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Lecture Comments (15)

1 answer

Last reply by: Professor Selhorst-Jones
Tue Jan 6, 2015 12:36 PM

Post by Andrew Demidenko on January 4, 2015

Professor, I am still not clear about Stacking Transformation.

2 answers

Last reply by: John K
Fri Aug 22, 2014 5:13 AM

Post by John K on August 21, 2014

Is vertical stretch and horizontal shrink the same thing?

1 answer

Last reply by: Professor Selhorst-Jones
Sat Nov 9, 2013 4:12 PM

Post by Damien O Byrne on November 9, 2013

If you horizontally stretch a function does that mean you vertically stretch the functions graph and visa versa. for example the inverse function 1/x transformed to 1/2x could this be interpreted as multiplying f(x) by 1/2 ( vertical shrink) and also interpretted as f(2x) horizontal shrink. is this just coincidence ?

its just visually if i stretch a graph horizontally shouldn't the y values decrease as in a vertical shrink and if I shrink horizontally shouldnt the y value increase.

1 answer

Last reply by: Professor Selhorst-Jones
Mon Oct 28, 2013 9:59 AM

Post by Charles Reinmuth on October 27, 2013

The vertical stretch and horizontal stretch look very similar to me. I see a difference in that there are parentheticals around the horizontal (eg. f(x) = (3x)^2 ...vs... f(x) = 3x^2)

Still, I don't think I understand fully what is going on. What exactly is the difference? Perhaps I missed something. Thankyou so much!!

1 answer

Last reply by: Professor Selhorst-Jones
Sun Jul 28, 2013 9:11 PM

Post by Jason Todd on July 26, 2013

Professor, in example 2 how did you differentiate vertical vs. horizontal flip possibilities? Thanks in advance.

1 answer

Last reply by: Professor Selhorst-Jones
Thu Jul 11, 2013 1:11 PM

Post by Sarawut Chaiyadech on June 28, 2013


1 answer

Last reply by: Professor Selhorst-Jones
Thu May 23, 2013 10:47 AM

Post by Matthew Chantry on May 22, 2013

These questions are for Example 2:

1. Shouldn't everything in the h(x) function after the - be in brackets?
2. Could this be seen as a horizontal flip as well? Would that look different?


Transformation of Functions

  • We often have to work with functions that are similar to ones we already know, but not precisely the same. Many times, this difference is the result of a transformation. A transformation is a shift, stretch, or flip of a function.
  • A vertical shift moves a function up or down by some amount. If we want to shift a function f by k units, we use
    f(x) + k.
    [If k is positive, it moves up. If negative, down.]
  • A vertical stretch/shrink "pulls/pushes" the function away from/toward the x-axis by some multiplicative factor. If we want to vertically stretch/shrink a function by a multiplicative factor a, we use
    a ·f(x).
    [If a > 1, the function stretches. If 0 < a < 1, it shrinks. If a=1, nothing happens.]
  • A horizontal shift moves a function left or right by some amount. If we want to shift a function f by k units, we use
    [If k is positive, the graph moves left. If k is negative, the graph moves right. (This may seem counter-intuitive, but remember that the shift is being caused by how f "sees" (x+k). Check out the video for an in-depth explanation of what's going on.)]
  • A horizontal stretch/shrink changes how fast the function "sees" the x−axis. If we want to horizontally stretch/shrink a function by a multiplicative factor a, we use
    f(a ·x).
    [If a > 1, it shrinks horizontally ("speeds up"). If 0 < a < 1, it stretches horizontally ("slows down"). (This may seem counter-intuitive, but remember that the stretch/shrink is being caused by how f "sees" (a·x). Check out the video for an in-depth explanation of what's going on.)]
  • To vertically flip a graph (mirror over the x-axis), we need to swap every output for the negative version. If we want to vertically flip, we use
  • To horizontally flip a graph (mirror over the y-axis), we need to "flip" how f "sees" the x-axis. We do this by plugging in −x (which is effectively a "flipped" x). If we want to horizontally flip a function, we use
  • If you want to do multiple transformations, just apply one transformation after another. However, order matters, so start by deciding on the order you want the transformations to occur in. Then apply them to the base function in that order.

Transformation of Functions

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
  • Vertical Shift 1:12
    • Graphical Example
    • A Further Explanation
  • Vertical Stretch/Shrink 3:34
    • Graph Shrinks
    • Graph Stretches
    • A Further Explanation
  • Horizontal Shift 6:49
    • Moving the Graph to the Right
    • Moving the Graph to the Left
    • A Further Explanation
    • Understanding Movement on the x-axis
  • Horizontal Stretch/Shrink 12:59
    • Shrinking the Graph
    • Stretching the Graph
    • A Further Explanation
    • Understanding Stretches from the x-axis
  • Vertical Flip (aka Mirror) 16:55
    • Example Graph
    • Multiplying the Vertical Component by -1
  • Horizontal Flip (aka Mirror) 18:43
    • Example Graph
    • Multiplying the Horizontal Component by -1
  • Summary of Transformations 22:11
  • Stacking Transformations 24:46
    • Order Matters
    • Transformation Example
  • Example 1 29:21
  • Example 2 34:44
  • Example 3 38:10
  • Example 4 43:46

Transcription: Transformation of Functions

Hi--welcome back to

Today, we are going to talk about transformations of functions.0002

By now, we are familiar with a variety of different functions--things like x, x2, √x, |x|, etc., etc.0005

