College Algebra: Graphing Exponential Functions

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Okay, so let's try to graph some slightly more exotic exponential functions to get a feel for how the exponential function works and manipulating that object. So, let's begin by graphing f(x) = 2^x+4. Now, how would I do that? Well, actually, there's two ways of thinking about this. Let me talk about both ways. One way is to notice that, in fact, I see this is just basically 2^x, but instead of x I have x + 4, so that's a shift in the x. Which way? Add to x, go west. So it should be a shift this way. So what I could do is the following: I could take the picture of just 2^x. Let me draw that in slightly. I'll dot that in. This is just for us. This is actually not part of the answer at all. This is a 2^x. Now what I do is take that picture and I'm going to shift it four units over this way. This point is 1, so where would that go. If I put in x = 0 here I'd see 2^4, so I'd see something quite large. So all I did there was take this picture and shift it four units this way. It's sort of an interesting optical illusion. It looks like it's sort of wide here, and then it's less wide here. That's because you're looking at the height. Look at the distance this way. They should always be the same. It's always four units. Even over here, it's four units. I took this picture and just slid it over four units.
So that's one way of seeing the graph of this, and that's correct, and if you like it, perfect. Another way, actually, would be to use some laws of exponents here. Let me just point this out as a little side note and to realize that I have 2 to a power that's a sum, I could just write that as 2^x(2^4). Because remember, what you do, when you have the same base you add exponents. And remember this little slogan with exponents--when in doubt, write it out. Think about this. I have 2 multiplied by itself x times, and then I have 2 multiplied by itself 4 times. How many 2's do I have? X + 4. So this is really okay. Well then you see this is just 2^4, which is 16, times 2^x. So then I could think about the graph in the following way. I take this graph and just multiply it by 16. And what does that do? It takes this point, 1, and moves it up to 16, and then it takes this thing and sort of bends it up and bends it over. So then this is sort of now a shrinking, basically, or expanding by multiplying this by 16.
So that's another way of thinking about it, which is sort of worth sort of noting, but I think the easy way to think about it is just think about it as a shift in the x four units this way.
Now, what I want to take a look at now--I want to write the next question here because I want you to see the difference. The next function I want to look at, I'll call it g, is going to be 2^x + 4, and notice that is dramatically different from this. This was 2^x+4. This is 2^x + 4. So I take all this stuff and add 4 to it. So what's that going to look like? There, that's going to be a shifting of the y's because I just take the answer I got with the x's and add 4 to everything. Whereas, this is a shifting of the x's. I've got to change the x by 4 and then figure out what y is. So here I start off with this picture, and what do I do to it? I just raise it four units up--one, two, three, four. So, in fact, that picture would look dramatically different. I would just take this picture and move it up four units. So that's a difference between this kind of thing and this. In fact, let me actually graph this right on the same axes. I move this up four units--this is by no means drawn to scale--I apologize. And this would be 2^x + 4, whereas the purple thing, I remind you, was 2^x+4. So to get to the green, what you did was you added four this way--everywhere. So took this picture and just raised it four units up. And this picture, I moved it four units this way. So make sure you see the difference between these two functions; they're dramatically different.
If I take an exponential and add 4 to the whole thing, that's going to be a shift upward. If I take 2^x+4 then that's going to be a shift to the left by four.
One final note. What happens if I look at something like this, like -3^x? Well, this is a little tricky. Because, for example, if I put in -3^2, that's okay, I know what that means. That's going to be 9. What if I look at -3^1/2? That's the square root of -3 and that's not a real number. And as you can see, any time I have a fractional exponent that has an even denominator, that will be some sort of root that I couldn't take. So, in fact, this is just complete garbage as a function--so junky function. So, in fact, we will never, ever look at functions that have a negative thing in here. And why? Because it wouldn't be defined in a lot of places--it would be a very holey function--it would be a religious function-- very, very holey, because there would be all these points where you just wouldn't have things there, so we don't even look at that. Never, never, never.
Now, for those of you who might say, "Wait a minute, isn't this just equal to this?" Well, those people are sadly, sadly, sadly mistaken. These are not equal at all. This is -3^x, this is 3^-x. This we will never graph; this is false. This we will never graph, this we can always graph, because that's just 1/3^x. So the only kind of basis we're allowed to have, that it makes sense to have, are positive bases. They can be less than 1, they can be bigger than 1, but they can't be negative.
All right. I'm glad I got that out of my system.
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Graphing Exponential Functions: More Examples Page [1 of 2]