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Lecture Comments (10)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Apr 17, 2015 4:22 PM

Post by Micheal Bingham on April 17, 2015

During your fourth example, you began listing x^n with n=2 for the even "n" functions.  What would happen if you compared even lower, yet still even (or odd) values of n.  For example, the graphs of: x^-2 or x^0 (constant function) and/or x^-5 do not seem easily comparable to the other positive integer (n) functions.  Is there still a way in which these negative numbered exponent graphs relate to their positive integer ones (with even/odd)?

1 answer

Last reply by: Professor Selhorst-Jones
Sun Mar 9, 2014 3:37 PM

Post by Linda Volti on February 24, 2014

For the square root function, why can't an output be negative? This would mean there are two outputs for one input and that, therefore, it would not be a function. Isn't the square root of, say, 4 +/- 2?

1 answer

Last reply by: Professor Selhorst-Jones
Thu Jul 11, 2013 12:17 PM

Post by Jonathan Traynor on July 1, 2013


3 answers

Last reply by: Professor Selhorst-Jones
Tue Jun 4, 2013 11:26 PM

Post by Rajendran Rajaram on June 4, 2013

I have an important question does a cube and a rectangular prism that has equal surface area have equal volumes.
thank you

Function Petting Zoo

  • An important step in learning mathematics is becoming accustomed to how various functions behave and what their graphs looks like. While long-term experience is the surest way to familiarize yourself with functions, this lesson will help give you a head-start on developing your "function intuition".
  • Other teachers or books might call a similar lesson a Function Library or Parent Functions. The point is the same, though: to introduce/review a wide variety of fundamental functions, along with their characteristics and graphs.
  • As we work through these functions and see their graphs, don't forget: axes matter. Pay attention to the "graphing window" so you can interpret what you're seeing. For the majority of these graphs, we'll use x: [−10, 10], y: [−10, 10], but a few graphs will be different.
  • We can't really explain or show the graphs here in the notes, so make sure to check out the video if you need to familiarize yourself with these fundamental functions. Here is a list of the functions we examine in this lesson:
    • The Constant Function: f(x) = k
    • The Identity Function: f(x) = x
    • The Square Function: f(x) = x2
    • The Cube Function: f(x) = x3
    • The Square Root Function: f(x) = √x
    • The Reciprocal Function: f(x) = 1/x
    • The Absolute Value Function: f(x) = |x|
    • The Trigonometric Functions: f(x) = sin(x) & g(x) = cos(x)
    • The Exponential and Logarithmic Functions: f(x) = 10x & g(x) = log10(x)

Function Petting Zoo

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:04
  • Don't Forget that Axes Matter! 1:44
  • The Constant Function 2:40
  • The Identity Function 3:44
  • The Square Function 4:40
  • The Cube Function 5:44
  • The Square Root Function 6:51
  • The Reciprocal Function 8:11
  • The Absolute Value Function 10:19
  • The Trigonometric Functions 11:56
    • f(x)=sin(x)
    • f(x)=cos(x)
    • Alternate Axes
  • The Exponential and Logarithmic Functions 13:35
    • Exponential Functions
    • Logarithmic Functions
    • Alternating Axes
  • Transformations and Compositions 16:08
  • Example 1 17:52
  • Example 2 18:33
  • Example 3 20:24
  • Example 4 26:07

Transcription: Function Petting Zoo

Hi--welcome back to

Today, we are going to have our function petting zoo.0002

An important step in learning mathematics is becoming accustomed to how various functions behave and what their graphs look like.0005

While long-term experience is the surest way to familiarize yourself with functions,0011

and you have probably already gotten some of that experience (when you see x2 show up,0014

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.0017

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.0023

And you will get the chance to develop it; we will talk about various properties, and that sort of thing.0029

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."0033

So, we are going to graph various fundamental functions; and we will talk briefly about their key points.0041

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.0045

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).0053

But really, what this is about is exposing you to these so that you are ready to understand things better down the road.0059

And other teachers or books might call a similar lesson a function library or parent functions.0065

I personally think it is kind of fun to call this a function petting zoo,0071

because really, we are just going out and meeting a bunch of different functions.0074

We are getting the chance to interact and play with one function, another function, another function...just for a little bit at a time.0077

The point of any of these names, though, is the same thing.0083

Function library, parent functions, function petting zoo--all of these things are just to introduce or review0085

a wide variety of fundamental functions, and then also to talk about their characteristics and their graphs,0090

to get you the chance to develop that function intuition, so that when you see a function,0096

you know what to expect out of it, even if you haven't worked with that precise one before.0099

