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Exponential Functions and their Graphs  Problem 1
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!
There are two different kinds of exponential graphs, there's one where our base number remember the number in the bottom is greater than one, and there's another graph where the base is between 0 and 1. So for this we're going to start with the graph that is a base larger than 1.
Now I'm just going to pick 2 as our sort of base graph, the other ones are very similar, we'll talk a little later about the settled differences. So the graph I just want to fill in a little table. Starting with x is equals to 2, so what that tells us is our f(x) is going to be 2 to the 2, remember that a negative exponent is going to flip our fraction, so this is really 1 over 2² or 1/4. 2 to the negative first again the negative flips it, just going to end up with onehalf, 2 to the 0, anything to the zero power is going to be 1, so this will give us 1, 1 to the first is 2, 2² is 4.
So we have some key points, we now go to plot these points and see what this graph looks like. X is 2 we have 1/4, so our number pretty close to 0, so we'll just throw that in right there. 1, 1/2 still pretty small, but a little bigger 0,1, 1,2 and lastly 2 and I need one more thick mark 2,4.
Connecting the dots what we end up with is a graph that looks something like this. Think about what happens if we plug in a large value for x, the graph is just going to go up and if we put in a large negative number say like 1000, we're going to have 1 over 2 to the 1/1000 which is a very small number but still a positive number, so what actually ends up happening here is we have a horizontal asymptote remember when the graph is really close, but doesn't actually touch at y equals 0 and it goes up.
So for this particular graph let's talk about the domain, domain corresponds to the x value, are there any values of x we can plug in here? No, we can plug in everything. So our domain here is going to be all reals or negative infinity to infinity if you like interval notation and our range is going to be our y values.
We don't have any y values down here, so it's just going to be from 0 to infinity, we have a horizontal asymptote is 0, so the graph that's really very close, but never touches, so we actually don't include 0 in this case.
So this is the general graph for 2 to the x. The base in this case doesn't change too much of the shape basically what's going to happen if I said 10 to the x, we're still going to have the point 0,1 that's going to be a very key point for us because any value, any base value will always give us 1, but now if we plugged in 1, we're going to end up with the point up here, if we plugged in 1, we end up with a point closer to 0. So what the graph looks like is basically similar graph but closer to the axis, both the x axis here and the y axis here. So anytime your base gets bigger, the graph just sort if gets squeezed closer to the axis, but in terms of the shape, they're pretty similar all things to consider.
So anytime you have a exponential function where your base is larger than 1, you're going to have a graph that looks like this close to the point 0,1 and it's what I call increasing the graph is always going to be going up.
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Carl Horowitz
B.S. in Mathematics University of Michigan
He knows how to make difficult math concepts easy for everyone to understand. He speaks at a steady pace and his stepbystep explanations are easy to follow.
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