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 talk about transformations of functions.
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By now, we are familiar with a variety of different functions--things like x, x², √x, |x|, etc., etc.
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We have seen a bunch of different fundamental parent functions--that function petting zoo we visited in the last lesson.
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However, we often have to work with or graph functions that are similar to these fundamental functions, but not precisely the same.
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They are not the ones we are already familiar with, exactly.
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Many times, the difference is the result of a transformation.
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A **transformation** is a shift, a stretch, or a flip of a function.
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And when I say that, I mean that graphically; it has been moved either left/right or up/down;
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it has been stretched either horizontally or vertically; or it has been flipped vertically or horizontally.
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Understanding transformations is useful for working with functions or building our own functions, if we want to build one from scratch.
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For this lesson, we will begin by looking at vertical shifts and stretches, because they are the easiest ones to graph.
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It is easiest to understand moving things around vertically.
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Then, we will learn how horizontal shifts and stretches work, and what we have understood in vertical
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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!
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The very first one is vertical shift; this is the easiest one of all--to shift vertically, you simply add to a function.
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Positive numbers shift up; negatives shift down; so let's see an example.
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Consider f(x) = x² graphed with f(x) + 2 = x² + 2 and f(x) - 7, which equals x² - 7.
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f(x) = x², our base function, is the red part of our graph.
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We want to see what f(x) + 2 is; that is the blue graph; and finally, the green graph is f(x) - 7.
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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.
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But if we take the function and subtract 7, f(x) - 7, which would be x² - 7, it goes down by 7.
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It goes lower off its base of normally starting at (0,0); x² has its home base, in a way, at (0,0).
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It drops down by 7; or we can raise it up; so we can move it up and down with this vertical shift.
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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.
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So, let's say, hypothetically, we want to change the point (1,2); we want to move it up by 5.
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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).
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That is what we would get if we wanted to move it up by 5.
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To move up one point, we would just add k or subtract k; but let's think of it as adding a negative k.
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We add k, and if k is positive, it goes up; if k is negative, it goes down; that moves one point up.
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If we want to move all of the points in a graph, then we have to add this to all of the vertical components.
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Now remember: the vertical components of a function's graph are the function's output.
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So, the function's output...we just need to make the output be k more, or k less, if it is a negative number.
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So, we add k to it; we just change the output everywhere by adding k to the function.
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So, if we start with f(x), and we want to vertically shift by f, we just use f(x) + k.
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If k is positive, it moves up; if k is negative, it moves down--as simple as that.
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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.
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Next, the vertical stretch/shrink: we want to vertically stretch or shrink a graph--we want to pull it/stretch it,
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or we want to shrink it--we want to squish it.
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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.
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If a > 1, the graph stretches.
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Let's see an example: consider f(x) = x²; that is the one in red--that is our basic function,
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that we are starting with to have a sense of how things are going to go.
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And then, we compare that to 3 times f(x): it has a multiplicative factor of 3 hitting it.
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So, in this case, we have the graph stretching, because it is 3 times f(x); so a is greater than 1.
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And indeed, this one right here has been stretched up; we have the parabola normally,
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but we have grabbed it and pulled it up higher.
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If we look at this, every point here is 3 higher; here is 1, 2, 3 higher to get up to there.
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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.
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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.
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It has been squished by a factor of 1/3.
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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.
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We will talk about negatives later.
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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.
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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.
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Its height is just 2; so if a = 1/2, then we get (1,2(1/2)), so we get (1,1).
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We have halved the point's height; if we were at (1,10), it would become (1,5).
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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.
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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.
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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.
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To do it to all of the function's output, we need to multiply the function itself by a.
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So, that will apply it to all of the outputs, because the function tells us where the outputs go.
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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.
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So, to stretch/shrink a function by a multiplicative factor, a, we use a times our function, a times f(x).
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If a is greater than 1, the function stretches; if a is between 0 and 1, it shrinks.
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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.
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We are exactly on the middle, and we are just left with the function as it was before; it has no effect.
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We have grabbed it, and then just immediately let go, instead of pulling it and pushing it together.
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Horizontal shift: this is a little more complex than vertical shift, but we are ready to talk about this now.
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To shift horizontally, we need to change where the function sees x = 0.
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We do this by plugging something different than x into the function.
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What I mean by that: take, for example, a normal line, a normal f(x) = x kind of line.
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Well, if we wanted, we could say that in a way, its home base is this 0 point where it crosses that axis.
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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.
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What has happened there is: we have taken this home base, and we have shifted it over by some amount.
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So, we have the same picture, but it has been moved to the right.
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We are seeing the home base, the effective x = 0, in a different place; that is the idea we want to bring to this.
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Consider f(x) = x² (once again, we see that in red) graphed with f(x) - 4 = (x - 4)².
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That one is in blue; it has been shifted four units to the right.
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And f(x) + 2...that one is in green: (x + 2)² has been shifted two units to the left.
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Now, that might seem counterintuitive at first: -4 causes us to shift to the right, but + 2 causes us to shift to the left.
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To understand what is going on, we need to think about this a little more deeply.
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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.
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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.