All right, before we get into this, though, don't forget that axes matter.0103

Remember from our very first lesson on graphs: how the axes are set up has a huge impact on what the graph looks like.0108

Whenever you look at a graph, pay attention to how the axes are set up.0115

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.0119

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 10.0125

Notice that these are square axes; the length of the horizontal is the same as the vertical length.0134

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."0139

These axes will help us see how each one of these compares to the other one.0147

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.0151

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.0157

So, things to notice about this: for example, in this one, we have that k is a little bit more than 3.0170

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 -8.0175

Whatever we put in, it all gets mapped to k.0184

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.0187

And then finally, what does it put out? It always outputs a horizontal line at k.0200

We plug in 5; it gives out k; we plug in 20; it gives out k; we plug in -47; it gives out k.0207

When we look at all of that together, graphically we are seeing a horizontal line at height k; great.0215

The identity function, f(x) = x: it is called the identity function because whatever we plug in is what we get out of it.0224

The input is the same as output; if we plug in 7, we get out 7; if we plug in -47, we get out -47.0232

So, what we get out of this is this nice, straight line that just cuts perfectly between the x-axis and the y-axis.0248

We see a slope of m = 1 because of this: slope equals 1, because it has to cut evenly between them.0257

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.0268

Whatever goes in is what comes out--all right, that is the identity function.0275

The square function, f(x) = x2: the first thing to notice is that, as it goes to the right...0281

notice how it starts to go up faster and faster; and as it goes to the left, it goes up faster and faster.0288

The height increase, how fast it is going up, its rate of change--height increase speeds up farther out.0294

The farther we get away from the middle, the faster it is moving up.0304

Height increase is speeding up farther out.0309

The other important thing to point out is that, on the right, we are going up; and on the left, we are going up.0311

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.0318

This is an important idea about the square function: we can trust the fact that it will cup in the same direction.0335

The right side goes off in the same way as the left side.0341

The cube function is similar to the square function: its height increase speeds up, the farther we get out.0345

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 4.0360

But when we get to 2 on x3, we are at an output of 8, 2 times 2 times 2.0367

So, its height increase speeds up, and it speeds up even faster than it does on x2.0372

The other thing to notice is that, if we were to continue the graph, one side goes up, and the other side goes down.0377

The ends of the graph point in opposite directions; unlike in x2, where they point in the same direction,0383

the ends of the graph in x3 point in opposite directions.0390

The ends of the graph when we are doing the cube function will point one way up and one way down--opposite directions.0398

All right, the next one is the square root function: for the square root function, notice:0410

the farther we go out, the slower it is increasing in height.0416

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 16.0420

So, height increase slows down the farther we get out--height increase slows down.0430

Also, notice the fact that there is nothing over here; there is nothing on the left side of the graph,0440

because if we try to plug in a negative number, it is not in the domain; so negatives are not in the domain.0445

So, in a way, √x only looks like half of a graph, because it doesn't really keep going.0460

Part of it just stops, because if we try to plug in negative numbers, square root fails to work.0465

There is no number that you can square (no real number, at least) that will make a negative number.0471

A negative number, squared, gives you a positive number; a positive number, squared, gives you a positive number.0477

So, there is no number that you can square that will give you a negative number, at least in the real numbers.0481

So, the negatives are not in the domain of the square root function.0487

The reciprocal function, 1/x, is called the reciprocal because the reciprocal of a number is just one over that number.0492

So, in this one, height increase, as we get farther and farther out, slows down.0498

Height increase slows down when we are far away from the y-axis.0506

The farther we get from the y-axis, the slower the height increase becomes.0518

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...0525

well, I guess decrease--it increases as we go to the left; but the point is that the change in the height,0531

"height change"--I will change that formally--height change slows down away from the y-axis.0536

But as we get close to the y-axis, as we get closer and closer, height will "blow out."0545

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.0551

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.0568

However, there is one point that simply isn't allowed: 0 is not in domain--why? because 1/0 would be not allowed.0578

We are not allowed to take dividing by 0; so 0 is not in domain.0587

We can talk about what happens to 0.00000001, but we can't talk about what happens to 0 itself.0591

So, 0 is not in the domain; but near 0, as we approach 0, it goes out to either positive or negative infinity.0599