We have seen a bunch of different fundamental parent functions--that function petting zoo we visited in the last lesson.0013

However, we often have to work with or graph functions that are similar to these fundamental functions, but not precisely the same.0019

They are not the ones we are already familiar with, exactly.0025

Many times, the difference is the result of a transformation.0027

A transformation is a shift, a stretch, or a flip of a function.0031

And when I say that, I mean that graphically; it has been moved either left/right or up/down;0036

it has been stretched either horizontally or vertically; or it has been flipped vertically or horizontally.0041

Understanding transformations is useful for working with functions or building our own functions, if we want to build one from scratch.0049

For this lesson, we will begin by looking at vertical shifts and stretches, because they are the easiest ones to graph.0056

It is easiest to understand moving things around vertically.0061

Then, we will learn how horizontal shifts and stretches work, and what we have understood in vertical0064

will give us a slightly better understanding of what is going on horizontally; and finally, we will look at flips; all right, let's go!0067

The very first one is vertical shift; this is the easiest one of all--to shift vertically, you simply add to a function.0073

Positive numbers shift up; negatives shift down; so let's see an example.0079

Consider f(x) = x2 graphed with f(x) + 2 = x2 + 2 and f(x) - 7, which equals x2 - 7.0083

f(x) = x2, our base function, is the red part of our graph.0092

We want to see what f(x) + 2 is; that is the blue graph; and finally, the green graph is f(x) - 7.0096

Notice: if we just take a function, and we add 2 to it, it gets raised by 2 units; the height goes up to 2; it goes up to here.0104

But if we take the function and subtract 7, f(x) - 7, which would be x2 - 7, it goes down by 7.0113

It goes lower off its base of normally starting at (0,0); x2 has its home base, in a way, at (0,0).0121

It drops down by 7; or we can raise it up; so we can move it up and down with this vertical shift.0129

What is going on? Think about it like this: if we wanted to move one point vertically, we would add k to its vertical component, its y-value.0135

So, let's say, hypothetically, we want to change the point (1,2); we want to move it up by 5.0141

So, if we want to move it up by 5, we would just add 5; we would have (1,2 + 5), which would be (1,7).0148

That is what we would get if we wanted to move it up by 5.0158

To move up one point, we would just add k or subtract k; but let's think of it as adding a negative k.0160

We add k, and if k is positive, it goes up; if k is negative, it goes down; that moves one point up.0167

If we want to move all of the points in a graph, then we have to add this to all of the vertical components.0172

Now remember: the vertical components of a function's graph are the function's output.0179

So, the function's output...we just need to make the output be k more, or k less, if it is a negative number.0184

So, we add k to it; we just change the output everywhere by adding k to the function.0190

So, if we start with f(x), and we want to vertically shift by f, we just use f(x) + k.0194

If k is positive, it moves up; if k is negative, it moves down--as simple as that.0201

So, f(x) + k: we take our original function, and we add k to it, and we have a vertical shift of k units; great.0207

Next, the vertical stretch/shrink: we want to vertically stretch or shrink a graph--we want to pull it/stretch it,0214

or we want to shrink it--we want to squish it.0221

We multiply the function by a multiplicative factor, a: if 0 is less than a, which is less than 1 (a is between 0 and 1), the graph will shrink.0223

If a > 1, the graph stretches.0232

Let's see an example: consider f(x) = x2; that is the one in red--that is our basic function,0235

that we are starting with to have a sense of how things are going to go.0241

And then, we compare that to 3 times f(x): it has a multiplicative factor of 3 hitting it.0244

So, in this case, we have the graph stretching, because it is 3 times f(x); so a is greater than 1.0250

And indeed, this one right here has been stretched up; we have the parabola normally,0258

but we have grabbed it and pulled it up higher.0263

If we look at this, every point here is 3 higher; here is 1, 2, 3 higher to get up to there.0266

If we compare that to the shrinking in the green graph, we have a = 1/3; so 1/3 is between 0 and 1, so it has been shrunk; it has been squished down.0276

Where we are at the red...we go to 1/3 of where we are at the red, and we find ourselves on the green one.0289

It has been squished by a factor of 1/3.0296

We can either stretch it with a larger-than-one factor, or squish it with a less-than-one factor, but not in the negatives.0299

We will talk about negatives later.0305

What is going on? First, when we say stretch/shrink, it means we are grabbing a point and pulling or pushing it away from or toward the x-axis.0308

So, for example, let's say we have the point (1,2), and we want to apply some multiplicative factor to the point's height.0317

Its height is just 2; so if a = 1/2, then we get (1,2(1/2)), so we get (1,1).0326

We have halved the point's height; if we were at (1,10), it would become (1,5).0338

We are squishing by a factor of 1/2; we could also expand by a factor of a = 7, and it would get multiplied by 7.0344

So, we have this multiplicative factor either pulling it apart if it is greater than 1, or squishing it together if it is less than 1.0351

If we want to do this to all of the points in the graph of a function, we need to apply that multiplicative factor to all of the function's outputs.0357

To do it to all of the function's output, we need to multiply the function itself by a.0363

So, that will apply it to all of the outputs, because the function tells us where the outputs go.0371

So, if we multiply something against the function, it will have done it to all of the possible outputs that would come out of that function.0374

So, to stretch/shrink a function by a multiplicative factor, a, we use a times our function, a times f(x).0380

If a is greater than 1, the function stretches; if a is between 0 and 1, it shrinks.0387

And finally, if a is equal to 1, then it doesn't do anything; it has no effect, because we are between stretching and shrinking.0395