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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.
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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.
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A graph shows how the function is seeing all of the x-axis at the same time.
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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.
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1, 2, 3; -1, 2, 3; it is exactly what we are used to; it is your normal number line.
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But we could move this number line around; we can move this home base.
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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.
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How do we move around that home base? We move it around like this.
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Normally, once again, our normal plane, x--the normal number line--we can move it around by plugging in x + k.
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For example, if k = -2, then our normal home base now is x - 2, so it is 0 - 2; and we get -2.
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1 - 2...we get -1; 2 - 2...we get 0; look, our home base has shifted 2 units to the right.
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We have gone over 1, 2 clicks over to the right.
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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.
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But what we have gotten by doing this is: we have effectively shifted over the location of home base.
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We have effectively shifted over the location of x = 0 by subtracting 2 from everything.
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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.
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So, I want to say this once again: The graph shifts to the right with a negative.
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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,
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because we are taking that much away from all of the numbers.
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And so, if we are taking them away, it has to be the high numbers, the traditionally right-side numbers,
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that are now going to have our new home base in them.
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Shifting to the right: horizontal shifts to the right happen with a negative value for k.
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What if we wanted to shift to the left? Well, we could plug in a different x + k.
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If we plug in k = 3, then if we have + 3, we now have 3 for our old 0.
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We have to go to -3 for us to get our home base back.
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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.
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And everything is going to get hit by this + 3, which is why we have 6 over here and all these sorts of things.
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So, by adding a positive number, it is the lower numbers, the negative numbers,
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that are now going to end up taking the place of having the home base be on that side.
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The home base will move to the left if we have a positive k.
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If we want to shift to the left, we use positive for k in that x + k; great.
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Therefore, we can shift around where the function sees x = 0...this idea of seeing x = 0,
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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.
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By shifting around the perceived x-axis, this perceived home base, the graph will move horizontally (move to the left/move to the right).
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Now, notice: this doesn't actually change the x-axis; it just changes the way the function sees it.
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The x-axis is still going to look totally normal to us; it will be that normal x-axis that we are used to.
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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.
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So, to horizontally shift the function by k units, we use f(x + k).
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And if k is a positive, we shift left; if k is a negative, we shift right.
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Now, remember: that seems a little bit counterintuitive at first; but if you think about why it is the case--
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if we put in a positive number, it has to be the negative side that establishes the new home base, the new 0--
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if we put in a negative number, then it has to be the positive side, the right side,
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that establishes the new home base, the new 0--that seems a little counterintuitive,
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but if you think about those slides, those ideas we just saw, you will think, "oh, yes, it makes sense
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that I am plugging in the positive to go left and plugging in the negative to go right."
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All right, horizontal stretch: this idea is similarly complex.
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But now that we understand how horizontal shifting works, this will probably make more sense.
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To horizontally stretch or shrink a function--that is, to pull it apart or to squish it together--
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we need to change how fast the function sees the x-axis.
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Once again, it is not really, literally seeing it; the function isn't a living, breathing thing.
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But the way that it is going to interact with it, we can effectively personify it and pretend that it is alive for this.
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Effectively, we need to stretch/shrink how the function perceives x.
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We need to change the way that the function will interact with that x-axis.
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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.
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We have put that multiplicative factor on the x, and we get (3x)² on our blue graph.
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We plug in 1/2 times x...(1/2x)² on our green graph.
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All right, so let's understand what is going on here--what is going on?
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This idea is very much like what we did with horizontal shift.
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We are playing with how the function sees the x-axis.
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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.
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Great, that is just like normal.
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But we can speed this up; we can speed up or slow down the experience of this number line.
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If we plug in an x-axis that has been stretched or shrunk by a multiplicative factor, a;
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if we have a times x--for example, if we have 3 times x--we speed it up.
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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.
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But they have gotten shrunk down; we are speeding up how fast we are moving through the numbers.
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So, each number effectively has been multiplied by 3; we are moving through the numbers faster, which will condense the graph.
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The graph will be happening faster horizontally, so it is going to go through what it would do normally, faster.
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We can also expand it by slowing it down with a small a.
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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.
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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.
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We are slowed down by a factor of 2.
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OK, by applying a multiplicative factor, a, to x, we can change how fast the x-axis looks to the function.
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This change in horizontal speed either stretches or shrinks the graph horizontally.
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If it looks faster, the graph will compress, because it has the same amount of things happening in a shorter amount of x-time.
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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.
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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.
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So, to horizontally stretch/shrink a function by a factor of a, we use f(ax); we play with how that x-axis works.
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We change around the speed that that x is moving at.
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So, if a is greater than 1, it will shrink horizontally, because we have sped up.
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So, a > 1: it shrinks horizontally, because it is speeding up how fast the x-axis goes.
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We want to think about that in terms of speeding up; it makes it easier to understand what is going on.
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So, if we speed it up by putting in a large a, a > 1, we are going to shrink horizontally,
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because more stuff will happen in the same period of "time."
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0 < a < 1: we are going to stretch horizontally; it will slow down the x-axis,
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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.
