WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to have our function petting zoo.
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An important step in learning mathematics is becoming accustomed to how various functions behave and what their graphs look like.
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While long-term experience is the surest way to familiarize yourself with functions,
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and you have probably already gotten some of that experience (when you see x² show up,
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you get a sense that you are going to see a parabola), this lesson is here to give you a head start on developing your function intuition.
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Maybe we will see some that you haven't really talked about before; I am actually sure that you will see a couple that you haven't really talked about before.
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And you will get the chance to develop it; we will talk about various properties, and that sort of thing.
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We can just get a sense of "when I see functions in this type, I know to expect a graph like this; I know to expect certain kinds of behavior."
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So, we are going to graph various fundamental functions; and we will talk briefly about their key points.
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Don't worry about memorizing this information; it is not here because you are going to have to know it because you are going to be drilled on this.
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It is never going to be tested directly, probably (I suppose there might be a couple of teachers out there who would test directly on it).
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But really, what this is about is exposing you to these so that you are ready to understand things better down the road.
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And other teachers or books might call a similar lesson a function library or parent functions.
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I personally think it is kind of fun to call this a function petting zoo,
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because really, we are just going out and meeting a bunch of different functions.
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We are getting the chance to interact and play with one function, another function, another function...just for a little bit at a time.
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The point of any of these names, though, is the same thing.
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Function library, parent functions, function petting zoo--all of these things are just to introduce or review
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a wide variety of fundamental functions, and then also to talk about their characteristics and their graphs,
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to get you the chance to develop that function intuition, so that when you see a function,
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you know what to expect out of it, even if you haven't worked with that precise one before.
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All right, before we get into this, though, don't forget that axes matter.
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Remember from our very first lesson on graphs: how the axes are set up has a huge impact on what the graph looks like.
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Whenever you look at a graph, pay attention to how the axes are set up.
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For consistency, and to help us see how various functions behave differently, all the graphs we are about to see will be on the same axes.
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So, the graphs we are about to see are going to be on the same axes; we will see them on the x, horizontal, -10 to 10, and the vertical -10 to 10.
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Notice that these are square axes; the length of the horizontal is the same as the vertical length.
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So, it is going to not give any sort of weird curving to it; we won't be squishing the picture from its "natural size" or its "natural shape."
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These axes will help us see how each one of these compares to the other one.
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We will have a general template to understand how the shape of this one is different from the shape of this one, is different from the shape of this one.
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All right, let's get going: the very first one is the **constant function**, f(x) = k, where k is a constant--just some constant number.
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So, things to notice about this: for example, in this one, we have that k is a little bit more than 3.
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The input doesn't matter; no matter what we put in, 2 gets mapped to the same thing as 10, gets mapped to the same thing as -8.
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Whatever we put in, it all gets mapped to k.
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Output is thus always the same; whatever we are putting in, once again, it doesn't matter what we put in; it always gives the same thing out.
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And then finally, what does it put out? It always outputs a horizontal line at k.
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We plug in 5; it gives out k; we plug in 20; it gives out k; we plug in -47; it gives out k.
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When we look at all of that together, graphically we are seeing a horizontal line at height k; great.
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The identity function, f(x) = x: it is called the **identity function** because whatever we plug in is what we get out of it.
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The input is the same as output; if we plug in 7, we get out 7; if we plug in -47, we get out -47.
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So, what we get out of this is this nice, straight line that just cuts perfectly between the x-axis and the y-axis.
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We see a slope of m = 1 because of this: slope equals 1, because it has to cut evenly between them.
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Otherwise, it wouldn't be giving an identity, where the same thing that goes in...if 6 goes in, then 6 has to come out.
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Whatever goes in is what comes out--all right, that is the identity function.
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The **square function**, f(x) = x²: the first thing to notice is that, as it goes to the right...
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notice how it starts to go up faster and faster; and as it goes to the left, it goes up faster and faster.
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The height increase, how fast it is going up, its rate of change--height increase speeds up farther out.
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The farther we get away from the middle, the faster it is moving up.
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Height increase is speeding up farther out.
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The other important thing to point out is that, on the right, we are going up; and on the left, we are going up.
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On the square function, the ends go in the same direction; the ends of graph go in the same direction--they point in the same direction.
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This is an important idea about the square function: we can trust the fact that it will cup in the same direction.