We are exactly on the middle, and we are just left with the function as it was before; it has no effect.0400

We have grabbed it, and then just immediately let go, instead of pulling it and pushing it together.0405

Horizontal shift: this is a little more complex than vertical shift, but we are ready to talk about this now.0409

To shift horizontally, we need to change where the function sees x = 0.0415

We do this by plugging something different than x into the function.0419

What I mean by that: take, for example, a normal line, a normal f(x) = x kind of line.0422

Well, if we wanted, we could say that in a way, its home base is this 0 point where it crosses that axis.0431

It is where it sees 0 on the x-axis; but we could also talk about another graph where it is the same line, but shifted over to the right.0439

What has happened there is: we have taken this home base, and we have shifted it over by some amount.0448

So, we have the same picture, but it has been moved to the right.0459

We are seeing the home base, the effective x = 0, in a different place; that is the idea we want to bring to this.0461

Consider f(x) = x2 (once again, we see that in red) graphed with f(x) - 4 = (x - 4)2.0469

That one is in blue; it has been shifted four units to the right.0478

And f(x) + 2...that one is in green: (x + 2)2 has been shifted two units to the left.0483

Now, that might seem counterintuitive at first: -4 causes us to shift to the right, but + 2 causes us to shift to the left.0492

To understand what is going on, we need to think about this a little more deeply.0498

What is going on? Think about it like this: the graph of a function is a way to look at how the function sees the x-axis.0501

A graph is where an input gets placed as an output; that is at least one way to interpret a graph, and it is how we are doing it with functions.0508

If we are looking at a graph, it is how it sees the entire x-axis--at least, all of the x-axis in our viewing window--at once.0516

That x-axis is mapped to some sort of curve; it takes in each x-value in the x-axis, and it outputs a y-value.0522

A graph shows how the function is seeing all of the x-axis at the same time.0529

Now, normally we plug in just x; so the x-axis looks like your normal number line: 0 in the middle, 1, 2, 3 out, -1, 2, 3 out in the other direction.0532

1, 2, 3; -1, 2, 3; it is exactly what we are used to; it is your normal number line.0542

But we could move this number line around; we can move this home base.0547

Currently, our home base of 0 (think about it as a home base for now)--0 is in the middle when we plug in just x.0550

How do we move around that home base? We move it around like this.0557

Normally, once again, our normal plane, x--the normal number line--we can move it around by plugging in x + k.0561

For example, if k = -2, then our normal home base now is x - 2, so it is 0 - 2; and we get -2.0569

1 - 2...we get -1; 2 - 2...we get 0; look, our home base has shifted 2 units to the right.0579

We have gone over 1, 2 clicks over to the right.0590

And everything has been moved by this amount: we have the -2 on everything, and so that is how we are getting all of these new points.0593

But what we have gotten by doing this is: we have effectively shifted over the location of home base.0601

We have effectively shifted over the location of x = 0 by subtracting 2 from everything.0607

Since everything is now 2 lower, we had to go where we originally had 2 to now just have 0--to be back in our usual home base.0613

So, I want to say this once again: The graph shifts to the right with a negative.0621

So, if we plug in a negative for k, this x + k here...if we plug in a negative, we get shifting to the right,0627

because we are taking that much away from all of the numbers.0632

And so, if we are taking them away, it has to be the high numbers, the traditionally right-side numbers,0635

that are now going to have our new home base in them.0640

Shifting to the right: horizontal shifts to the right happen with a negative value for k.0643

What if we wanted to shift to the left? Well, we could plug in a different x + k.0649

If we plug in k = 3, then if we have + 3, we now have 3 for our old 0.0655

We have to go to -3 for us to get our home base back.0662

So, it is now 1, 2, 3 over to the left for us to get from our old home base to our new home base.0666

And everything is going to get hit by this + 3, which is why we have 6 over here and all these sorts of things.0675

So, by adding a positive number, it is the lower numbers, the negative numbers,0680

that are now going to end up taking the place of having the home base be on that side.0684

The home base will move to the left if we have a positive k.0688

If we want to shift to the left, we use positive for k in that x + k; great.0691

Therefore, we can shift around where the function sees x = 0...this idea of seeing x = 0,0699

seeing our home base, and the rest of the x-axis in turn, is by plugging in x + k instead of just plugging in x.0705

By shifting around the perceived x-axis, this perceived home base, the graph will move horizontally (move to the left/move to the right).0712

Now, notice: this doesn't actually change the x-axis; it just changes the way the function sees it.0720

The x-axis is still going to look totally normal to us; it will be that normal x-axis that we are used to.0725

But the way that the function will interact with it is now based off of this new home base, because of plugging in that x + k.0730

So, to horizontally shift the function by k units, we use f(x + k).0737

And if k is a positive, we shift left; if k is a negative, we shift right.0741

Now, remember: that seems a little bit counterintuitive at first; but if you think about why it is the case--0749

if we put in a positive number, it has to be the negative side that establishes the new home base, the new 0--0753

if we put in a negative number, then it has to be the positive side, the right side,0759

that establishes the new home base, the new 0--that seems a little counterintuitive,0763

but if you think about those slides, those ideas we just saw, you will think, "oh, yes, it makes sense0768

that I am plugging in the positive to go left and plugging in the negative to go right."0773

All right, horizontal stretch: this idea is similarly complex.0778

But now that we understand how horizontal shifting works, this will probably make more sense.0782