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a > 1 shrinks horizontally; it speeds up; if a is between 0 and 1, it stretches horizontally; it slows down.
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All right, vertical flip, which we might also call mirroring vertically, or a vertical mirror:
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to vertically flip a graph around the x-axis, we simply multiply the function by -1; it is as simple as that.
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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.
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-f(x), which is -x²...it flips to pointing in the opposite direction.
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What is going on here? Well, if we wanted to flip a single point around the x-axis--say we have (1,2) again--
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if we wanted to flip it around the x-axis to the opposite height, then we would just multiply the vertical component,
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the y-value, by -1; so vertical times -1 would become (1,2(-1)), or (1,-2).
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We have flipped that point to the opposite vertical location, the opposite height.
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This sends it to the opposite side; but it still has the same height in terms of distance from the x-axis.
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It is now a negative height; or if it started negative, it will now be positive.
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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.
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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.
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The vertical components are the outputs of the function, so we need to make the function output the negative version everywhere.
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We do this by just multiplying the whole function by -1.
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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;
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it is flipped vertically; so to flip a function vertically around the x-axis, we use -f(x); great.
00:18:43.900 --> 00:18:49.900
To horizontally flip a graph around the y-axis, we change how the function sees the x-axis to its opposite.
00:18:49.900 --> 00:18:56.900
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.
00:18:56.900 --> 00:18:58.800
We are going to do that for horizontal flip.
00:18:58.800 --> 00:19:05.300
We do this by plugging in -x; let's see an example--consider f(x) = √x.
00:19:05.300 --> 00:19:10.500
If we have √x, and that is the red graph (we couldn't use x², because its horizontal flip
00:19:10.500 --> 00:19:16.700
will just look like the exact same thing); we will graph that with f(-x).
00:19:16.700 --> 00:19:22.400
So, we plug in -x, and we will get -√x, which ends up pointing in the exact opposite direction.
00:19:22.400 --> 00:19:24.000
Why is it pointing in the exact opposite direction?
00:19:24.000 --> 00:19:37.500
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.
00:19:37.500 --> 00:19:45.200
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.
00:19:45.200 --> 00:19:53.700
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.
00:19:53.700 --> 00:19:57.200
All right, what is going on here--how is this working?
00:19:57.200 --> 00:19:59.700
We are reversing how the function sees the x-axis.
00:19:59.700 --> 00:20:04.800
Normally, once again, we see x going off to the right and to the left, just like usual.
00:20:04.800 --> 00:20:10.500
But if we plug in -x, it reverses; let's put some color here, so we can see what I am talking about.
00:20:10.500 --> 00:20:16.700
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,
00:20:16.700 --> 00:20:25.000
when we plug in -x, we see that it goes to the right in blue, and the left in red.
00:20:25.000 --> 00:20:36.400
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.
00:20:36.400 --> 00:20:43.800
As opposed to going from negative to positive, it now goes from positive to negative; we have flipped the order that it occurs in.
00:20:43.800 --> 00:20:48.900
To flip a point horizontally around the y-axis, we need to just multiply the horizontal component by -1.
00:20:48.900 --> 00:21:00.300
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.
00:21:00.300 --> 00:21:07.800
So, we would go to (-1,2); so this will move the point to the opposite horizontal location.
00:21:07.800 --> 00:21:12.100
If we plug -x into a function, it reverses how it sees the x-axis throughout.
00:21:12.100 --> 00:21:18.900
So, we will be plugging in opposite horizontal locations everywhere; so everything will flip to the opposite horizontal location.
00:21:18.900 --> 00:21:24.000
All of the points are going to show up in the opposite horizontal location, because we have plugged in this -x.
00:21:24.000 --> 00:21:29.500
To flip a function horizontally around the y-axis, we use f(-x).
00:21:29.500 --> 00:21:35.500
What does that mean? Once again, say we have some smiley-face over here.
00:21:35.500 --> 00:21:44.300
Smiley-face, sadly, doesn't have anything left-right; he is a perfect left/right thing.
00:21:44.300 --> 00:21:50.700
So, let's make smiley-face--it is now Ms. Smiley-face, and Ms. Smiley-face has a little bow.
00:21:50.700 --> 00:21:58.800
So, if we flip her around the y-axis--we flip her horizontally--she will show up on the other side.
00:21:58.800 --> 00:22:05.200
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.
00:22:05.200 --> 00:22:12.000
All right, so she shows up on the opposite side now; she has been mirrored horizontally around the y-axis.
00:22:12.000 --> 00:22:16.500
Here is a summary of transformations; I know it is a lot of transformations that we have seen at this point.
00:22:16.500 --> 00:22:19.900
So, don't worry if you have to refer back to this list later on.
00:22:19.900 --> 00:22:23.900
Also, if you currently have some sort of book that you are working on along with this course in,
00:22:23.900 --> 00:22:26.700
or if you have another teacher who is working in a book, you are almost certainly
00:22:26.700 --> 00:22:32.100
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.
00:22:32.100 --> 00:22:36.100
This table is really useful, because it can be a little hard to remember all of them immediately.
