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.[br]Study the resulting equation.[br][br]To individually rescale the x- or y-axis, pinch (touchscreen) or shift-click-drag (keyboard/mouse) on the axis.[br][br]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?[br]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?[br]What needs to be true about the equation in order to make the graph pass through the third and fourth quadrants?