To horizontally stretch or shrink a function--that is, to pull it apart or to squish it together--0785

we need to change how fast the function sees the x-axis.0790

Once again, it is not really, literally seeing it; the function isn't a living, breathing thing.0794

But the way that it is going to interact with it, we can effectively personify it and pretend that it is alive for this.0799

Effectively, we need to stretch/shrink how the function perceives x.0805

We need to change the way that the function will interact with that x-axis.0808

Let's look at some examples first: f(x) = x2, our normal red graph--let's try if we put in 3 times x instead.0813

We have put that multiplicative factor on the x, and we get (3x)2 on our blue graph.0822

We plug in 1/2 times x...(1/2x)2 on our green graph.0828

All right, so let's understand what is going on here--what is going on?0833

This idea is very much like what we did with horizontal shift.0836

We are playing with how the function sees the x-axis.0838

Once again, remember: if we plug in just x, we get this one, our normal number line with 0 in the middle, -1, -2, -3 to the left, and 1, 2, 3 to the right.0840

Great, that is just like normal.0850

But we can speed this up; we can speed up or slow down the experience of this number line.0851

If we plug in an x-axis that has been stretched or shrunk by a multiplicative factor, a;0859

if we have a times x--for example, if we have 3 times x--we speed it up.0863

Instead of 0 to 1, it is now 0 to 3; so it is 0 to 3, and then 3 to 6; and of course, 1 and 2 are still in there.0869

But they have gotten shrunk down; we are speeding up how fast we are moving through the numbers.0879

So, each number effectively has been multiplied by 3; we are moving through the numbers faster, which will condense the graph.0884

The graph will be happening faster horizontally, so it is going to go through what it would do normally, faster.0890

We can also expand it by slowing it down with a small a.0897

If we put in a small a, like 1/2 times x, we go from 0 to 1; and now, we are going 0 to 1/2.0901

We have to take two steps forward before we even manage to make it to 1; so now, we are going at a speed that is 1/2 the speed of originally.0908

We are slowed down by a factor of 2.0914

OK, by applying a multiplicative factor, a, to x, we can change how fast the x-axis looks to the function.0917

This change in horizontal speed either stretches or shrinks the graph horizontally.0925

If it looks faster, the graph will compress, because it has the same amount of things happening in a shorter amount of x-time.0929

If it is stretched, then if we slow it down, it will be stretched, because it takes more x-time to be able to get through the same information.0936

And like before, this doesn't actually change the x-axis that we see; it is just how the function will interact with the x-axis.0945

So, to horizontally stretch/shrink a function by a factor of a, we use f(ax); we play with how that x-axis works.0951

We change around the speed that that x is moving at.0959

So, if a is greater than 1, it will shrink horizontally, because we have sped up.0962

So, a > 1: it shrinks horizontally, because it is speeding up how fast the x-axis goes.0968

We want to think about that in terms of speeding up; it makes it easier to understand what is going on.0973

So, if we speed it up by putting in a large a, a > 1, we are going to shrink horizontally,0977

because more stuff will happen in the same period of "time."0984

0 < a < 1: we are going to stretch horizontally; it will slow down the x-axis,0987

because we now have to go through a longer interval, a longer amount of x-time, for us to be able to get the same information through.0994

a > 1 shrinks horizontally; it speeds up; if a is between 0 and 1, it stretches horizontally; it slows down.1003

All right, vertical flip, which we might also call mirroring vertically, or a vertical mirror:1015

to vertically flip a graph around the x-axis, we simply multiply the function by -1; it is as simple as that.1019

We just multiply the function; so if we have f(x) = x2 in red, then we can flip it vertically by just multiplying by a -1.1025

-f(x), which is flips to pointing in the opposite direction.1034

What is going on here? Well, if we wanted to flip a single point around the x-axis--say we have (1,2) again--1039

if we wanted to flip it around the x-axis to the opposite height, then we would just multiply the vertical component,1045

the y-value, by -1; so vertical times -1 would become (1,2(-1)), or (1,-2).1051

We have flipped that point to the opposite vertical location, the opposite height.1063

This sends it to the opposite side; but it still has the same height in terms of distance from the x-axis.1068

It is now a negative height; or if it started negative, it will now be positive.1074

For example, we could have, say, (3,-7); and that would flip to (3,7); so we are flipping from one side to the other side.1077

If we want to do this to all the points on the graph of a function, we need to apply it to the entire function.1089

The vertical components are the outputs of the function, so we need to make the function output the negative version everywhere.1094

We do this by just multiplying the whole function by -1.1100

So, to flip a function vertically around the x-axis, that is if we have smiley-face here, then it will become reverse-smiley-face here;1103

it is flipped vertically; so to flip a function vertically around the x-axis, we use -f(x); great.1115

To horizontally flip a graph around the y-axis, we change how the function sees the x-axis to its opposite.1124

We need to flip its perception of the x-axis, just like we changed perception of the x-axis with horizontal shift and horizontal stretch.1130

We are going to do that for horizontal flip.1137

We do this by plugging in -x; let's see an example--consider f(x) = √x.1139

If we have √x, and that is the red graph (we couldn't use x2, because its horizontal flip1145

will just look like the exact same thing); we will graph that with f(-x).1150

So, we plug in -x, and we will get -√x, which ends up pointing in the exact opposite direction.1157

Why is it pointing in the exact opposite direction?1162

Well, if we tried to plug in a positive number, like, say, positive 6...√-6...if we plug in x = 6, it is going to get us √-6, which does not exist.1164

