Unit
Inverse, Exponential and Logarithmic Functions
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!
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.