Transformations - Video Tutorials & Practice Problems

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1

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Intro to Transformations

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Hey, everyone and welcome back up to this point, we've spent a lot of time talking about functions. And in this video, we're going to be taking a look at a transformation of functions. Now, this topic can seem a bit complicated at first because there's a lot of different types of transformations that you'll see. But in this video, we're going to learn that transformations really only boil down to three basic transformations which were going to cover. And after taking a look at these, I think you'll find that this concept is a lot less abstract and a lot more clear. So let's get right into this. Now, a transformation occurs when a function is manipulated such that it changes position or shape. Now, an example of this and actually there are three main examples are the three examples we have listed down here. So the three basic types of transformations that you're going to see are reflections, shifts and stretches for a reflection. This occurs when a function is folded over a certain axis. So if we were to take this function that we see right here and reflect it over the X axis, it would look something like this. Notice how we literally just took this graph and folded it. That's a reflection. Now, another type of transformation is a shift and a shift occurs when you move a function. So if we were to take this function, which is currently at the position 00, the origin and we were to move it to some new location, the graph would look something like this. Notice how we literally took the function and moved it somewhere else. That's a ship's transformation. And the last transformation we're going to look at is a stretch and the stretch occurs when you imagine squeezing a function. So if you imagine taking this function and stretching it vertically, such that it squeezes the function together, that's the idea of a stretch. So a stretch would look something kind of like this from our original function. Now, these are the three basic transformations you're going to see throughout this course. And we will cover these in more detail as we go through this series on transformations. But it's also important to know how the function notation is going to change in these certain situations. So when you have a reflection, a reflection is going to become negative when you reflect over the X axis. Notice how we started with F of X and this became negative F of X A shift is going to turn into this function F of X minus H plus K. And in this notation, the H represents the horizontal shift and the K represents the vertical shift. Now, for a stretch transformation, the function is going to look like this where you have some constant multiplied by the function F of X. So C is the constant responsible for causing this kind of squeeze on the graph or basically the vertical stretch. Now, let's see if we can actually apply this knowledge to an example. So in this example, we're given the function F of X is equal to the absolute value of X. And we are also given the corresponding graph. Now, what we're asked to do is match the following functions P of XQ of X and R of X to the correct corresponding graph. Because all of these functions that we have here are transformations of our original function, the absolute value of X. So let's take a look at these. Now, for this first function that I see P of X is equal to absolute value of X minus three plus two, we need to figure out which one of these graphs this is associated with. And my question to you would be what type of transformation do we have here? Because these are the three situations that we have transformations. And if I look at this, this actually looks the most like a shift transformation because notice how we have this X minus H plus K and here we have this X minus three plus two. So what I'm gonna do is look for whichever one of these graphs appears to be a shift. And if I look at all of these number two looks a lot like a shifted version of our original graph because we started here at the origin and then we finish somewhere up here. So I'm gonna say that graph two matches with function A. Now let's take a look at function B we have that Q of X is equal to the negative absolute value of X. Now we first need to figure out what transformation from our original function. This looks the most like. And if I look at these transformations, this very much seems like a reflection. And a reflection is a situation where you're folding the graph over a certain axis. But we have a little bit of a problem here because notice both graphs one and three have been folded. We see reflections happening in both of these because our graph is originally pointed up. And we can see for both of these examples, the graph appears to be pointed down. But the difference is is that for graph three, it looks like the graph has also been squeezed. Whereas for graph one, it's just been flipped. And so looking at these graphs and looking at our function, I see that there's really no factor in front here that's going to cause it to be squeezed. So that means that graph number one is going to match with option B. So that's our second function. Now, I can tell just by process of elimination that for our third function function C this is going to match with graph number three. But I want us to understand why these two graphs match together as well. So notice that we have this negative sign in our function which is causing the fold or reflection over the X axis. But I also noticed this graph has gone through a vertical stretch. And that actually makes sense because we have a constant being multiplied by the front here as well. So because a constant and a negative sign, this is causing both a vertical stretch and a reflection transformation. And it's very common that you're going to see multiple transformations happen to a single function. So that's just something you want to be aware of. So overall, these are the three functions that match with the three graphs below. That is the basic idea of a transformation of function. So hopefully, you found this helpful, let me know if you have any questions and thanks for watching.