So, it doesn't exist on the right side, just like √x, normal square root of positive x, doesn't exist on the left side.1177

So, our blue graph has to go in the opposite direction, because it sees -6 as being the same height as the red one sees +6.1185

All right, what is going on here--how is this working?1194

We are reversing how the function sees the x-axis.1197

Normally, once again, we see x going off to the right and to the left, just like usual.1200

But if we plug in -x, it reverses; let's put some color here, so we can see what I am talking about.1205

So, if we have, on our normal, positive x version, that it goes to the right in red, and it goes to the left in blue,1210

when we plug in -x, we see that it goes to the right in blue, and the left in red.1217

If we hit 3 by -1, it becomes -3; -3 by -1 becomes positive 3; so we have flipped the order that the x-axis occurs in.1225

As opposed to going from negative to positive, it now goes from positive to negative; we have flipped the order that it occurs in.1236

To flip a point horizontally around the y-axis, we need to just multiply the horizontal component by -1.1244

For example, our point (1,2): if we want to flip horizontally, then we are just going to look at the negative version of the x-axis.1249

So, we would go to (-1,2); so this will move the point to the opposite horizontal location.1260

If we plug -x into a function, it reverses how it sees the x-axis throughout.1268

So, we will be plugging in opposite horizontal locations everywhere; so everything will flip to the opposite horizontal location.1272

All of the points are going to show up in the opposite horizontal location, because we have plugged in this -x.1279

To flip a function horizontally around the y-axis, we use f(-x).1284

What does that mean? Once again, say we have some smiley-face over here.1289

Smiley-face, sadly, doesn't have anything left-right; he is a perfect left/right thing.1295

So, let's make smiley-face--it is now Ms. Smiley-face, and Ms. Smiley-face has a little bow.1304

So, if we flip her around the y-axis--we flip her horizontally--she will show up on the other side.1311

And her face will look the same, because her face is mirror-symmetric horizontally; but now her bow is going to be on the opposite side.1319

All right, so she shows up on the opposite side now; she has been mirrored horizontally around the y-axis.1325

Here is a summary of transformations; I know it is a lot of transformations that we have seen at this point.1332

So, don't worry if you have to refer back to this list later on.1336

Also, if you currently have some sort of book that you are working on along with this course in,1340

or if you have another teacher who is working in a book, you are almost certainly1344

going to be easily able to find a table of these in any section where they would be teaching the same things in that book.1347

This table is really useful, because it can be a little hard to remember all of them immediately.1352

Vertical shift is f(x) + k; k is positive; that causes us to go up; k is negative--that causes us to go down.1356

Vertical stretch is a(f(x)); if a is between 0 and 1, it shrinks it; if a is greater than 1, it stretches it.1362

Horizontal shift is plugging in x + k; if k is positive, we go to the left; if k is negative, we go to the right.1369

Horizontal stretch is...0 to 1 means we slow down, and slowing down means we stretch out.1378

a greater than 1...did I say horizontal stretch?...f(a) times x...I am not quite sure I said that...1384

a greater than 1 causes us to go faster, which means we squish together.1390

Vertical flip: we flip over vertical; that is -f(x); horizontal flip, f(-x), causes us to flip horizontally.1394

One thing to notice: all the vertical stuff happens outside the function.1402

If it is vertical shift, it is f(x) + k; if it is vertical stretch, it is a(f(x)); if it is vertical flip, it is negative times f(x).1415

Everything is doing it on the outside of the function.1427

However, horizontal things happen inside: horizontal shift is where you plug in x + k.1430

x + k goes into the function; horizontal stretch is a times x, which goes into the function.1440

Horizontal flip is where -x goes into the function.1447

So, horizontal things will happen inside the function; it happens to what we are plugging into the function,1452

whereas vertical things happen on the outside of the function--we don't have to worry about it being plugged in.1457

OK, that is a summary of transformations; don't worry if you have to refer to this.1461

But you can also probably think about this sort of thing, now that we have an understanding of where this stuff is coming from.1465

You can probably actually figure out that it makes sense, and just re-figure it out, re-derive it, in your own head,1469

without even having to refer to these lists.1474

Horizontal shift, horizontal stretch, horizontal flip--they might be a little bit more difficult.1476

But remember that idea where we are shifting around our home base--we are shifting around the experience of the x-axis.1480

That is what we are shifting with those.1485

Stacking transformations: if you want to do multiple transformations on one function, you just apply one transformation after another.1488

But order matters; unlike when we multiply and divide and multiply...if I multiply 5 by 3 by 7 by 8, I will multiply all of those numbers together.1495

It doesn't matter what order I multiply them in; but in transformations, it matters what order you put the transformations on in.1504

Be careful: the order you apply your transformations in can affect the results.1510

The order you apply will affect how it comes out--not always, but a lot of the time.1516

Decide on the order you want before you do it; decide on the order, then apply them to your base function in that order.1521

The order that they hit that base function--the order that the do their transformations in--will change what happens.1528

There are some cases where it won't matter what order you put it on in.1535

But other times, you are going to get totally different results; and we will look at an example in just a second.1538

This means you have to think about order when doing multiple transformations.1542

So, make sure you think about order if you are doing multiple transformations,1545

because if you don't think about it, you can really get confused and get completely the wrong answer.1549

All right, let's look at an example of why it matters how we stack our transformations--the order that we put our transformations on in.1553

For example, let's consider f(x) = √x.1558

And just in case you have forgotten what that looks like, it starts with the origin and just goes up like that, and slowly increases the farther it goes up.1561