We will talk about this more when we talk about vertical asymptotes.0605

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."0610

All right, the next one is the absolute value function.0618

For this one, it is only going to output positives; why? because absolute value only gives out positive numbers.0621

If you put in -3, it becomes positive 3; if you put in positive 3, it becomes positive 3.0633

Whatever you put in, it is stripped of negative numbers; it has to come out as a positive number.0637

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.0642

That is the thing to notice--that in a way, it flips its direction upon touching the x-axis.0655

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.0665

What I mean by that: let's imagine that we start over here at -10; we would get positive 10.0681

We are moving this way; we now plug in -8; we get positive 8; we plug in -4; we get positive 4.0687

We are going this way; we are going this way; we are going this way; we are going this way.0691

All of a sudden, we hit a height of 0, and it bounces up; it flips to going this way right here.0694

And this is because it only outputs positives; so when we would get below the x-axis, it has to bounce off,0702

because otherwise we would be outputting a negative.0708

So, it flips the direction that it is going in upon touching the x-axis.0710

All right, now the trigonometric functions: these ones...there is a good chance you haven't seen these.0715

Or if you have, they are pretty new to you at this point.0720

The trigonometric functions--you will learn a lot about these in trigonometry.0723

But right now, the main thing I want to point out is the fact that they repeat.0726

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.0732

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.0742

They are slightly different in how they are set up, but they are repeating functions.0748

Let's also look at it in different axes, so that you can understand what is going on better.0752

This is the classic -10 to 10 axes that we did for everything else; but how about some other ones?0755

It turns out that it does its variance between 0 and 2π; don't worry about actually understanding what is going on here.0761

I just want to have you see this stuff, so it is not totally new when you see it later.0767

You will get it all very well in trigonometry.0772

From 0 to 2π, we have one repetition; from -2π to 0, we have another repetition.0775

It repeats itself every 2π; it also varies between 1 and -1, both for sine and cosine.0782

It varies between 1 and -1, and it also has repetitions on a 2π basis, both for cosine and sine.0790

We will see why this is the case when we actually study trigonometry.0797

But the main thing to get out of this right now is that trigonometric functions are these repeating functions,0800

that they are a way of being able to see the same thing happen after we go down far enough.0804

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.0808

Exponential and logarithmic functions: all right, we are back to our -10 to 10, our standard axes that we were used to before.0816

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.0822

We get outside of being able to see this out of our viewing window so quickly.0835

By the time we have made it to 1, we are at 10; at 2, we are at 100; at 3, we are at 1,000.0839

So, exponential functions (10x is only one possible exponential function)--they are going to go out really, really quickly.0845

They are going to just shoot up, having absolutely massive height.0851

Blowing out probably isn't the perfect word, since we used "blowing out" for asymptotes.0854

Let's instead say its height grows really fast.0859

Over here at log10(x), look at how long it takes to even get up to 1; it takes us out to 10 to get it.0865

We can see that the height is slowing down; its increase in height slows down.0870

Its height increase slows massively; it grows really, really slowly.0874

That is probably the main important thing to get out of these.0885

Don't worry about actually understanding what is going on precisely right now.0888

We will have an entire section on this, when we talk about exponential and logarithmic functions in detail for an entire section.0890

But right now, I just want you to say, "Oh, exponential functions get really big really fast,0896

and logarithmic functions stay pretty low for a very long time."0901

Just to make a point of how long these sorts of things are, how slow and how fast they are,0905

for logarithmic and exponential, respectively, let's look at it with new axes.0912

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 1,000.0917

1,000 is how big we have managed to get.0926

And it doesn't actually get to 0; it just looks like that--it is approaching it, because 10-3 would be 1/103, 1/1000.0929

So, it is just really close to the x-axis.0937

Once again, don't worry about understanding this perfectly right now; we will talk about it later.0939

Logarithmic functions: they are going to take forever to even get to reasonable numbers.0942

We have to get to 1,000 before we even manage to make it up to a height of 3.0948

So, they grow really, really slowly; the height growth on logarithmic functions is really, really slow,0954

and slows down massively the farther you get out; whereas exponential functions are really, really fast,0959

and increase massively the farther you get out.0964

All right, of course, the functions we see--that makes it for our petting zoo--we have completed our petting zoo;0967

but when we see functions out in the wild, they are normally not going to end up being in their pure form.0973

They are not going to be x2 or √x.0979

Normally, they have had other things put on them or added on them; they have been shifted, stretched, or flipped.0982