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The right side goes off in the same way as the left side.
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The **cube function** is similar to the square function: its height increase speeds up, the farther we get out.
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In fact, its height increase is going to speed up even faster; when we get to 2 on x², we are only on an output of 4.
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But when we get to 2 on x³, we are at an output of 8, 2 times 2 times 2.
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So, its height increase speeds up, and it speeds up even faster than it does on x².
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The other thing to notice is that, if we were to continue the graph, one side goes up, and the other side goes down.
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The ends of the graph point in opposite directions; unlike in x², where they point in the same direction,
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the ends of the graph in x³ point in opposite directions.
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The ends of the graph when we are doing the cube function will point one way up and one way down--opposite directions.
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All right, the next one is the **square root function**: for the square root function, notice:
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the farther we go out, the slower it is increasing in height.
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To get to a 2 of height, it has to put in a 4; to get to a 3 in height, it has to put in a 9; to get to a 4 in height, it has to put in a 16.
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So, height increase slows down the farther we get out--height increase slows down.
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Also, notice the fact that there is nothing over here; there is nothing on the left side of the graph,
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because if we try to plug in a negative number, it is not in the domain; so negatives are not in the domain.
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So, in a way, √x only looks like half of a graph, because it doesn't really keep going.
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Part of it just stops, because if we try to plug in negative numbers, square root fails to work.
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There is no number that you can square (no real number, at least) that will make a negative number.
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A negative number, squared, gives you a positive number; a positive number, squared, gives you a positive number.
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So, there is no number that you can square that will give you a negative number, at least in the real numbers.
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So, the negatives are not in the domain of the square root function.
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The **reciprocal function**, 1/x, is called the reciprocal because the reciprocal of a number is just one over that number.
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So, in this one, height increase, as we get farther and farther out, slows down.
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Height increase slows down when we are far away from the y-axis.
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The farther we get from the y-axis, the slower the height increase becomes.
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And this makes sense, because at 2, we are at 1/2; at 4, we are at 1/4; at 6, we are at 1/6; so the height increase...
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well, I guess decrease--it increases as we go to the left; but the point is that the change in the height,
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"height change"--I will change that formally--height change slows down away from the y-axis.
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But as we get close to the y-axis, as we get closer and closer, height will "blow out."
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The height is going to "blow out": near 0, the function "blows out," and by that I mean that it blows out to either positive or negative infinity.
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And as we approach 0 from the right side, we go out to positive infinity; as we approach from the left side, we go out to negative infinity.
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However, there is one point that simply isn't allowed: 0 is not in domain--why? because 1/0 would be not allowed.
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We are not allowed to take dividing by 0; so 0 is not in domain.
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We can talk about what happens to 0.00000001, but we can't talk about what happens to 0 itself.
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So, 0 is not in the domain; but near 0, as we approach 0, it goes out to either positive or negative infinity.
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We will talk about this more when we talk about vertical asymptotes.
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All right, that is spelled kind of weirdly, just in case you are curious to look at it right now: asymptotes--a weird spelling, but pronounced "aa-sim-tohts."
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All right, the next one is the **absolute value function**.
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For this one, it is only going to output positives; why? because absolute value only gives out positive numbers.
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If you put in -3, it becomes positive 3; if you put in positive 3, it becomes positive 3.
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Whatever you put in, it is stripped of negative numbers; it has to come out as a positive number.
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Notice that it is also kind of similar to f(x) = x: when we have that normal f(x) = x, it would keep going like this.
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That is the thing to notice--that in a way, it flips its direction upon touching the x-axis.
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It flips direction when touching..."upon touching"--I will make it exactly correct...upon touching the x-axis; it flips the direction that it is going in.
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What I mean by that: let's imagine that we start over here at -10; we would get positive 10.
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We are moving this way; we now plug in -8; we get positive 8; we plug in -4; we get positive 4.
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We are going this way; we are going this way; we are going this way; we are going this way.
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All of a sudden, we hit a height of 0, and it bounces up; it flips to going this way right here.
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And this is because it only outputs positives; so when we would get below the x-axis, it has to bounce off,
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because otherwise we would be outputting a negative.
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So, it flips the direction that it is going in upon touching the x-axis.
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All right, now the **trigonometric functions**: these ones...there is a good chance you haven't seen these.