All right, let's say we want to move it two units right and flip it horizontally.1567

We move it two units right by plugging in x - 2 into where we have x.1573

And we flip horizontally by plugging that -x in here as well.1579

That is how we do the two different things: we plug in -x into the function, or we plug in x - 2.1583

Look at how order matters; we get very different things, depending on the order we put this in.1588

So, if we move, and then we flip, first we would move; we would plug in x - 2 first, so we would get √(x - 2).1592

And then, the next action is flipping; so we then plug in -x next; so -x will go into where we have x, so we will get √(-x - 2).1599

That gives us the function g(x) = √(-x - 2), which would look like this graph right here.1611

And that makes sense, because I think this is the direction of right for you (for me, it is my left, but oh well).1617

If we have a square root going out like this, and then we pick it up and we move it over,1625

well, the middle is still here; it used to be coming directly out, but we picked it up, and we moved it over.1630

So, when we flip it, it is going to be a farther distance over now, and going out in the opposite direction.1634

And that is what we see here on this red graph.1639

We see that it is away from the y-axis, because it moved away from the y-axis, and then it flipped.1642

It turned all the way over; it basically grabbed it like a pole and spun to the other side.1647

So, now it is 2 away, but -2.1651

What if we did flip and then move? If we did flip and then move, the first thing we would do is plug in the -x.1655

So, -x would go in first; and then, into that x, we would plug in x - 2.1662

x - 2 goes in there; so we have x - 2, which is now replacing the x, because we are plugging into the function.1668

We are just plugging inside of that x; remember, it is just a placeholder.1676

It doesn't really mean that we have just x allowed to be there; it is just a placeholder for f(_) = √_.1681

So, if we want to flip horizontally, we plug in -x; if we want to move units right, we move x - 2.1688

Flip, then move: we get √-x; and then, the next thing--we put in the move, so we plug into that x an x - 2 instead of x.1694

So, we have -(x - 2); that gives us the function h(x) = √(-x + 2), because the negative cancels out the minus sign.1702

That is going to start at positive 2, and then move off to the right.1713

It starts at positive 2 and moves to the right; and once again, this makes sense.1717

We have this sort of centerpost of the y-axis, and it moves off to the right--the normal square root moves off to the right.1719

So, if we start by flipping it so that it is going this way, and then we move it two units to the right, we are going to still being going right of that y-axis.1726

We will move right of that y-axis.1735

So, when we move and then flip, we move this way, but then we flip into this location.1739

But when we flip and then move, we start going this way, but then we flip into back this way; but then we move after that.1745

So, that is why we are seeing two totally different things; so the order we put the transformation on...1754

we get totally different answers; so order really matters with our transformations.1757

All right, let's start working some examples.1761

We want to give three transformations of f(x) = x3 - 2x + 2: first, we shift it down by 5.1763

So, we will do this in red: shift it down by 5; remember, f(x) - 5...f(x) + k, so if we want to go down 5, it is -5 that we plug in for k.1771

So, f(x) - 5 will give us some new function; let's name this g(x) is equal to...1786

I will rewrite it, so we can see more easily what is going on.1797

g(x) = f(x) - 5, which would be x3 - 2x + 2, our normal function, and then just - 5.1800

So, we have g(x) = x3 - 2x - 3 (two minus five); and that is what we get for shifting down by 5.1813

If we want to shift right by 2, then let's make a new function; we will call this one h(x), and h(x) is going to be equal1826

to f, our original function, shifted right by 2; we do that by x + k; going to the right causes a negative k,1835

because we have to get that 0 to show up on the right now.1843

So, x + k is x - 2, since we are shifting to the right; that equals...we plug in for our old x.1847

Our old placeholder is now replaced by x - 2; so (x - 2)3 - 2(x - 2) + 2.1856

And if we wanted to, we could expand and simplify; but that is really not what the point of this lesson is about.1865

Expanding and simplification--I am pretty sure you can handle that.1874

And if you can't, we will have other lessons where we are doing that more carefully in polynomials.1876

We could expand if we wanted to--if we had to for the problem.1881

Then finally, in green, we shift it left by 1, then up by 4, then flip vertically.1885

This is the most complicated one of all: we are going to start by using g and h, and then finally we will make it to k.1891

We will treat each of these as one function after another.1897

We start at f(x) = x3 - 2x + 2.1900

All right, now we are going to have a transformation that is going to be left by 1.1906

We do that by plugging in x + 1, positive k, to move left (if we are plugging in horizontally).1914

So, we have a new function, g(x), that is equal to f plugged in...x + 1; and this is what happens if we just shift to the left.1926

It equals (x + 1)3 - 2(x + 1) + 2; great.1934

We could expand if we wanted to; we are not going to worry about that right now.1942

Next, we move up by 4; so up by 4 is function + 4.1946

Let's call it a new function; so it is h(x), and now we are doing this to g(x), so h(x) = g(x) + 4.1956

And since g(x) was equal to f(x) + 1, it is f(x) + 1 + 4, so we have what we had before, (f(x) + 1)3 - 2(x + 1) + 2.1964

Great; the final one--we are going to name this function k; and now it is a flip vertically.1979

It is a negative version of the function--just multiplying the function by -1.1987

So, k(x) is the vertical flip of h(x); remember, this has to happen in order, so we are doing it to h(x).1993

So, it is -h(x); now, what was h(x)? Well, h(x) is negative quantity..what did we have for h(x)?2000

