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Exponential Functions and their Graphs - Problem 2

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

A graph of an exponential function when our base is between 0 and 1. So we have exponential base 1/2. The graph of this is going to look pretty similar for any value between 0 and 1, we're just going to look at 1/2 for a reference graph. So easiest way to start graphing this is just by plugging in some points.

First thing I want to plug in is -2, so if I have 1/2 to the -2 remember negative exponents flip this, so this is the same thing as 2², we end up with 4. 1/2 to the -1, the -1 again flips, we just end up with 2 anything to the 0 power is 1, anything to the first power stays the same, so this is going to be 1/2 and 1/2 squared is going to give us 1/4.

So we have some key points, let's graph those out. -2, 4 so we go back 2 up 4 1, 2 0, 0 1, 1/2 and then 2, 1/4, 1/4 is a fairly small positive number just plug in close to your x axis. Connect that to see what roughly this graph looks like and we get something like this. So let's look at domain and range.

Remember domain is the values for x and do we have any restrictions on what we can put in for x? We can put in 1/2 to a huge number that's okay 1/2 is a really small number so that's okay, so we really have everything for x. So our domain is going to be all reals.

Our range is our y values, so if you look at this what restrictions do we have on y? Do we have ever end up with ys down here? No so that tells us we have a restricted range and everything is positive so we go from zero to infinity, 0 is actually a horizontal asymptote, we can plug in really big numbers for x, 1/2 to the million is a a really, really small number, so we really are going to get close to 0, but we're never going to touch it, so we don't include 0 for our range.

So like I said this is the basic graph for any exponential function with a base between 0 and 1. If you want to do the comparison, we did 1/10 to the x, if x was -1, we would end up with 10, so we're going to put the point up here, if x was 1 we would end with 1/10, x is 0 it's still go to this point 0, 1 so what we'd end up with is a graph that looks roughly the same just closer to the x axis and closer to the y axis, so it's just sort of squeezed out, but the general shape is going to be very, very similar.

So anytime we have a graph with it's base between 0 and 1, we're going to end up with a decreasing curve, the graph is always going down of varying steepness.

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