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Or if you have, they are pretty new to you at this point.
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The trigonometric functions--you will learn a lot about these in trigonometry.
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But right now, the main thing I want to point out is the fact that they repeat.
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Sine of x just does the same thing: see, this interval here is the same as this interval here, is the same as this chunk here, is the same as this chunk here.
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We are just seeing it repeat: cosine of x, this chunk to this chunk, is the same as this chunk to this chunk; we are just seeing it repeat.
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They are slightly different in how they are set up, but they are repeating functions.
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Let's also look at it in different axes, so that you can understand what is going on better.
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This is the classic -10 to 10 axes that we did for everything else; but how about some other ones?
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It turns out that it does its variance between 0 and 2π; don't worry about actually understanding what is going on here.
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I just want to have you see this stuff, so it is not totally new when you see it later.
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You will get it all very well in trigonometry.
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From 0 to 2π, we have one repetition; from -2π to 0, we have another repetition.
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It repeats itself every 2π; it also varies between 1 and -1, both for sine and cosine.
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It varies between 1 and -1, and it also has repetitions on a 2π basis, both for cosine and sine.
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We will see why this is the case when we actually study trigonometry.
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But the main thing to get out of this right now is that trigonometric functions are these repeating functions,
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that they are a way of being able to see the same thing happen after we go down far enough.
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We go down a certain amount, and it becomes the same thing; go down a certain amount, and it becomes the same thing; they repeat themselves.
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**Exponential and logarithmic functions**: all right, we are back to our -10 to 10, our standard axes that we were used to before.
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The thing to notice here: on exponential, it blows out really fast; look at how fast this manages to go out of our axis windows.
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We get outside of being able to see this out of our viewing window so quickly.
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By the time we have made it to 1, we are at 10; at 2, we are at 100; at 3, we are at 1,000.
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So, exponential functions (10^x is only one possible exponential function)--they are going to go out really, really quickly.
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They are going to just shoot up, having absolutely massive height.
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Blowing out probably isn't the perfect word, since we used "blowing out" for asymptotes.
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Let's instead say its height grows really fast.
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Over here at log<font size="-6">10</font>(x), look at how long it takes to even get up to 1; it takes us out to 10 to get it.
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We can see that the height is slowing down; its increase in height slows down.
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Its height increase slows massively; it grows really, really slowly.
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That is probably the main important thing to get out of these.
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Don't worry about actually understanding what is going on precisely right now.
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We will have an entire section on this, when we talk about exponential and logarithmic functions in detail for an entire section.
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But right now, I just want you to say, "Oh, exponential functions get really big really fast,
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and logarithmic functions stay pretty low for a very long time."
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Just to make a point of how long these sorts of things are, how slow and how fast they are,
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for logarithmic and exponential, respectively, let's look at it with new axes.
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So, for our exponential, we are going from only -3 to 3; by the time we have made it up to 3, we have hit 1,000.
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1,000 is how big we have managed to get.
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And it doesn't actually get to 0; it just looks like that--it is approaching it, because 10^-3 would be 1/10³, 1/1000.
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So, it is just really close to the x-axis.
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Once again, don't worry about understanding this perfectly right now; we will talk about it later.
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Logarithmic functions: they are going to take forever to even get to reasonable numbers.
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We have to get to 1,000 before we even manage to make it up to a height of 3.
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So, they grow really, really slowly; the height growth on logarithmic functions is really, really slow,
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and slows down massively the farther you get out; whereas exponential functions are really, really fast,
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and increase massively the farther you get out.
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All right, of course, the functions we see--that makes it for our petting zoo--we have completed our petting zoo;
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but when we see functions out in the wild, they are normally not going to end up being in their pure form.
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They are not going to be x² or √x.
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Normally, they have had other things put on them or added on them; they have been shifted, stretched, or flipped.
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They have been transformed in some way.
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Still, it helps to know the general shape for a function before transformation.
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Beyond these shiftings, stretchings, and flippings, we can still have a pretty good idea of what is going on.
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Other times, functions will be mixed with other functions.
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We might have things like f(x) = x³ + x²: that is not just one pure function--that is x³ and x².
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Or x² times √x--once again, that is not just one pure function; that is two functions mixed together.
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Or h(x) = |x³|: once again, that is not just one pure function; it is two things put together; it is absolute value and x³.