We had g(x) + 4; so it is -(g(x) + 4); and then, what was g(x) + 4?2008

That was -(f(x) + 1 + 4); so we get, finally, -((x + 1)3 - 2(x + 1)...2019

oops, on the very one above that--sorry about that--I forgot to add on the 4; so we get + 2 + 4, or + 6; sorry about that one.2039

So, plus 6; so it is - 2(x + 1 + 6); great.2048

And there we are; we could continue to expand that if we wanted to; but that is ultimately what it is going to be.2055

We would have to expand a bunch of things--expand the cube in there, and then distribute out that negative sign.2059

But that is pretty much what is going on; we just now have to do that all in the order that we are supposed to.2065

But it is important that we do it in the order of shift to the left by 1, then the shift up by 4, then flip vertically.2069

If we break that order, it is not going to end up working out; we are going to get a different answer.2077

Great; all right, the next example: we have a parent function of cube root of x, and that is this one on the left.2081

Now, we want to give the function for the graph on the right.2090

We need to figure out what things happen to the graph on the right.2093

So, the first thing it looks like to me is...notice how we have that the right side is down and the left side is up.2096

The way we would do that is: at first, we have a vertical flip; and what comes after that?2104

The "home base" moves; where was it originally?2111

It was originally at (0,0); we will make a home and say, "That seems like a reasonable place to say where its home used to be."2119

Its home used to be at (0,0); and over the course of becoming the second graph, it goes to...where is its home?2125

Here is (2,1); so it is at (2,1); so the home moves to (2,1), which means that we have two things coming out of this.2132

We have a shift right by 2, and a shift up by 1.2142

So, we can come up with a function for this by just applying these transformations.2154

First, we have f(x)...let's do our first transformation, g(x) =...the vertical flip is -f(x) = cube root of x.2160

This is our first part; but next, we have the second part that comes in.2170

The second part...we will call this one h(x), so that will be what our final function actually is.2176

h(x) equals...we have shifting right; shift right happens by x - 2; remember, it is x + k, but we put in negatives to shift to the right.2180

Shift up is just our function + 1; those two we can actually put in in any order; they will never end up interacting with each other,2189

since the + 1 happens completely outside of the function--outside of everything.2198

So, -f(x)...we have here that h(x) is equal to g; plugging in x - 2, and also shifting it up by 1--2203

now, if g(x) is equal to -f(x), we have -f(x); g(x) becomes f(x), so this here is just the same thing as -f(x).2214

So, it is going to be -f(x) is 3√x (oops, it should be a negative sign there).2236

So, g(x - 2) is -f(x - 2) + 1, which is equal to -3√(x - 2) + 1.2244

So, the function that we are seeing over here is h(x) = -3√(x - 2) + 1; great.2276

For this example, we want to give the parent function, f, and what transformations were applied, and the order they were applied in to create g.2292

So, g(x) = 7 - x3 for the first one; so the first one...we will do it in red; what is the parent function that makes this up?2299

Well, the parent function for g(x) in red is going to be f(x) = x3.2307

We see that x3 there, so it seems reasonable that that is going to be it.2314

What had to happen to be able to get 7 - x3?2317

Well, the first thing that had to happen is a vertical flip, and then a vertical shift up.2320

We could also have a vertical shift down, and then a vertical flip of everything.2325

But it is easier to see it as a vertical flip, and then the second thing as shift up by 7.2329

And that is why we get -x3 from the vertical flip, and then shift up 7 will be -x3 + 7; so we get 7 - x3.2341

And that is how we get our red g(x).2349

Now, our blue g(x), 10√(x + 5); this f(x) will start from the basic function of √x.2353

That is our fundamental function; so what has happened in here?2361

Let's say that the first seems like it is easier to move horizontally than to have to do a vertical stretch first.2364

And actually, it won't matter what order we do it in.2372

But let's say we will do it in first order of shift; so we have x + 5 in there.2374

So, x + 5...x + k means that it is a positive, so shift by 5 left.2382

It is a horizontal shift, and it goes to the left, because it is a positive k.2391

And then, the second thing we do is multiply the entire function by 10; so it is a vertical stretch by a = 10.2396

We actually could do that in the completely opposite order; we could do vertical stretch by a = 10,2409

and we get 10√x, and then plug in x + 5; and we get 10(√x + 5).2413

So, 1 or doesn't actually matter which one goes first, unlike the red function, which actually...2418

it did matter that we had a vertical flip, and then shifted up by 7.2424

We would have to do a slightly different thing if we wanted to do it in a different order for the red function.2426

But the blue function--anything works.2430

Finally, the green function: its basic parent function, what is creating it, its base function, is |x|.2431

Now, for this one, it is a little bit harder to see which one this has to be.2440

It is much easier to start by shifting to the right by 3, because we see right here this 2x.2444

So, 2x means that how the x-axis is being affected is that it has been "sped up."2450

But if we speed up, and then move by a different thing...we are used to moving;2455

all of our theory about moving is based off of "move first"; our theory of moving how we experience the x-axis2459

was all done on the principle of it starting as x, and not starting as 2x or 5x or 1/2x.2466

It was all based on x + k, not 2x + k.2474

So, we want to start by shifting--doing our shifting--dealing with that first.2477

We will shift right by 6; and we know it is to the right, because we have a -6.2482

Now, here is actually the thing: it is not by 6; this is a confusion.2491

It seems to be 6 at first, because of that + k; but notice what is really there.2495