00:22:36.100 --> 00:22:42.300
Vertical shift is f(x) + k; k is positive; that causes us to go up; k is negative--that causes us to go down.
00:22:42.300 --> 00:22:49.500
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.
00:22:49.500 --> 00:22:57.800
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.
00:22:57.800 --> 00:23:04.600
Horizontal stretch is...0 to 1 means we slow down, and slowing down means we stretch out.
00:23:04.600 --> 00:23:09.700
a greater than 1...did I say horizontal stretch?...f(a) times x...I am not quite sure I said that...
00:23:09.700 --> 00:23:13.700
a greater than 1 causes us to go faster, which means we squish together.
00:23:13.700 --> 00:23:22.300
Vertical flip: we flip over vertical; that is -f(x); horizontal flip, f(-x), causes us to flip horizontally.
00:23:22.300 --> 00:23:34.900
One thing to notice: all the vertical stuff happens outside the function.
00:23:34.900 --> 00:23:47.000
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).
00:23:47.000 --> 00:23:50.000
Everything is doing it on the outside of the function.
00:23:50.000 --> 00:23:59.700
However, horizontal things happen inside: horizontal shift is where you plug in x + k.
00:23:59.700 --> 00:24:07.100
x + k goes into the function; horizontal stretch is a times x, which goes into the function.
00:24:07.100 --> 00:24:11.800
Horizontal flip is where -x goes into the function.
00:24:11.800 --> 00:24:16.800
So, horizontal things will happen inside the function; it happens to what we are plugging into the function,
00:24:16.800 --> 00:24:21.500
whereas vertical things happen on the outside of the function--we don't have to worry about it being plugged in.
00:24:21.500 --> 00:24:24.800
OK, that is a summary of transformations; don't worry if you have to refer to this.
00:24:24.800 --> 00:24:29.400
But you can also probably think about this sort of thing, now that we have an understanding of where this stuff is coming from.
00:24:29.400 --> 00:24:34.500
You can probably actually figure out that it makes sense, and just re-figure it out, re-derive it, in your own head,
00:24:34.500 --> 00:24:36.500
without even having to refer to these lists.
00:24:36.500 --> 00:24:40.100
Horizontal shift, horizontal stretch, horizontal flip--they might be a little bit more difficult.
00:24:40.100 --> 00:24:45.600
But remember that idea where we are shifting around our home base--we are shifting around the experience of the x-axis.
00:24:45.600 --> 00:24:48.300
That is what we are shifting with those.
00:24:48.300 --> 00:24:55.000
Stacking transformations: if you want to do multiple transformations on one function, you just apply one transformation after another.
00:24:55.000 --> 00:25:04.600
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.
00:25:04.600 --> 00:25:10.400
It doesn't matter what order I multiply them in; but in transformations, it matters what order you put the transformations on in.
00:25:10.400 --> 00:25:16.300
Be careful: the order you apply your transformations in can affect the results.
00:25:16.300 --> 00:25:21.400
The order you apply will affect how it comes out--not always, but a lot of the time.
00:25:21.400 --> 00:25:28.500
Decide on the order you want before you do it; decide on the order, then apply them to your base function in that order.
00:25:28.500 --> 00:25:35.300
The order that they hit that base function--the order that the do their transformations in--will change what happens.
00:25:35.300 --> 00:25:38.100
There are some cases where it won't matter what order you put it on in.
00:25:38.100 --> 00:25:41.800
But other times, you are going to get totally different results; and we will look at an example in just a second.
00:25:41.800 --> 00:25:45.500
This means you have to think about order when doing multiple transformations.
00:25:45.500 --> 00:25:48.800
So, make sure you think about order if you are doing multiple transformations,
00:25:48.800 --> 00:25:53.000
because if you don't think about it, you can really get confused and get completely the wrong answer.
00:25:53.000 --> 00:25:58.400
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.
00:25:58.400 --> 00:26:01.400
For example, let's consider f(x) = √x.
00:26:01.400 --> 00:26:07.600
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.
00:26:07.600 --> 00:26:12.700
All right, let's say we want to move it two units right and flip it horizontally.
00:26:12.700 --> 00:26:19.000
We move it two units right by plugging in x - 2 into where we have x.
00:26:19.000 --> 00:26:23.000
And we flip horizontally by plugging that -x in here as well.
00:26:23.000 --> 00:26:28.500
That is how we do the two different things: we plug in -x into the function, or we plug in x - 2.
00:26:28.500 --> 00:26:32.400
Look at how order matters; we get very different things, depending on the order we put this in.
00:26:32.400 --> 00:26:39.600
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).
00:26:39.600 --> 00:26:50.900
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).
00:26:50.900 --> 00:26:57.500
That gives us the function g(x) = √(-x - 2), which would look like this graph right here.
00:26:57.500 --> 00:27:05.000
And that makes sense, because I think this is the direction of right for you (for me, it is my left, but oh well).
00:27:05.000 --> 00:27:10.100
If we have a square root going out like this, and then we pick it up and we move it over,
00:27:10.100 --> 00:27:14.200
well, the middle is still here; it used to be coming directly out, but we picked it up, and we moved it over.