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Once again, though, it helps to know each function's general form before trying to figure out how they interact.
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If we understand how |x| works, and we understand how x³ works, it will make sense to us
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when we work on the graph of the absolute value of x³.
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We will have a better understanding of what is going on--what we are seeing.
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We will learn about both of the above ideas in the lessons Transformations of Functions--the transformations
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where we shift, stretch, and flip will be in the Transformations of Functions lesson; and we will talk
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about composite functions--we will talk about arithmetic combinations (these first two are arithmetic combinations--
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once again, don't worry if you don't know what these things mean precisely; we have lessons for that);
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and then finally, an actual composite function, where we combine the way that two functions are working.
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We will learn about these in much greater detail in those two respective sections.
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All right, great; we are ready for some examples.
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Here are two graphs without axes; they are the graphs of what functions?
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Well, this one is going up; it looks like a slope of 1; it is pretty stable; it is just increasing continuously.
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It doesn't bounce; so this is almost certainly f(x) = x; great.
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And this one: one side goes up; we see this sort of blowing out--it is approaching -∞ (that is what it is going down to here);
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and it is going up to positive ∞ up here; so what blew out and what got really, really slow in its height change?
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Oh, yes, it is the reciprocal function: f(x) = 1/x; great.
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Next, here are two functions that have been shifted, stretched, and/or flipped; what are the base functions making them up?
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For this first one, the red graph, we think, "Oh, well, it looks kind of like a parabolic arc, but a parabolic arc on its side."
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Oh, look over here: there is nothing over on the left side.
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So, if there is nothing on the left side, it has cut out the negative side; it has cut out the left side.
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What cut out one side? √x--we have that √x, and its height increase is still slowing.
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Height increase slows the farther it gets out; so that is both of the identifying marks of being a √x function.
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It is very different than the normal √x; but the basic function that is making this up is √x.
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It has been shifted; it has been stretched; but it is still √x.
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What about this one over here? Well, it is going down on both sides.
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And its height increase gets faster...well, "height increase" is incorrect; it is not height increase, because we are going down.
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But it is a height change, and that is the more fundamental idea about x².
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It isn't necessarily that it has to be going up, but that the change, the rate that it is going up
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or the rate that it is going down, possibly, is continuing to increase.
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Height change speeds up the farther we get out from the center; the center has been moved in this one.
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The height change will speed up; we get faster and faster changes in our height.
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So once again, this is f(x) = x²: it has been shifted, it has been flipped, and it has been stretched;
00:20:17.300 --> 00:20:25.500
but we can still recognize that that is a parabola; it must be, at heart, coming from that same idea as behind x²; great.
00:20:25.500 --> 00:20:33.300
Example 3: we want to graph f(x) = x³, g(x) = x², and h(x) = x, all on the same axes.
00:20:33.300 --> 00:20:39.000
And we also want to set up the axes such that we go from -10 to 10 on the horizontal,
00:20:39.000 --> 00:20:45.900
and none of the graphs are cut off vertically; so that means we can't lose any vertical information.
00:20:45.900 --> 00:20:48.100
To help us understand what is going on here, let's make a table.
00:20:48.100 --> 00:20:53.500
Now, we know what x³ looks like, what x² looks like, and what x looks like, in general.
00:20:53.500 --> 00:20:55.500
So, we can use that information to help us out.
00:20:55.500 --> 00:20:58.400
Let's see what the extreme values are and what the middle values are.
00:20:58.400 --> 00:21:08.700
So, here is our table: x, f(x)...here is x at -10 and x at positive 10.
00:21:08.700 --> 00:21:18.200
f(x)...let's make all three of them: x³, x², and x.
00:21:18.200 --> 00:21:25.000
Let's actually use colors for these: x² will be in red, and blue will be x; great.
00:21:25.000 --> 00:21:31.700
So, if we plug in -10, we are going to get -10 cubed, which is -1,000.
00:21:31.700 --> 00:21:41.700
x² will become 100, positive, because the negatives cancel out; and x will become -10.
00:21:41.700 --> 00:21:53.200
If we go to the other extreme at 10, that will be at positive 1,000; x² will also be at positive 100, and x will be at positive 10.
00:21:53.200 --> 00:21:55.500
So, let's try some other things: let's look at what happens in the middle.