2

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Reflections of Functions

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Welcome back, everyone. In the last video, we talked about transformations of functions. We discussed how there are three types of transformations you're going to see throughout this course. Now, for this video, we're specifically going to focus on the reflection transformation and with reflections, they can be kind of tricky because there are multiple different types of reflections that you're going to see and multiple different ways that your graph is going to fold. But in this video, we're going to be covering all those different situations. So hopefully this topic won't seem as tricky and will seem a lot more clear. So let's get right into this. A reflection transformation is a situation where the function appears to be folded. And we've discussed that a bit in the previous video, how it's like folding your graph in some way. But it's possible for the function to be folded over the X axis or the Y axis and how the function is folded is important for both the graph and the equation and what they're going to end up looking like. So let's say we have a fold or reflection over the X axis. When doing this, you can imagine taking your graph and increasing it at the x axis, like it's a piece of paper, you're then going to take it and fold it in this fashion and that's how your graph is going to form. So if we were to take this graph and fold it, it would end up looking like this. So this would be a reflection over the x axis. But now let's say we have a reflection over the Y axis. When reflecting over the Y axis, you're going to treat your graph at like a sheet of paper just like before. But now you're going to crease the graph at the Y axis. So the fold is going to be more like this kind of like opening or closing a book. So when doing this, your function is going to go from this position to that position, notice how it reflected over the Y because we folded it this way. So that's a reflection over the Y axis. Now, it's important to also notice how the function is going to change depending on how it reflects. So if we look at a reflection over the X axis, notice that we originally started at the point negative 11. And then after the reflection, we finished at the point negative one negative one. So notice how the X values stayed the same, but the Y values changed signs, we went from positive to negative one. So because of this, we could say that for our graph, when we reflect over the X axis, it's the Y values that change signs. So we basically go from positive Y to negative Y when reflecting over the X axis. So we could also say that our function goes from F of X to negative F of X because it's the outputs that become negative. But now let's take a look at a reflection over the Y axis. Notice how, when we reflected over the Y axis, we went from a point of negative 11 which we have on this left side and we reflected over to positive 11. So notice that our Y value stayed the same but our X values change signs. So because of this, we could say that we went from X to negative X. So it's actually going to be the inside of our function because we have X on the inside that's going to change signs. So this is how the function is going to change. But let's actually see if we can take this concept and apply it to an example. So in this example, we're given the function F of X is equal to X plus two. And we're told to let H of X be the function F of X with a reflection over the X axis. We're asked to sketch the graph of H of X and determine which of the functions below matches the new function H of X. So what I'm first going to do is see if I can sketch the reflection H of X on this graph over here. So this is the original function that we see F of X sketched for us. And I see in our example that it's given we're gonna have a reflection over the X axis. You can imagine taking this graph and increasing it at the X, you're then going to fold it and whatever gets folded down or up is going to be the new function. So if I take this portion of the graph and I fold it down, we're going to go from here all the way down through here and notice for this graph that we started at A Y value of positive two and finished at A Y value of negative two. That's because whenever you reflect over the X axis, it's the Y value that changes signs. Now I also need to fold this bottom portion of the graph up because we're doing a reflection over the X axis. So this bottom portion of the graph is going to extend up here like this after we reflect it. So this is going to be the reflected function H of X from our original function F of X. But we're also asked to figure out what the new equipment is going to look like based on the four options below. Well, our function H of X is simply F of X with a reflection over the X axis and we're told that when we reflect over the X axis, we need to make our entire function negative. So that means down here, our function is going to be negative F of X. So to draw this out, we're given the F of X is X plus two. So this whole thing is going to be negative X plus two. And if I go ahead and distribute this negative sign into the parentheses, we're going to end up with negative X minus two. So this is the equation for our reflected function H of X. And if I look at the four options below this matches with option C. So after reflecting our function over the X axis, our graph is going to look like this and our new equation is going to look like that. So this is the idea of reflection transformations when dealing with functions. Thanks for watching and let me know if you have any questions.