00:27:14.200 --> 00:27:19.400
So, when we flip it, it is going to be a farther distance over now, and going out in the opposite direction.
00:27:19.400 --> 00:27:21.700
And that is what we see here on this red graph.
00:27:21.700 --> 00:27:26.900
We see that it is away from the y-axis, because it moved away from the y-axis, and then it flipped.
00:27:26.900 --> 00:27:31.600
It turned all the way over; it basically grabbed it like a pole and spun to the other side.
00:27:31.600 --> 00:27:35.100
So, now it is 2 away, but -2.
00:27:35.100 --> 00:27:41.900
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.
00:27:41.900 --> 00:27:48.100
So, -x would go in first; and then, into that x, we would plug in x - 2.
00:27:48.100 --> 00:27:56.000
x - 2 goes in there; so we have x - 2, which is now replacing the x, because we are plugging into the function.
00:27:56.000 --> 00:28:00.800
We are just plugging inside of that x; remember, it is just a placeholder.
00:28:00.800 --> 00:28:08.500
It doesn't really mean that we have just x allowed to be there; it is just a placeholder for f(_) = √_.
00:28:08.500 --> 00:28:14.600
So, if we want to flip horizontally, we plug in -x; if we want to move units right, we move x - 2.
00:28:14.600 --> 00:28:22.600
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.
00:28:22.600 --> 00:28:32.800
So, we have -(x - 2); that gives us the function h(x) = √(-x + 2), because the negative cancels out the minus sign.
00:28:32.800 --> 00:28:36.700
That is going to start at positive 2, and then move off to the right.
00:28:36.700 --> 00:28:39.100
It starts at positive 2 and moves to the right; and once again, this makes sense.
00:28:39.100 --> 00:28:46.000
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.
00:28:46.000 --> 00:28:55.400
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.
00:28:55.400 --> 00:28:59.200
We will move right of that y-axis.
00:28:59.200 --> 00:29:05.200
So, when we move and then flip, we move this way, but then we flip into this location.
00:29:05.200 --> 00:29:13.800
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.
00:29:13.800 --> 00:29:17.500
So, that is why we are seeing two totally different things; so the order we put the transformation on...
00:29:17.500 --> 00:29:21.500
we get totally different answers; so order really matters with our transformations.
00:29:21.500 --> 00:29:23.500
All right, let's start working some examples.
00:29:23.500 --> 00:29:30.700
We want to give three transformations of f(x) = x³ - 2x + 2: first, we shift it down by 5.
00:29:30.700 --> 00:29:45.900
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.
00:29:45.900 --> 00:29:56.700
So, f(x) - 5 will give us some new function; let's name this g(x) is equal to...
00:29:56.700 --> 00:30:00.000
I will rewrite it, so we can see more easily what is going on.
00:30:00.000 --> 00:30:13.400
g(x) = f(x) - 5, which would be x³ - 2x + 2, our normal function, and then just - 5.
00:30:13.400 --> 00:30:26.100
So, we have g(x) = x³ - 2x - 3 (two minus five); and that is what we get for shifting down by 5.
00:30:26.100 --> 00:30:35.400
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 equal
00:30:35.400 --> 00:30:42.700
to f, our original function, shifted right by 2; we do that by x + k; going to the right causes a negative k,
00:30:42.700 --> 00:30:46.700
because we have to get that 0 to show up on the right now.
00:30:46.700 --> 00:30:56.400
So, x + k is x - 2, since we are shifting to the right; that equals...we plug in for our old x.
00:30:56.400 --> 00:31:05.300
Our old placeholder is now replaced by x - 2; so (x - 2)³ - 2(x - 2) + 2.
00:31:05.300 --> 00:31:13.700
And if we wanted to, we could expand and simplify; but that is really not what the point of this lesson is about.
00:31:13.700 --> 00:31:16.400
Expanding and simplification--I am pretty sure you can handle that.
00:31:16.400 --> 00:31:21.300
And if you can't, we will have other lessons where we are doing that more carefully in polynomials.
00:31:21.300 --> 00:31:25.500
We could expand if we wanted to--if we had to for the problem.
00:31:25.500 --> 00:31:31.200
Then finally, in green, we shift it left by 1, then up by 4, then flip vertically.
00:31:31.200 --> 00:31:36.800
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.
00:31:36.800 --> 00:31:40.000
We will treat each of these as one function after another.
00:31:40.000 --> 00:31:46.500
We start at f(x) = x³ - 2x + 2.
00:31:46.500 --> 00:31:53.900
All right, now we are going to have a transformation that is going to be left by 1.
00:31:53.900 --> 00:32:05.900
We do that by plugging in x + 1, positive k, to move left (if we are plugging in horizontally).
00:32:05.900 --> 00:32:14.200
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.
00:32:14.200 --> 00:32:21.900
It equals (x + 1)³ - 2(x + 1) + 2; great.
00:32:21.900 --> 00:32:26.100
We could expand if we wanted to; we are not going to worry about that right now.
00:32:26.100 --> 00:32:36.500
Next, we move up by 4; so up by 4 is function + 4.
00:32:36.500 --> 00:32:44.200
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.
