")} is \textcolor{#000099}{") * sub2) * "}.}")) → text16_1
  71. LaTeX((((((("\text{When \textcolor{#000099}{" * val2) * "} is substituted for \textcolor{#000099}{x},the value of \textcolor{#000099}{f(") * val2) * ")} is \textcolor{#000099}{") * sub3) * "}.}")) → text16_2
  72. LaTeX((((((("\text{When \textcolor{#000099}{" * val3) * "} is substituted for \textcolor{#000099}{x},the value of \textcolor{#000099}{f(") * val3) * ")} is \textcolor{#000099}{") * sub4) * "}.}")) → text16_3
  73. LaTeX((((((((("\text{\textcolor{#000099}{" * sub1) * ", ") * sub2) * ",") * sub3) * ", and \; ") * sub4) * "}.}")) → text31
  74. LaTeX("\text{Collect the set of outputs for the inputs.Recall that the outputs are the \textcolor{#000099}{f(x)} values,and the inputs are the}") → text18

AI Drawing Prompt Reference

Draw the graph of the quadratic function f(x)=4x²-3 over the discrete domain {0,1,2,3}, showing the mapping from input values to output values. Start by marking the points 0 to 10 on the x-axis and labeling them. Then, for each input value, draw line segments connecting the input points to their corresponding function output points to form the function graph. Finally, label the function expression and highlight the set of function values, which is the range of the function over the given domain.
The prompt above was compiled from the original drawing commands and is for reference only.