General Exponentials Worksheet

In this worksheet, we will examine the behavior of different general exponential functions by comparing their graphs. HOW TO USE THIS WORKSHEET: As you read, graph the curves on your own graphing calculator or software. Try changing: the function (especially the base), the x_min, and the x_max. Ask: How did the range (the maximum and minimum y-values) change? How did the shape of the curve change? What does that tell me about the function? This is the best way to get a correct gut feeling for general exponential functions! HOW TO READ THIS WORKSHEET: This was written using Sage (a free computer algebra software system). To plot the function y=f(x) on the domain (x_min, x_max), we will write plot(f(x),x_min,x_max) To plot the graphs of f(x) and g(x) on the same axes, we will write plot(f(x),x_min,x_max,linestyle='dashed')+plot(g(x),x_min,x_max,linestyle='solid') We can tell the two functions apart because the graph of y=f(x) will be a dashed curve, and the graph of y=g(x) will be a solid curve. 
       
PART I. To begin, let's look at exponential functions y=a^x where the base a>1. Notice that as the base of the exponential increases, the function grows faster (as we move from smaller x-axis values to larger x-axis values). 
       
plot(2^x,0,10,linestyle='solid')+plot(e^x,0,10,linestyle='dashed')+plot(3^x,0,10,linestyle='dotted') 
       
It is clear by looking at the graph that e^x and 3^x both increase quickly toward infinity. It is less clear from the above graph what the function 2^x does. In fact, 2^x also increases quickly toward infinity. To see this, lets graph the slowest growing of the three (y=2^x) on a slightly longer interval of the x-axis: 
       
plot(2^x,0,15,linestyle='solid') 
       
Notice that the curve y=2^x does grow toward infinity very quickly, it just takes longer for the curve to really take off. 
       
PART II. Now, let's look at what happens to the graph of y=a^x in three cases: a>1, a=1, and 0<a<1 
       
plot(1.001^x,-250,1000,linestyle='dashed')+plot(1^x,-250,1000,linestyle='solid')+plot(.999^x,-250,1000,linestyle='dotted') 
       
Note that as x goes to infinity: a^x goes to infinity when a>1, a^x is constant when a=1, and a^x goes to 0 when 0<a<1 
       
PART III. Finally, let's look at exponential functions y=a^x where the base 0<a<1. Notice that when the base of the exponential is bigger, the function drops off to 0 faster (as we move from small x-values to large x-values). 
       
plot(.1^x,-.5,10,linestyle='dotted')+plot(.5^x,-.5,10,linestyle='dashed')+plot(.9^x,-.5,10,linestyle='solid') 
       
In the above graph, it is easy to see that .1^x and .5^x both approach 0, first quickly and then evening off. By graphing y=.9^x on a larger interval, we can see that .9^x also approaches 0, first quickly and then evening off. 
       
plot(.9^x,-5,50,linestyle='solid') 
       
 
       
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