00:32:44.200 --> 00:32:59.000
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.
00:32:59.000 --> 00:33:06.700
Great; the final one--we are going to name this function k; and now it is a flip vertically.
00:33:06.700 --> 00:33:12.700
It is a negative version of the function--just multiplying the function by -1.
00:33:12.700 --> 00:33:19.700
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).
00:33:19.700 --> 00:33:27.700
So, it is -h(x); now, what was h(x)? Well, h(x)...it is negative quantity..what did we have for h(x)?
00:33:27.700 --> 00:33:39.100
We had g(x) + 4; so it is -(g(x) + 4); and then, what was g(x) + 4?
00:33:39.100 --> 00:33:59.200
That was -(f(x) + 1 + 4); so we get, finally, -((x + 1)³ - 2(x + 1)...
00:33:59.200 --> 00:34:08.100
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.
00:34:08.100 --> 00:34:15.300
So, plus 6; so it is - 2(x + 1 + 6); great.
00:34:15.300 --> 00:34:19.200
And there we are; we could continue to expand that if we wanted to; but that is ultimately what it is going to be.
00:34:19.200 --> 00:34:25.600
We would have to expand a bunch of things--expand the cube in there, and then distribute out that negative sign.
00:34:25.600 --> 00:34:29.400
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.
00:34:29.400 --> 00:34:37.100
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.
00:34:37.100 --> 00:34:41.100
If we break that order, it is not going to end up working out; we are going to get a different answer.
00:34:41.100 --> 00:34:50.400
Great; all right, the next example: we have a parent function of cube root of x, and that is this one on the left.
00:34:50.400 --> 00:34:53.000
Now, we want to give the function for the graph on the right.
00:34:53.000 --> 00:34:55.800
We need to figure out what things happen to the graph on the right.
00:34:55.800 --> 00:35:03.800
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.
00:35:03.800 --> 00:35:11.400
The way we would do that is: at first, we have a vertical flip; and what comes after that?
00:35:11.400 --> 00:35:18.800
The "home base" moves; where was it originally?
00:35:18.800 --> 00:35:25.000
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."
00:35:25.000 --> 00:35:32.300
Its home used to be at (0,0); and over the course of becoming the second graph, it goes to...where is its home?
00:35:32.300 --> 00:35:41.700
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.
00:35:41.700 --> 00:35:54.100
We have a shift right by 2, and a shift up by 1.
00:35:54.100 --> 00:36:00.300
So, we can come up with a function for this by just applying these transformations.
00:36:00.300 --> 00:36:10.300
First, we have f(x)...let's do our first transformation, g(x) =...the vertical flip is -f(x) = cube root of x.
00:36:10.300 --> 00:36:16.300
This is our first part; but next, we have the second part that comes in.
00:36:16.300 --> 00:36:20.600
The second part...we will call this one h(x), so that will be what our final function actually is.
00:36:20.600 --> 00:36:29.600
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.
00:36:29.600 --> 00:36:38.200
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,
00:36:38.200 --> 00:36:43.500
since the + 1 happens completely outside of the function--outside of everything.
00:36:43.500 --> 00:36:54.100
So, -f(x)...we have here that h(x) is equal to g; plugging in x - 2, and also shifting it up by 1--
00:36:54.100 --> 00:37:15.700
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).
00:37:15.700 --> 00:37:24.000
So, it is going to be -f(x) is ³√x (oops, it should be a negative sign there).
00:37:24.000 --> 00:37:56.300
So, g(x - 2) is -f(x - 2) + 1, which is equal to -³√(x - 2) + 1.
00:37:56.300 --> 00:38:12.200
So, the function that we are seeing over here is h(x) = -³√(x - 2) + 1; great.
00:38:12.200 --> 00:38:19.100
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.
00:38:19.100 --> 00:38:26.900
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?
00:38:26.900 --> 00:38:34.200
Well, the parent function for g(x) in red is going to be f(x) = x³.
00:38:34.200 --> 00:38:37.400
We see that x³ there, so it seems reasonable that that is going to be it.
00:38:37.400 --> 00:38:40.300
What had to happen to be able to get 7 - x³?
00:38:40.300 --> 00:38:45.500
Well, the first thing that had to happen is a vertical flip, and then a vertical shift up.
00:38:45.500 --> 00:38:48.800
We could also have a vertical shift down, and then a vertical flip of everything.
00:38:48.800 --> 00:39:01.600
But it is easier to see it as a vertical flip, and then the second thing as shift up by 7.
00:39:01.600 --> 00:39:09.500
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.
00:39:09.500 --> 00:39:12.700
And that is how we get our red g(x).
00:39:12.700 --> 00:39:21.000
Now, our blue g(x), 10√(x + 5); this f(x) will start from the basic function of √x.
00:39:21.000 --> 00:39:24.500
That is our fundamental function; so what has happened in here?
00:39:24.500 --> 00:39:32.000
Let's say that the first thing...it seems like it is easier to move horizontally than to have to do a vertical stretch first.
00:39:32.000 --> 00:39:34.500
And actually, it won't matter what order we do it in.