00:21:55.500 --> 00:22:02.200
Well, in the middle, x³ is at 0; x² is at 0; and x is also at 0.
00:22:02.200 --> 00:22:10.600
Let's see what happens in the middle between the middles: if we plug in, say, -5 and 5:
00:22:10.600 --> 00:22:16.700
-5 times -5 is positive 25, times another -5...we get -125.
00:22:16.700 --> 00:22:25.800
Over here, we will have positive 125; the red...in this, we will get squaring, so we will be at positive 25, positive 25.
00:22:25.800 --> 00:22:29.600
And we will have -5 and positive 5.
00:22:29.600 --> 00:22:37.000
And finally, at 1 and -1, -1 will go to -1; positive 1 will go to positive 1 for x³.
00:22:37.000 --> 00:22:44.900
For x², -1 will get cancelled to a positive; a positive is still for the positive; and then, -1 and positive 1.
00:22:44.900 --> 00:22:49.500
The thing to notice here is that, when it is close to 0, they are not that different.
00:22:49.500 --> 00:22:53.400
But the farther we get from 0, the more their differences become apparent.
00:22:53.400 --> 00:23:01.700
So, let's make our graph; we will do it in blue, which hopefully won't be too confusing,
00:23:01.700 --> 00:23:08.000
even though blue is connected to x; so what is the maximum vertical height that we have to have?
00:23:08.000 --> 00:23:15.100
The maximum vertical height that we have to have is a huge 1,000, because we get up to -1,000 and positive 1,000.
00:23:15.100 --> 00:23:19.800
So, we have to be at -1,000 and positive 1,000 as the vertical extremes.
00:23:19.800 --> 00:23:29.900
Here is the middle at -500, positive 500; and then, we will actually go to an extreme
00:23:29.900 --> 00:23:33.600
of 10 and -10 horizontally, because that is what we were told to do.
00:23:33.600 --> 00:23:42.200
And here is our middle at 5, and middle at -5; and here is 1 and positive 1; great.
00:23:42.200 --> 00:23:46.300
All right, so at this point, let's graph x³; this is probably going to be one of the easiest to graph.
00:23:46.300 --> 00:23:51.200
At 0, it is at 0; at 1, it manages to be at 1, so it has barely even gotten off the x-axis.
00:23:51.200 --> 00:23:57.400
At 5, it is at 125, so it is a little bit over 1/5 of the way to the 500; so let's say it is around there.
00:23:57.400 --> 00:24:00.300
And then, 10 is going to be all the way up at 1,000.
00:24:00.300 --> 00:24:11.400
-1 to -1 is barely off of the x-axis; -5 will be at -125, so we are little past, but close to...probably a little too far down, actually...
00:24:11.400 --> 00:24:18.900
we are a little over 1/5 of the way to the 500; and then finally, at -10, we are all the way at a huge -1,000.
00:24:18.900 --> 00:24:36.300
Its curve is going to look like this; it manages to grow massively very, very quickly, as it gets farther and farther away from the y-axis--
00:24:36.300 --> 00:24:38.900
as it gets farther and farther from the center of its graph.
00:24:38.900 --> 00:24:47.200
What about x²? We get 1 and 1 in the same location; at 5, it is at a 25--a meager, tiny jump above;
00:24:47.200 --> 00:24:53.300
and then, at 10, it is at 100, so it is a little bit below the height of x³ at 5.
00:24:53.300 --> 00:25:07.800
The same on the reverse side: the parabola...when we look at it this far out, it is growing fast;
00:25:07.800 --> 00:25:14.900
it manages to get to 100 by the time it has gotten to 10; but it is still tiny, tiny;
00:25:14.900 --> 00:25:19.400
it looks so stout---it looks so short--compared to x³.
00:25:19.400 --> 00:25:23.600
Finally, we look at what happens to x, just the plain identity function.
00:25:23.600 --> 00:25:29.400
And at 10, it manages to only be 10 above; we are talking about there; at -10, it is only -10.
00:25:29.400 --> 00:25:43.200
So, we have it barely, barely growing off of that x-axis; it is barely breaking away, when compared to these giants like x² and x³.
00:25:43.200 --> 00:25:50.800
So, when we are really, really close to actually being near 0, when we are really close to the center of these graphs, they are very similar.
00:25:50.800 --> 00:25:56.200
But when we look at them on a larger scale, not even that big--just -10 to 10--suddenly the differences become apparent.
