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GeoGebraGeoGebra Classroom

Exponential or Logistic Function that passes through two given points

Visualize the exponential function that passes through two points, which may be dragged within the x-y plane. The points will snap to the grid points (with integer x- and y-values). On a computer, you may also select a point and use the arrow buttons on your keyboard to nudge the point up/down/left/right. Study the resulting equation. To individually rescale the x- or y-axis, pinch (touchscreen) or shift-click-drag (keyboard/mouse) on the axis. The "exponential/logistic" button toggles between those two types of functions. When set to "logistic," the carrying-capacity line may be dragged to any desired location above the two labeled points.
For the exponential function f(x) = a b^x, what is the significance of the value of a in the graph? When the function is increasing from left to right, what do you notice is always true about the equation? How about when the function is decreasing? What needs to be true about the equation in order to make the graph pass through the third and fourth quadrants?