00:39:34.500 --> 00:39:41.800
But let's say we will do it in first order of shift; so we have x + 5 in there.
00:39:41.800 --> 00:39:51.500
So, x + 5...x + k means that it is a positive, so shift by 5 left.
00:39:51.500 --> 00:39:55.900
It is a horizontal shift, and it goes to the left, because it is a positive k.
00:39:55.900 --> 00:40:08.800
And then, the second thing we do is multiply the entire function by 10; so it is a vertical stretch by a = 10.
00:40:08.800 --> 00:40:13.100
We actually could do that in the completely opposite order; we could do vertical stretch by a = 10,
00:40:13.100 --> 00:40:18.200
and we get 10√x, and then plug in x + 5; and we get 10(√x + 5).
00:40:18.200 --> 00:40:23.800
So, 1 or 2...it doesn't actually matter which one goes first, unlike the red function, which actually...
00:40:23.800 --> 00:40:26.400
it did matter that we had a vertical flip, and then shifted up by 7.
00:40:26.400 --> 00:40:29.800
We would have to do a slightly different thing if we wanted to do it in a different order for the red function.
00:40:29.800 --> 00:40:31.200
But the blue function--anything works.
00:40:31.200 --> 00:40:39.800
Finally, the green function: its basic parent function, what is creating it, its base function, is |x|.
00:40:39.800 --> 00:40:44.500
Now, for this one, it is a little bit harder to see which one this has to be.
00:40:44.500 --> 00:40:50.000
It is much easier to start by shifting to the right by 3, because we see right here this 2x.
00:40:50.000 --> 00:40:54.800
So, 2x means that how the x-axis is being affected is that it has been "sped up."
00:40:54.800 --> 00:40:59.100
But if we speed up, and then move by a different thing...we are used to moving;
00:40:59.100 --> 00:41:06.600
all of our theory about moving is based off of "move first"; our theory of moving how we experience the x-axis
00:41:06.600 --> 00:41:13.700
was all done on the principle of it starting as x, and not starting as 2x or 5x or 1/2x.
00:41:13.700 --> 00:41:17.200
It was all based on x + k, not 2x + k.
00:41:17.200 --> 00:41:22.300
So, we want to start by shifting--doing our shifting--dealing with that first.
00:41:22.300 --> 00:41:31.400
We will shift right by 6; and we know it is to the right, because we have a -6.
00:41:31.400 --> 00:41:35.400
Now, here is actually the thing: it is not by 6; this is a confusion.
00:41:35.400 --> 00:41:40.000
It seems to be 6 at first, because of that + k; but notice what is really there.
00:41:40.000 --> 00:41:52.500
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.
00:41:52.500 --> 00:42:00.200
So, we have to get it to 1x first; so we pull out the 2, and we get 2 times (x - 3).
00:42:00.200 --> 00:42:07.800
The shift to the right is actually by this 3 here; so we shift right by 3, because we have k at -3.
00:42:07.800 --> 00:42:24.800
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.
00:42:24.800 --> 00:42:28.200
Any horizontal interval will squish to half.
00:42:28.200 --> 00:42:35.600
And then finally, we have this + 1 here; so it shifts up by 1.
00:42:35.600 --> 00:42:40.900
Now, it actually turns out that the 3 could be at the top, or it could be at the bottom; it doesn't matter.
00:42:40.900 --> 00:42:44.300
Shifting up by 1--that could happen at the very beginning; it could happen at the very end.
00:42:44.300 --> 00:42:50.600
But because of the shift right and the horizontal speed-up, we have to have it in this x + k form.
00:42:50.600 --> 00:42:54.700
We can't get it out of 2x - 6, because it is 2x + k; that is not the same form.
00:42:54.700 --> 00:42:57.800
We have to have it as x + k; so we have to pull that 2 out first.
00:42:57.800 --> 00:43:01.500
There is a way where we could have the horizontal speed-up go first, and then shift.
00:43:01.500 --> 00:43:05.000
But it is much easier to think in terms of the shift right, and then the horizontal speed-up.
00:43:05.000 --> 00:43:07.400
And if that one seems a little confusing, I wouldn't worry about it too much.
00:43:07.400 --> 00:43:12.700
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,
00:43:12.700 --> 00:43:19.200
until you are in college--or if you are not just in college, but taking an advanced-level math class in college.
00:43:19.200 --> 00:43:24.700
So, don't really worry about this right here; this is a fairly difficult kind of problem.
00:43:24.700 --> 00:43:26.800
But this is the sort of thing you want to be thinking about it with.
00:43:26.800 --> 00:43:32.600
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?"
00:43:32.600 --> 00:43:35.200
And you have to follow it carefully; what does it have to fit in?
00:43:35.200 --> 00:43:41.000
It has to fit in things of the form x + k; and you notice, 2x is different--it is not in that same form.
00:43:41.000 --> 00:43:45.200
So, you have to get it into that form before you can use these things that we talked about before.
00:43:45.200 --> 00:43:52.600
All right, the final example: How is vertically stretching the graph of f(x) = x² the same as horizontally stretching it?