00:25:56.200 --> 00:26:03.100
They become massive, huge differences: the difference between x³ and x² at just 10 out is a difference of 900.
00:26:03.100 --> 00:26:07.700
There are huge differences between these; and they get even bigger, the farther we go out.
00:26:07.700 --> 00:26:17.900
The final example, Example 4: Think about functions of the form f(x) = x^n, where n is a positive integer--it is contained in the natural numbers.
00:26:17.900 --> 00:26:22.200
Then distinguish the difference between when n is odd and when n is even.
00:26:22.200 --> 00:26:30.800
Let's look at some examples for when n is odd: n is odd would be like x, or maybe x³ or x⁵.
00:26:30.800 --> 00:26:35.100
So, we know what x looks like; it is just like that.
00:26:35.100 --> 00:26:41.800
x³ blows out pretty quickly; x⁵ blows out even faster.
00:26:41.800 --> 00:26:48.000
By the time it makes it to 2, it is at 2, 4, 8, 16, 32 height.
00:26:48.000 --> 00:26:57.700
So, by the time x⁵ has an input of 2, it is getting an output of 32; so it blows out really, really fast.
00:26:57.700 --> 00:27:05.200
Let's compare some even ones; say we have x²...well, we know what that one looks like; it just looks like a parabola.
00:27:05.200 --> 00:27:10.700
x⁴...well, it is like a parabola, but it grows even faster.
00:27:10.700 --> 00:27:15.200
By the time we get to positive 2, instead of just being at 4, we are at 16.
00:27:15.200 --> 00:27:20.400
So, it grows really fast--not quite as fast as x⁵, but faster than x³.
00:27:20.400 --> 00:27:30.200
And as we go to the other side, since (-2)² is positive 4, (-2)⁴ is positive 4 times positive 4, or positive 16.
00:27:30.200 --> 00:27:33.800
So, as long as we are even, we are going to cancel out those negatives.
00:27:33.800 --> 00:27:39.100
And that is the idea that we see right here: look, when we are even, we go off in the same direction.
00:27:39.100 --> 00:27:43.500
If we were to do this for x⁶, it would be the same thing, but growing even faster.
00:27:43.500 --> 00:27:52.800
We would just be growing even faster; so what we are seeing here is that, when n is odd,
00:27:52.800 --> 00:27:59.500
that means that the ends of the graph go in opposite directions.
00:27:59.500 --> 00:28:06.000
So, the ends of the graph go in opposite directions if n is odd.
00:28:06.000 --> 00:28:16.300
But if n is even, the ends go in the same direction.
00:28:16.300 --> 00:28:25.900
That is the major difference between these; in many ways, they are very similar--the higher the n, the faster we have this growth rate.
00:28:25.900 --> 00:28:31.400
But depending on if we are odd or if we are even, that changes whether or not the two ends will point in the same direction.
00:28:31.400 --> 00:28:34.700
If it is even, they are both pointing up, as long as there isn't a negative in front.
00:28:34.700 --> 00:28:38.600
And if it is odd, one of them is going to be pointing down; the first one is going to be pointing down,
00:28:38.600 --> 00:28:44.200
as long as there is not a negative in front, because a negative raised to an odd number remains a negative.
00:28:44.200 --> 00:28:53.900
So, if n is odd, we have opposite directions; if n is even, we have the same direction.
00:28:53.900 --> 00:28:58.400
All right, great; that finishes up for the examples; I hope you have a good idea of the various functions out there.
00:28:58.400 --> 00:29:02.600
There are a lot of functions out there; but at this point, you probably have a reasonable understanding of what they are doing.
00:29:02.600 --> 00:29:06.100
And the more you continue to do math, functions are just going to make more and more sense.
00:29:06.100 --> 00:29:09.700
Just pay attention to what you are doing, and say, "Oh, yes, I have seen this one before,"
00:29:09.700 --> 00:29:13.600
or "Oh, I haven't seen this one before"--pay attention to what it looks like, and then,
00:29:13.600 --> 00:29:16.700
the next time you see something like that one, you will be able to apply that information
00:29:16.700 --> 00:29:18.900
and have a better idea of how to draw that curve.
00:29:18.900 --> 00:29:20.000
All right, I will see you at Educator.com later--goodbye!