3

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Reflections of Functions Example 1

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So let's see if we can solve this problem. So written below, we have the function F of X, which is this curve that you see or the green dotted lines basically. And we're asked to sketch a graph of G of X where G of X is the function F of X after it has been reflected over the X axis. Now, this is a reflection transformation like we've been learning about. And for this situation, what's going to happen is we're going to have our graph over the X. And when this happens or called it reflecting over the X axis is like taking this entire graph like a sheet of paper, creasing it at the X and folding it. So if you were to take this portion of the graph and you were to fold it down over the X axis, the graph would end up looking something like this. So this is what your graph is going to look like where you can literally imagine that this whole thing just gets flipped down and folded over like sheet of paper. Now, this portion of the graph is going to fold up because it's like folding up over the X axis. Again, you're creasing this like a piece of paper. So this graph is going to look something like that. So once we have reflected over the X axis, this is what the new function G of X is going to look like with respects to our original function F of X this green dotted curve. So this is how you can solve the problem. Hope you found this helpful.

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Problem

Problem

Written below (green dotted curve)is a graph of the function $f\left(x\right)=\sqrt{x-2}$. If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x)?

A

$g\left(x\right)=\sqrt{-x-2}$

B

$g\left(x\right)=\sqrt{-x}-2$

C

$g\left(x\right)=\sqrt{x-2}$

D

$g\left(x\right)=\sqrt{x}-2$

5

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Shifts of Functions

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Hey, everyone and welcome back. So up to this point, we've been talking about transformations of functions. In the last video, we looked at reflections and in this video, we're specifically going to be taking a look at shifts. Now, shifts can be a bit complicated because rather than just dealing with negative signs, like we were with reflections, you're going to be dealing with actual numbers inside and or outside your function. So the notation and graphs can get a little bit complicated. But don't worry because in this, we're going to be going over some examples and scenarios that will hopefully make this concept crystal clear. So let's get right into this. A shift occurs when a function is moved either vertically or horizontally from its original position. Now, something that's important to note is that you will often have situations where the graph is moved both vertically and horizontally because the shift is just moving your graph to a new location or whatever function you originally had. Now, the general form for shifts is when you have your original function F of X that goes to F of X minus H plus K after you've shifted the function in this notation, the H here represents the horizontal shift. Whereas the K represents the vertical shift now to understand this concept a bit better, let's take a look at the strictly vertical shift. This means that you're only going to move your graph vertically. So an example of a vertical shift would be if we took this parabola that we here, this green curve and we shifted it up here. So it looked something like this. Notice how we went from a position of 00 to a position of 02, the X values stayed the same but the Y value changed by two. So when having the vertical shift, we would say that it's the Y values that change. But now let's take a look at the horizontal shift. An example of a horizontal shift is whenever your graph moves strictly horizontally. So we could say that we have our original curve right here at 00, the green curve. And then let's say that this gets shifted over to the right. So our graph went from 00 to this position of 30. And because the X values changed and the Y value stayed the same, we would say that for the horizontal shift, it's the X's that change. Now, it's also important to know how the function is going to behave when dealing with the shift transformation. So notice in both cases, we have the same general form that we saw at the start here. But for the vertical shift, notice that our H value is always going to be zero. So we can just ignore this H value that we see in the equation. So our function starts as F of X, this is the curve that we have in here. And then this gets transformed to F of X plus K where K is the vertical shift. And since we shift it up by two, we would say that K is two and notice how whenever we have a plus two, the graph shifts up. So in this situation, if you ever see F of X plus K, you're always going to shift up. Whereas if you see F of X minus K, the graph is going to shift down. Since we had a positive two, we went up by two. That's the idea of a vertical shift. Now, for the horizontal shift in this situation, it's the K value that's equal to zero. So we can take this K and pretend like it's not even there when dealing with a strictly horizontal shift. So we started with our initial function F of X just like before. But then we went to F of X minus H and because H represents the horizontal shift and we ended up at an X value of three, we would say we have F of X minus three and that's the horizontal shift. Now, whenever you see F of X minus H, this means your graph is going to shift to the right. Whereas if you see F of X plus H your graph shifts to the left. And we can see that because F of X minus three because caused our graph to shift to the right by three. But this seems a bit counterintuitive because isn't the minus sign typically associated with the left side of the graph. Well, here's the reason why this happens, notice how there's already a negative sign or a minus sign inside the equation that we started with. So if we wanted to shift over to the left, let's say at negative two, our graph would look something like this. And that means that the H value would be negative two. So we would end up having our original function F of X become F of X minus negative two. But we know that two negative signs are going to cancel each other. So this would actually become F of X plus two. So notice how, even though we have a plus sign here, we actually shift to the left because realistically, we actually plugged in a negative two here for the H value. So this is just something you want to keep in mind, pay very close attention to what this sign is in front of the H value. So now that we've gotten to look at the vertical and the horizontal shifts, let's see if we can apply this to an example. So in this example, we're given the function F of X and we're asked to sketch the transformation F of X minus two three. We know that this is the standard equation when dealing with a shift where H is the horizontal shift and K is the vertical shift. Looking at the equation that we have, I can see that this H value here matches with the two that we have right there. And I can see that there's a minus sign in front of both with which means we're going to shift to the right. So I can keep this as positive two for our H value. I can see also that we have a K value of positive three. So what this means is since we have positive two for H and positive three for OK, that means we're going to shift two units to the right from our original position and we're going to shift three units up. So we started here at 00 and we're going to finish at this point which is at 23. So our new function is going to look something like this, notice how we have the same overall shape that we had before. But now we've been shifted to this new location based on the transformation we were given. So this is how you can handle situations where you have a shift transformation. Hopefully, you found this video helpful and let me know if you have any questions

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Problem

Problem

The green dotted line in the graph below represents the function $f\left(x\right)$. The blue solid line represents the function $g\left(x\right)$, which is the function $f\left(x\right)$ after it has gone through a shift transformation. Find the equation for $g\left(x\right)$.

A

$g\left(x\right)=f\left(x-2\right)+3$

B

$g\left(x\right)=f\left(x-2\right)-3$

C

$g\left(x\right)=f\left(x+2\right)-3$

D

$g\left(x\right)=f\left(x\right)-3$

7

concept

Graphs of Shifted & Reflected Functions

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Hey, everyone and welcome back. So up to this point, we've been talking about transformations of functions. Now, in the previous couple of videos, we took a look at reflections and shifts. Now, in this video, we're going to see how we can graph shifted and reflected functions. Now, it's very common throughout this course that you're going to see a combination of transformations to a single function. And this process can oftentimes be a bit tricky because we're not really used to seeing multiple transformations at once. But in this video, we're going to take a look at some scenarios and examples and I think that you'll find this process is actually pretty straightforward. So without further ado let's get right into this. Now, we've already taken a look at the reflection transformation where you can imagine folding your graph in some kind of way. We've also taken a look at the shift where you imagine taking your graph and literally just moving it to another location. So we started here at 00 and moved to 32. This would be an example of a shift. Now, what you can do is take both the reflection and the shifts and you can combine them into a single transformation or I should say really a combination of transformations to a single function. And what this would look like is literally just the two transformations combined. So notice how we have our graph here, which started by pointing up and then it got reflected over the X axis. So it was pointing down, then this graph got shifted to a new location specifically the location that we had over here. So this would be an example of a combination of multiple transformations to a single function. And the way that the notation is going to change is actually pretty straightforward as well because notice for the reflection, our function negative when we reflected over the X axis and our shift to 32 was shown in the function where we had our horizontal shift of three and our vertical shift of two. And notice for the combined reflection and shift, we had this negative sign which showed up for the reflection. But then we had this three and this two which showed up for the horizontal and vertical transformations respectively as well when we did the shift. So notice how the function notation and the graph is actually pretty straightforward when you combined multiple transformations. But let's actually see if we can do an of this ourselves. So this example says if G of X is a transformation of the function F of X is the absolute value of X write the equation for G of X. Now notice on this graph, we have the original function F of X and we also have the transformation G of X. So based on this graph, we're trying to figure out how G of X is going to change F of X or basically what the final equation for G of X is going to look like and to solve this. Well, to figure out GE X, we need to figure out how F of X has changed. So we know that G of X is a transformation of our original function F of X. And let's see what we notice about this graph. Well, notice that this graph was initially pointing up and now it's pointing down, this is an example where we reflect over the X axis. So because of this, whenever we reflect over the X axis, our function becomes negative. So what that means is that in the example we have down here, our F of X is going to become negative F of X when we do this transformation. But this is not the only thing that happened to this graph because notice this graph has also been shifted. So we were originally going to be at this position after the reflection, but now this graph got shifted two units to the left. So we started here at 00, the origin and we finished centered over here at negative 20. Now whenever we have a shift transformation, you're going to have that your function becomes F of X minus the horizontal shift plus the vertical shift. But we don't have a vertical shift of any kind. So we only have to deal with the horizontal shift. And in this example, the horizontal shift which I'll call H is negative two. So that means we're going to have our function negative F of X minus negative two because that's our shift transformation. But notice that these negative signs will just cancel. So all we're going to end up having is negative F of X plus two. And by the way, recall that when you see this plus sign, that means that you shift to the left. And whenever you see this minus sign, that means you shift to the right. So just something to recall from our video on shifts. So this is what the transformation is going to ultimately look like. So if I were to put this all into our transform function G of XG of X is going to be the negative absolute value of X plus two because notice how we had X plus two as the inside function and we made our entire function negative to match all the transformations that happened on the graph over here. So our transform function is going to look like this. And this right here is the solution to the problem. So that is how you can deal with multiple transformations on a single function hope you found this video. Helpful. Thanks for watching.

8

example

Graphs of Shifted & Reflected Functions Example 1

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Let's see if we can solve this problem. So in this problem, we have a function H of X which is a transformation of the original function F of X is equal to X cubed. We're told in this transformation, the function is reflected over the X axis and then shifted down to units. We're asked to write an equation for H of X and then sketch a graph of the function H of X. So let's begin. Now, first off, we should recognize what type of transformations are happening here. And in this first case, I see that we have a reflection over the X axis recall from previous videos that we, when we reflect over the X axis, our function F of X becomes negative F of X. This is something we learned in the video on reflections. Now we can also see here we have a shift down to units. And since we're only shifted down, this means we're only being vertically shifted. When this happens, our original function F of X becomes F of X plus K where K represents the vertical shift. So these are the two situations we need to watch out for in this problem. Now we'll start things off by taking a look at this function here, which is F of X is equal to X cubed. And what I need to do first is take a look at the first transformation that happens, we see that we're reflected over the X axis. And when this happens, your function becomes negative. So that means X cubed is going to become negative X cubed. This is the first thing that we have where we reflect over the X axis. Now the next thing that happens is the shift down two units. So since we're shifting down two units, this K represents the vertical shift. And since we're shifted down, that means K is going to be negative two for the downward shift. So if we add K to this equation that we have here, our transformation is going to become negative X cubed and then we're going to have minus two because it's a negative two that we plug in for K. So this right here is going to be our transform function H of X. And this is what the new equation is going to look like. So this is how you can figure out what the equation is for H of X. And that's part a of this problem. But we're also asked to sketch a graph of H of X as well and to do this, well, let's just once again, look at the transformations we have, this is the original function right here. And first off, we have a reflection over the x axis, we call that when reflecting over the x axis, you can imagine folding your graph over the X axis, increasing it like a piece of paper. So this portion of the function is going to go kind of down like this and then or more like that and then this portion of the function is going going to fold up kind of like this. So this is what the function will look like when we reflect over the X axis. But we're not quite done yet because notice that we've also been shifted down to units. So this graph is not going to be centered at the origin. It's actually going to be centered down 12 units right here at negative two. So our final graph is going to look like this for the transform function. So this is our transformation H of X as a graph and then this is the corresponding equation. So that is how you can deal with multiple transformations on a single function. Hope you found this helpful.

9

concept

Stretches & Shrinks of Functions

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Welcome back everyone. So up to this point, we've been talking about transformations of functions. We've taken a look at reflections and shifts and in this video, we're going to focus on the stretch transformation or to be a bit more specific stretches and shrinks. Now, the nice thing about the stretch and shrink transformation is unlike shifts where we're dealing with two numbers in our function, the stretch and shrink only deals with one number. Now with stretches and shrinks. There are a few things that you'll have to track of and remember for how the graph is going to behave. But in this video, we're going to be going over a bunch of different scenarios and examples that will hopefully make this topic super clear. So let's get into this, a stretch or a shrink occurs when some constant is multiplied either inside or outside of the function. Now, to understand this a bit better, let's take a look at these two cases down here. Now, for this example, we have on the left, this is an example of a vertical stretch or compression and something I want to mention right off the bat is whenever you see the word compression or shrink. These words mean the exact same thing. So that's just something to keep in mind because these will often be used interchangeably. Now, in this example, here we have a function F of X and this function is plotted on this graph. It's this dotted line here. And we're looking at what this function would look like if it went through a vertical stretch and a vertical shrink. So if this function went through a vertical stretch, you can imagine taking this function and vertically stretching it, what that would do is cause the graph that we have right here, which is the vertical stretch. Now, if you were to take this function that we have in the middle and compress it, you would end up with a vertical shrink which looks like this graph in here. But now let's take a look at the horizontal case for the horizontal situation. We have the same function that we had before, which is the dotted curve that you see right here. And when going through a horizontal stretch, you can imagine taking the graph and stretching it horizontally. If you were to do that, you would get a stretch, which looked like this. So this is the function after you've stretched the graph horizontally. Whereas if you were to do a horizontal compression, the graph would end up looking like this. Notice how it looks like we just took our graph and squeezed it closer to the y axis. So that's the idea of a horizontal stretch or compression. Now, whether you see the vertical case or the horizontal case is going to depend on where the constant is multiplied. So if you see a vertical stretch or compression, this means the constant is multiplied outside the function. Whereas if you see a horizontal stretch or compression, the constant is multiplied on the in side of the function. And when it comes to the vertical case, whenever you have a situation where the constant that you have is between zero and one, the graph is going to vertically shrink. So you're going to see a vertical shrink if this happens. Whereas if you see your constant is greater than one, then this means the graph is going to stretch. Now, in the case for the horizontal stretcher compression, if you see that your constant is between zero and one, then your graph is actually going to do a horizontal stretch. Whereas if the constant is greater than one, the graph is going to horizontally shrink. So notice how the vertical stretch or compression is opposite of the horizontal stretch or compression when it comes to what constant stretches versus shrinks the graph. So that's something important to keep in mind when solving problems as well. Now, let's actually see if we can try an example where we have this type of situation. In this example, we are given the function F of X which is plotted. On this graph to the right. And we're asked to sketch the graphs of the following functions where we have some kind of stretch or compression happening to this function. So let's first start with this case that we see on the left here, which is a for case A, we have two times the function and the two is multiplied outside F of X. Whenever you have your constant multiplied outside of the function, this corresponds to the vertical case. And since our constant, we see is two, that's greater than one, which means we're going to have a vertical stretch. So what this means is our graph is going to vertically stretch by a factor of two. So we can see right here that we have the 0.22 but our Y values are going to stretch vertically. So we're going to end up actually at 24. And at this point where we have one negative one, we would stretch down here to one, negative two, right? Because we're vertically stretching. Likewise, at this point, negative 11, we would stretch up here to, to negative 12. And then at this point, we would stretch down here to negative four as a Y value. So our vertical stretch would look something like this where our graph would be stretched vertically. But now let's take a look at the second case where we have one half being multiplied on the outside of the function. Since this constant is outside of the function. We're still going to have the vertical case for the stretcher compression. But notice that the constant is now between zero and one, which means we're going to have a vertical shrink. So everything is going to shrink by a factor of one half. Rather than being at this 0.22 we're going to be at 21 because we're shrinking our graph. And rather than being here at uh at one negative one, we're actually going to be at one negative one half. Likewise, we'd be at a Y value of positive one half there. And then we would be at a Y value of negative one right here. So notice that in this case, we have a vertical shrink. Now, the last case, we're going to take a look at is this situation where we have a one half multiplied inside the function. So in this case, since we're on the inside of the function, this corresponds to the horizontal stretcher compression. And since the constant that we have is between zero and one, this corresponds to a horizontal stretch. Now this one can be a little bit tricky to do just by looking at the graph. But what you can basically imagine is since we have this X value here, that's being multiplied by one half on the inside. The way that we would get back to our original function is if we doubled all the X values because multiplying this by two would cancel the two that we have there. So we actually want to double all the X values that we see. So rather than our original function, which is in here, rather than it being at two, it would be over here at four. And down here, rather than being at an X value of one, we'd be at an X value of two. And rather than being at negative one, we would be here at negative two. And then rather than being at negative two, we would be over here at negative four. So in this case, our graph is going to horizontally stretch, notice how we basically stretched it on the horizontal axis. So this is the basic idea behind the stretch transformation. Hopefully, you found this video helpful and let me know if you have any questions.

10

example

Stretches & Shrinks of Functions Example 1

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4m

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OK. So let's give this problem a try. So here we have the function F of X is equal to C times X squared. And we're asked to graph our function F of X when C is equal to two and when C is equal to one half, now what we have on the graph already here is a curve which represents our function F of X is equal to X squared. So this is the general for a parabola, but we need to figure out what's going to happen if we take various constants and plug it in to this function. So let's see. Well, we're going to start for the case where our constant here is equal to two. So this means that F of X is going to be two times X squared. This is what the function is going to look like if we plug two into for the constant. Now, what I'm going to do is try a bunch of different X values. So I'll first try an X value of zero. If this were to happen, we would have two times zero squared and two times zero is 00, squared is just zero. So that means we're going to have a point at 00, that's gonna be one point we can plot here. Now, what I'm also going to do is plot a value of negative one. If I do this, we're going to get two times negative one squared just replacing this X with a negative one. In this case, two times negative one is negative two. So we'll have negative two squared and negative two squared is actually positive four. So that means that a value of negative two or excuse me, a negative one, I mean a negative one, we're going to be at a Y value of four. So this is another point that we could put on this graph. Now lastly, I'm going to try a point of positive one. So in this case, we'll have two times one squared, two times one is two and two squared is four. So at a value of one, we're going to or at an X value of one, we're going to have a Y value of four. So our graph is going to look something like this when we replace this constant that we see here with a two. And I think this actually makes sense because recall in the previous video, we discussed whenever you have a constant multiplied inside of your function, it's going to cause a horizontal stretch or shrink to your graph. In this case, we saw the, it caused a horizontal shrink and this happens whenever your constant is greater than one. So it makes sense that we would get this situation where the graph shrinks since two is greater than one. But now let's try this other situation where we have one half. This basically means our constant is between zero and one. So let's see what happens if I do this. Well, in this case, our function F of X is going to become one half X squared because now we're just going to take this constant and replace it with the one half that we have over here. Well, let's see how this behaves. What I'm first going to do is plug in an X value of zero like we did before. In which case, we have one half times zero squared and anything multiplied by zero is just zero. So we already know this whole thing will come out to zero, meaning we'll have the same origin 0.00. Now, next, what I'm going to do is I'm actually going to try X value of two. And the reason that I'm trying two specifically is because if I take this X and replace it with a two and by the way, it doesn't actually matter, you could try one again, but you would just get a fraction. Um But if I go ahead and replace this with a two notice, we're going to get one half times two squared and one half and two will actually cancel each other here because taking two and cutting it in half will just give you one. So really, you just end up with one squared, which is just one. So if you try an X value of positive two, you're going to end up here at A Y value of one. And likewise, if you were to try negative two, well, in this case, we would have one half times negative two squared. In which case, this two would cancel with that one giving us negative one squared and negative one squared is just positive one because negative one times negative one will cause the negative signs to cancel. So at an X value of negative two, we're going to be at a Y value of positive one. Again, meaning our function is going to look something like this. We multiply the inside by one half and notice in this case, we got a horizontal stretch because whenever our constant is between zero and one, we get a horizontal stretch. Whereas when our constant is greater than one, we get a horizontal shrink. Now, one more thing I want to mention before finishing this video is notice that when we had the horizontal shrink, it almost appeared like we had a vertical stretch because in this situation where we had the horizontal shrink, it looked like vertically, the graph almost stretched and like when we had a horizontal stretch, this kind of looked like we had a situation with a vertical compression or shrink. And that will often be the case when you see these types of graphs that have symmetry on them is that the horizontal stretches will oftentimes be similar to the vertical shrinks and the horizontal shrinks will be similar to the vertical stretches. So that's just something to keep in mind visually when looking at these graphs. But either way this is how you solve the problem and these are the answers. So hopefully you found this helpful, this is what the graph is going to look like.

11

concept

Domain & Range of Transformed Functions

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4m

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Welcome back everyone. So up to this point, we've talked about the three main types of transformations being reflections, shifts and stretches and shrinks. Now, in this video, we're going to take a look at how we can find the domain and range of a function. After it's been transformed, it's called that you're going to see scenarios where you have to find the domain and range of something where a transformation has acted on it. So it's important that we know how to solve these types of problems when we come across them. So let's get right into this. A transformation can change the domain and range of a function. Now, when finding the domain and range of a function that has been transformed, you can actually do this by observing whatever the new graph looks like after the transformation. Now, we talked about in previous videos how to find the domain and range of a graph. But just as a refresher, let's try finding the domain and range of this function F of X. So to find the domain, we can imagine taking our graph and squishing it down to the X axis if we were to squish this graph down, it would look like a line that goes from negative three to positive three. So this tells us our domain. Now, if we want to find the range of this graph, we can imagine taking this graph and squishing it down to the Y axis. If we squish the graph down to the Y, we're going to end up with a line on the Y axis that goes from negative three to positive three as well. And that's our range. So pretty straightforward for finding the domain and range of this graph. But what if we had a transformation that acted on this function? Would we have the same domain and range? Well, we discussed that this could change the domain and range. So let's see what happens. Notice we have the same overall shape but it's been shifted to a new location. Specifically, we've been shifted to 12, the position 12 from the position 00. So let's see what happens here. Well, by looking at this graph, if I go ahead and try and squish this thing down to the X axis, I'm going to get a line that goes from negative two to positive four. So our domain goes from negative 2 to 4. And if I want to find the range of this graph, I can squish this down to the y axis which will give me a range from negative one all the way up to positive five. So our range is go from negative 1 to 5 and notice how the domain and range that we got are different than the, than the domain and range we had in the original function. So this just goes to show you that a transformation can change our domain and range. Now to really solidify this concept. Let's try an example to see how we do. So here we're given a function F of X is equal to X squared. And we're asked to sketch a graph of the function G of X is equal to X minus three squared plus two and determine its domain and range. Now the function X squared is just going to be a parabola centered at the origin. And if I look at the transformation that we're given, I noticed that this looks to be in the form F of X minus H plus K which is a shift transformation. Now, I can see here that the H corresponds with this three right here because we have X minus H within the function and then inside the square function, we have X minus three. So I can tell that our H is going to be three. Now, I can also see that our K value is going to correspond to this positive two. So our K is positive two, since the H was positive, the graph will shift to the right. And since the K is positive, the graph will shift up. So our new parabola is going to go 123 units to the right and 12 units up, meaning we're going to be at this point right here. So notice we have the same Parabola, but it's been shifted to a new location, specifically, it's been shifted to 32. So this is the new position. Now, if I look at the domain and range of the Parabola, I can see here that the domain is going to be all real numbers because notice how this Parabola just expands in all directions to the left and right. So there's going to be no restrictions on our domain. So we can say our domain goes from negative infinity to positive infinity for this new function that we got G of X. But what about the range? Well, originally we have the, our range of the initial problem goes from zero to infinity because we can see here that on the y axis it goes from zero and continuously goes up. But after our transformation, notice that we went from this range to a range that goes like this. So our range is really going to go from positive two, all the way to infinity, meaning our range is going to go from two. And we need to include this value to infinity and this would be the range of our graph. So notice how the domain stayed the same but the range was different when we shifted our graph. So this is how transformations can change the domain and range of your function. Hopefully, you found this video helpful. Let me know if you have any questions.

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Problem

Problem

The green dotted curve below is a graph of the function $f\left(x\right)$. Find the domain and range of $g\left(x\right)$ (the blue solid curve), which is a transformation of $f\left(x\right)$.