The graph of the exponential function has a very interesting property. If you draw a vertical line (green in the graph to the right) from a point on the graph down to the point on the x-axis, then draw another line 1 unit to the left (red), to the point , and then finally complete the triangle by drawing the line through and (magenta), this final line will just touch the graph of at this latter point without passing through the graph; that is, this line is tangent to the graph of at the point . This property (with the base of the triangle having length 1) and the specification that the function has value 1 at completely and uniquely determine the exponential function .
The slope of this (magenta) line tells you how fast the function is growing. So this property of the exponential function can be summarized this way: At every point, how big the exponential function is and how fast it is growing are the same.