2x - 6...once again, we have things in the form x + k; 2x - 6 is not in the form x + k, because it has 2x; it is not just 1x.2500

So, we have to get it to 1x first; so we pull out the 2, and we get 2 times (x - 3).2512

The shift to the right is actually by this 3 here; so we shift right by 3, because we have k at -3.2520

And then, our second one is a horizontal speed-up by a = 2, which is to say it will squish to half of its original horizontal length.2528

Any horizontal interval will squish to half.2545

And then finally, we have this + 1 here; so it shifts up by 1.2548

Now, it actually turns out that the 3 could be at the top, or it could be at the bottom; it doesn't matter.2555

Shifting up by 1--that could happen at the very beginning; it could happen at the very end.2561

But because of the shift right and the horizontal speed-up, we have to have it in this x + k form.2564

We can't get it out of 2x - 6, because it is 2x + k; that is not the same form.2570

We have to have it as x + k; so we have to pull that 2 out first.2575

There is a way where we could have the horizontal speed-up go first, and then shift.2578

But it is much easier to think in terms of the shift right, and then the horizontal speed-up.2581

And if that one seems a little confusing, I wouldn't worry about it too much.2585

That is probably the absolute hardest kind of question of this type that you would ever see, at least for the next couple of years,2587

until you are in college--or if you are not just in college, but taking an advanced-level math class in college.2593

So, don't really worry about this right here; this is a fairly difficult kind of problem.2599

But this is the sort of thing you want to be thinking about it with.2605

You want to be thinking in terms of "What do I have to do here if I am following that formula table--if I am following that table?"2607

And you have to follow it carefully; what does it have to fit in?2612

It has to fit in things of the form x + k; and you notice, 2x is different--it is not in that same form.2615

So, you have to get it into that form before you can use these things that we talked about before.2621

All right, the final example: How is vertically stretching the graph of f(x) = x2 the same as horizontally stretching it?2625

Remember, a vertical stretch is done by a times f(x); a horizontal stretch is done by f(a times x).2632

Now, I think it is a little bit confusing to use a in two places.2656

So instead, we are going to call this b; so we say we are just using b to prevent confusion.2660

Don't worry about the fact that it is not what we were seeing before; it is not the same a multiplicative factor that we saw before.2670

It means the exact same thing; a and b are both just constants.2675

So, a and b are multiplicative constants; they are just how much we are stretching by--2680

whether it is a horizontal stretch or it is a vertical stretch--it is just how much we are stretching by.2693

All right, so let's see how this works on f(x) = x2.2698

a times f(x)...that is going to be a times x2; f(bx)...that means b times x will plug in, instead of the x;2702

so we will get bx; it is the quantity, squared.2710

So, over here we have ax2 and b2x2, because the "squared" will get put onto both of them.2714

So, we have these two things; how is it that they are the same?2724

Well, how is it that they are similar--what is the connection between them?2728

Well, think about this: a is just a constant, and b is just a constant.2733

a and b are both constants; but if b is a constant, then that means that b2 is also just a constant.2738

If b is 4, then b2 is just 16; so it will be a larger constant than b, but it is still just a constant.2751

It is not allowed to vary around; so b2 is also a constant.2758

So, what that means here is that in either case, whether it is a vertical stretch or a horizontal stretch,2767

it just has the effect of multiplying x2 by a constant.2777

So, what we are seeing here: the reason why, if we do a vertical stretch, and we do a horizontal stretch,2796

and if you go back and you look at what you saw when we saw a vertical stretch example and we saw the horizontal stretch example...2802

you will notice that they actually looked basically the exact same.2808

There were slight differences, but it is the same sort of stretching going on,2811

because when we compress it horizontally, it causes it to just sort of squirt up vertically.2814

And when we stretch it out vertically, it is the same thing as if we had compressed it horizontally.2819

So, in either case--whether it is a vertical stretch or a horizontal stretch--it is just the same thing as multiplying by a constant.2822

So, that is why they are so similar.2827

We, in actually, I think, all of the fundamental functions that we are used to using by this point...2830

all of them are already things where this is just the horizontal and the vertical will end up having the same effect.2836

It will do it by different amounts; but ultimately, it is just putting a constant into the mix--multiplying things by a constant.2843

The first time that you will end up seeing things--and right now, if you have even taken trigonometry,2850

the only thing you would see where you would be able to see the difference between a horizontal and a vertical stretch--2854

is trigonometric functions; if you look at sine and cosine, it actually is possible to horizontally stretch those2859

and vertically stretch those, and you will get totally different-looking things out of a vertical stretch versus a horizontal stretch.2864

Now, it is OK that we haven't really talked about trigonometric functions yet.2870

And you haven't seen them yet, probably; don't worry about that--that is OK.2873

And if you have seen trigonometric functions, or you have taken some trigonometry, that is all the better.2876

You probably are already exposed to this.2880

But just know that later on, you will see cases where there is a difference between horizontal stretch and vertical stretch.2882

But for some other functions, like f(x) = x2, it ends up being that there is not really a difference at all.2888

All right, I hope you now have a good understanding of all of the different transformations that are available to us.2892

I know that there are a lot of them; but if you think through what you are doing with each one,2896

you can probably figure it out without even having to resort to the table.2899

These are really useful, because they let us build a bunch of different functions2902

and understand how to graph functions that seem complex at first,2905

but are really just some basic function we are used to graphing, that has been stretched and squished and moved around.2908

All right, we will see you at later--goodbye!2913