They have been transformed in some way.0987

Still, it helps to know the general shape for a function before transformation.0990

Beyond these shiftings, stretchings, and flippings, we can still have a pretty good idea of what is going on.0994

Other times, functions will be mixed with other functions.1001

We might have things like f(x) = x3 + x2: that is not just one pure function--that is x3 and x2.1004

Or x2 times √x--once again, that is not just one pure function; that is two functions mixed together.1010

Or h(x) = |x3|: once again, that is not just one pure function; it is two things put together; it is absolute value and x3.1016

Once again, though, it helps to know each function's general form before trying to figure out how they interact.1024

If we understand how |x| works, and we understand how x3 works, it will make sense to us1030

when we work on the graph of the absolute value of x3.1035

We will have a better understanding of what is going on--what we are seeing.1038

We will learn about both of the above ideas in the lessons Transformations of Functions--the transformations1042

where we shift, stretch, and flip will be in the Transformations of Functions lesson; and we will talk1047

about composite functions--we will talk about arithmetic combinations (these first two are arithmetic combinations--1052

once again, don't worry if you don't know what these things mean precisely; we have lessons for that);1058

and then finally, an actual composite function, where we combine the way that two functions are working.1062

We will learn about these in much greater detail in those two respective sections.1066

All right, great; we are ready for some examples.1070

Here are two graphs without axes; they are the graphs of what functions?1074

Well, this one is going up; it looks like a slope of 1; it is pretty stable; it is just increasing continuously.1077

It doesn't bounce; so this is almost certainly f(x) = x; great.1083

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);1089

and it is going up to positive ∞ up here; so what blew out and what got really, really slow in its height change?1098

Oh, yes, it is the reciprocal function: f(x) = 1/x; great.1107

Next, here are two functions that have been shifted, stretched, and/or flipped; what are the base functions making them up?1115

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."1120

Oh, look over here: there is nothing over on the left side.1128

So, if there is nothing on the left side, it has cut out the negative side; it has cut out the left side.1134

What cut out one side? √x--we have that √x, and its height increase is still slowing.1139

Height increase slows the farther it gets out; so that is both of the identifying marks of being a √x function.1147

It is very different than the normal √x; but the basic function that is making this up is √x.1155

It has been shifted; it has been stretched; but it is still √x.1161

What about this one over here? Well, it is going down on both sides.1166

And its height increase gets faster...well, "height increase" is incorrect; it is not height increase, because we are going down.1174

But it is a height change, and that is the more fundamental idea about x2.1187

It isn't necessarily that it has to be going up, but that the change, the rate that it is going up1193

or the rate that it is going down, possibly, is continuing to increase.1197

Height change speeds up the farther we get out from the center; the center has been moved in this one.1201

The height change will speed up; we get faster and faster changes in our height.1207

So once again, this is f(x) = x2: it has been shifted, it has been flipped, and it has been stretched;1211

but we can still recognize that that is a parabola; it must be, at heart, coming from that same idea as behind x2; great.1217

Example 3: we want to graph f(x) = x3, g(x) = x2, and h(x) = x, all on the same axes.1225

And we also want to set up the axes such that we go from -10 to 10 on the horizontal,1233

and none of the graphs are cut off vertically; so that means we can't lose any vertical information.1239

To help us understand what is going on here, let's make a table.1246

Now, we know what x3 looks like, what x2 looks like, and what x looks like, in general.1248

So, we can use that information to help us out.1253

Let's see what the extreme values are and what the middle values are.1255

So, here is our table: x, f(x) is x at -10 and x at positive 10.1258

f(x)...let's make all three of them: x3, x2, and x.1269

Let's actually use colors for these: x2 will be in red, and blue will be x; great.1278

So, if we plug in -10, we are going to get -10 cubed, which is -1,000.1285

x2 will become 100, positive, because the negatives cancel out; and x will become -10.1292

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 10.1302

So, let's try some other things: let's look at what happens in the middle.1313

Well, in the middle, x3 is at 0; x2 is at 0; and x is also at 0.1315

Let's see what happens in the middle between the middles: if we plug in, say, -5 and 5:1322

-5 times -5 is positive 25, times another -5...we get -125.1330

Over here, we will have positive 125; the this, we will get squaring, so we will be at positive 25, positive 25.1337

And we will have -5 and positive 5.1346

And finally, at 1 and -1, -1 will go to -1; positive 1 will go to positive 1 for x3.1349

For x2, -1 will get cancelled to a positive; a positive is still for the positive; and then, -1 and positive 1.1357

The thing to notice here is that, when it is close to 0, they are not that different.1365

But the farther we get from 0, the more their differences become apparent.1369

So, let's make our graph; we will do it in blue, which hopefully won't be too confusing,1373

even though blue is connected to x; so what is the maximum vertical height that we have to have?1382

The maximum vertical height that we have to have is a huge 1,000, because we get up to -1,000 and positive 1,000.1388

So, we have to be at -1,000 and positive 1,000 as the vertical extremes.1395

Here is the middle at -500, positive 500; and then, we will actually go to an extreme1400

of 10 and -10 horizontally, because that is what we were told to do.1410

And here is our middle at 5, and middle at -5; and here is 1 and positive 1; great.1413

All right, so at this point, let's graph x3; this is probably going to be one of the easiest to graph.1422

At 0, it is at 0; at 1, it manages to be at 1, so it has barely even gotten off the x-axis.1426

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.1431

And then, 10 is going to be all the way up at 1,000.1437

-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...1440

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 -1,000.1451

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--1459

as it gets farther and farther from the center of its graph.1476

What about x2? We get 1 and 1 in the same location; at 5, it is at a 25--a meager, tiny jump above;1479

and then, at 10, it is at 100, so it is a little bit below the height of x3 at 5.1487

The same on the reverse side: the parabola...when we look at it this far out, it is growing fast;1493

it manages to get to 100 by the time it has gotten to 10; but it is still tiny, tiny;1508

it looks so stout---it looks so short--compared to x3.1515

Finally, we look at what happens to x, just the plain identity function.1519

And at 10, it manages to only be 10 above; we are talking about there; at -10, it is only -10.1523

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 x3.1529

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.1543

But when we look at them on a larger scale, not even that big--just -10 to 10--suddenly the differences become apparent.1551

They become massive, huge differences: the difference between x3 and x2 at just 10 out is a difference of 900.1556

There are huge differences between these; and they get even bigger, the farther we go out.1563

The final example, Example 4: Think about functions of the form f(x) = xn, where n is a positive integer--it is contained in the natural numbers.1568

Then distinguish the difference between when n is odd and when n is even.1578

Let's look at some examples for when n is odd: n is odd would be like x, or maybe x3 or x5.1582

So, we know what x looks like; it is just like that.1591

x3 blows out pretty quickly; x5 blows out even faster.1595

By the time it makes it to 2, it is at 2, 4, 8, 16, 32 height.1602

So, by the time x5 has an input of 2, it is getting an output of 32; so it blows out really, really fast.1608

Let's compare some even ones; say we have x2...well, we know what that one looks like; it just looks like a parabola.1618

x4...well, it is like a parabola, but it grows even faster.1625

By the time we get to positive 2, instead of just being at 4, we are at 16.1631

So, it grows really fast--not quite as fast as x5, but faster than x3.1635

And as we go to the other side, since (-2)2 is positive 4, (-2)4 is positive 4 times positive 4, or positive 16.1640

So, as long as we are even, we are going to cancel out those negatives.1650

And that is the idea that we see right here: look, when we are even, we go off in the same direction.1654

If we were to do this for x6, it would be the same thing, but growing even faster.1659

We would just be growing even faster; so what we are seeing here is that, when n is odd,1663

that means that the ends of the graph go in opposite directions.1673

So, the ends of the graph go in opposite directions if n is odd.1679

But if n is even, the ends go in the same direction.1686

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.1696

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.1706

If it is even, they are both pointing up, as long as there isn't a negative in front.1711

And if it is odd, one of them is going to be pointing down; the first one is going to be pointing down,1715

as long as there is not a negative in front, because a negative raised to an odd number remains a negative.1718

So, if n is odd, we have opposite directions; if n is even, we have the same direction.1724

All right, great; that finishes up for the examples; I hope you have a good idea of the various functions out there.1734

There are a lot of functions out there; but at this point, you probably have a reasonable understanding of what they are doing.1738

And the more you continue to do math, functions are just going to make more and more sense.1742

Just pay attention to what you are doing, and say, "Oh, yes, I have seen this one before,"1746

or "Oh, I haven't seen this one before"--pay attention to what it looks like, and then,1750

the next time you see something like that one, you will be able to apply that information1753

and have a better idea of how to draw that curve.1757

All right, I will see you at later--goodbye!1759