00:43:52.600 --> 00:44:16.200
Remember, a vertical stretch is done by a times f(x); a horizontal stretch is done by f(a times x).
00:44:16.200 --> 00:44:20.300
Now, I think it is a little bit confusing to use a in two places.
00:44:20.300 --> 00:44:29.800
So instead, we are going to call this b; so we say we are just using b to prevent confusion.
00:44:29.800 --> 00:44:35.600
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.
00:44:35.600 --> 00:44:39.900
It means the exact same thing; a and b are both just constants.
00:44:39.900 --> 00:44:52.900
So, a and b are multiplicative constants; they are just how much we are stretching by--
00:44:52.900 --> 00:44:57.800
whether it is a horizontal stretch or it is a vertical stretch--it is just how much we are stretching by.
00:44:57.800 --> 00:45:01.800
All right, so let's see how this works on f(x) = x².
00:45:01.800 --> 00:45:10.600
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;
00:45:10.600 --> 00:45:13.900
so we will get bx; it is the quantity, squared.
00:45:13.900 --> 00:45:24.200
So, over here we have ax² and b²x², because the "squared" will get put onto both of them.
00:45:24.200 --> 00:45:27.800
So, we have these two things; how is it that they are the same?
00:45:27.800 --> 00:45:32.700
Well, how is it that they are similar--what is the connection between them?
00:45:32.700 --> 00:45:38.600
Well, think about this: a is just a constant, and b is just a constant.
00:45:38.600 --> 00:45:51.200
a and b are both constants; but if b is a constant, then that means that b² is also just a constant.
00:45:51.200 --> 00:45:58.200
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.
00:45:58.200 --> 00:46:07.400
It is not allowed to vary around; so b² is also a constant.
00:46:07.400 --> 00:46:17.100
So, what that means here is that in either case, whether it is a vertical stretch or a horizontal stretch,
00:46:17.100 --> 00:46:36.600
it just has the effect of multiplying x² by a constant.
00:46:36.600 --> 00:46:42.100
So, what we are seeing here: the reason why, if we do a vertical stretch, and we do a horizontal stretch,
00:46:42.100 --> 00:46:48.400
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...
00:46:48.400 --> 00:46:51.500
you will notice that they actually looked basically the exact same.
00:46:51.500 --> 00:46:54.600
There were slight differences, but it is the same sort of stretching going on,
00:46:54.600 --> 00:46:58.800
because when we compress it horizontally, it causes it to just sort of squirt up vertically.
00:46:58.800 --> 00:47:02.500
And when we stretch it out vertically, it is the same thing as if we had compressed it horizontally.
00:47:02.500 --> 00:47:07.400
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.
00:47:07.400 --> 00:47:10.100
So, that is why they are so similar.
00:47:10.100 --> 00:47:16.500
We, in actually, I think, all of the fundamental functions that we are used to using by this point...
00:47:16.500 --> 00:47:23.300
all of them are already things where this is just the horizontal and the vertical stretch...it will end up having the same effect.
00:47:23.300 --> 00:47:30.000
It will do it by different amounts; but ultimately, it is just putting a constant into the mix--multiplying things by a constant.
00:47:30.000 --> 00:47:34.400
The first time that you will end up seeing things--and right now, if you have even taken trigonometry,
00:47:34.400 --> 00:47:38.900
the only thing you would see where you would be able to see the difference between a horizontal and a vertical stretch--
00:47:38.900 --> 00:47:44.500
is trigonometric functions; if you look at sine and cosine, it actually is possible to horizontally stretch those
00:47:44.500 --> 00:47:49.900
and vertically stretch those, and you will get totally different-looking things out of a vertical stretch versus a horizontal stretch.
00:47:49.900 --> 00:47:52.900
Now, it is OK that we haven't really talked about trigonometric functions yet.
00:47:52.900 --> 00:47:56.100
And you haven't seen them yet, probably; don't worry about that--that is OK.
00:47:56.100 --> 00:48:00.000
And if you have seen trigonometric functions, or you have taken some trigonometry, that is all the better.
00:48:00.000 --> 00:48:02.200
You probably are already exposed to this.
00:48:02.200 --> 00:48:07.800
But just know that later on, you will see cases where there is a difference between horizontal stretch and vertical stretch.
00:48:07.800 --> 00:48:12.500
But for some other functions, like f(x) = x², it ends up being that there is not really a difference at all.
00:48:12.500 --> 00:48:16.300
All right, I hope you now have a good understanding of all of the different transformations that are available to us.
00:48:16.300 --> 00:48:19.100
I know that there are a lot of them; but if you think through what you are doing with each one,
00:48:19.100 --> 00:48:22.300
you can probably figure it out without even having to resort to the table.
00:48:22.300 --> 00:48:25.000
These are really useful, because they let us build a bunch of different functions
00:48:25.000 --> 00:48:27.900
and understand how to graph functions that seem complex at first,
00:48:27.900 --> 00:48:33.000
but are really just some basic function we are used to graphing, that has been stretched and squished and moved around.
00:48:33.000 --> 00:48:35.000
All right, we will see you at Educator.com later--goodbye!