The graph of an exponential function of the form y = f(x) = ax passes through the points and . the graph lies the x-axis.

  the  exponential function f with base a  is defined by

f(x) = ax

the graphs of all exponential functions have these characteristics. they all contain the point (0, 1), because a0  = 1. the x-axis is always an asymptote. they are decreasing if 0 < a < 1, and increasing if 1 < a. the only thing that varies is the steepness.

given a point in the upper half plane, i.e. above the x-axis, that is not on the y-axis, then there is a unique exponential function whose graph contains that point. ( note that if the point lies on the horizontal line y = 1, then it lies on the graph of f(x) = 1x. this function is constant and hence not very interesting! ) the applet below illustrates this fact. initially the variable point is (1, 2), so the exponential function determined is f(x) = 2x, and this is indicated in the text box below the graph. as you drag the point, the function and graph are updated. note that unlike earlier applets, here you cannot drag the point (0,1), because the graphs of all exponential functions contain that point!

exponential function determined by a point.

drag the point (1, 2) to update.

in general, equations involving exponential functions are easier to graph than polynomial equations because the exponential functions themselves are easy to graph.

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step-by-step explanation: