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Graphs of y=a^x and y=(1/a)^x

Graphs of y=a^x and y=(1/a)^x
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2023-07-04 12:15
FunctionTransformationSlider AnimationGraphing
① This interactive GeoGebra example dynamically displays the graphs of the exponential functions y=a^x and y=(1/a)^x, allowing users to adjust the base a via a slider and observe how the two curves are mirror images about the y-axis. ② The core mathematical concepts include the monotonicity of exponential functions, the symmetry between reciprocal-base exponentials, and the identity (1/a)^x = a^(-x). ③ Construction details: The example uses a slider to control parameter a, defines the functions f(x)=a^x and g(x)=(1/a)^x, and employs the Mirror command to verify symmetry. It also incorporates commands such as Element, Segment, Textfield, TableText, and If, along with multiple buttons and an input box for interactive control. Objects include points A, B, C, D and their reflected counterparts, used to mark key coordinates. ④ This resource is suitable for high school mathematics lessons on exponential functions, helping students visualize how the base affects the graph and understand the concept of reflection symmetry.

GeoGebra Commands & Parameters

  1. Element(l7, n) → a
  2. Segment(C, D) → s
  3. Segment(Rotate[I, -90°, O], Rotate[I, 90°, O]) → k
  4. Mirror(I, O) → I'
  5. Textfield(f) → InputBox2
  6. TableText({tx, text3}, {tx3, If[o1, t3, ""]}, {tx2, If[o1, t2, ""]}, {tx1, If[o1, t1, ""]}, {tx0, If[o1, t0, ""]}, {-1, If[o1, t1', ""]}, {-2, If[o1, t2', ""]}, {-3, If[o1, t3', ""]}, "|_") → text16
  7. If(a ≠ 1, (log(a, y(Corner[3])), y(Corner[3])), (x(Corner[3]) - 2, 1)) → F
  8. TableText({tx, text3_{4}}, {tx3, If[o1', t3', ""]}, {tx2, If[o1', t2', ""]}, {tx1, If[o1', t1', ""]}, {tx0, If[o1', t0, ""]}, {-1, If[o1', t1, ""]}, {-2, If[o1', t2, ""]}, {-3, If[o1', t3, ""]}, "|_") → text16_{1}

AI Drawing Prompt Reference

Plot two exponential function graphs y=a^x and y=(1/a)^x where a is an adjustable parameter. Use a slider to change the value of a and observe how the two curves change. Specifically, demonstrate that these two curves are symmetric about the y-axis, showing that (1/a)^x equals a to the power of negative x. Mark some key points with coordinates to verify the symmetry. The visualization should dynamically reflect the monotonicity of exponential functions and how the base affects the graph shape.
The prompt above was compiled from the original drawing commands and is for reference only.