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Exponential Functions and their Graphs  Problem 3
Carl Horowitz
Carl Horowitz
University of Michigan
Runs his own tutoring company
Carl taught upperlevel math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Graphing a exponential function with transformations; so for this particular example, what we're doing is we're dealing with a exponential graph that has some movement going up.
So the first thing we're going to do is look at this and realize what the base function is of this graph and hopefully you can see that this is just going to be similar to 10 to the x, we're just moving it side to side up and down from 10 to the x. 10 to the x, we need to think back and think about the graph, our base is greater than 1, so therefore we know that it is a increasing exponential that passes through the point 0,1 and has a horizontal asymptote at y equals 0 or the x axis and what we're doing is basically moving this graph around.
What the 2 does, the x minus 2 is it moves it side to side two units, so we take and it depends on how many points your teacher is going to want you to do, I tend to just focus on the asymptote and the point 0,1. They may want you to follow a couple more, it's the same process for every single point.
So what we're doing is moving everything over two units to the positive, so what that does is move everything over to the point 0,1 is now 2,1 and we have one unit, two units it gets over to and what the +1 does is move everything up 1, so the most important part of that is our horizontal asymptote was at 0 got moved up.
So what I would do is I draw in my new asymptote it got moved up 1 and then take my key point which got moved 2 over, it was at 0, 1. It gets moved over two units and then up 1, so that's my key point now and then the exponential goes through it. Like I said if you want to follow this with another point, we know that the point 1, 10 is on this curve, you can do the same thing over 2 up 1 to plot two more points, it all depends on how precise you want to be. Like I said I'm just going to do one, I know this is the point, I know my horizontal asymptote, so the graph is going to look something like this.
So laws of transformation told you for a power function as well up and down, left and right just make sure you go the right direction and plot your new asymptote and you have the transformation of an exponential.
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Carl Horowitz
B.S. in Mathematics University of Michigan
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Sample Problems (8)
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f(x) = 10^{x − 2} + 1 
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