Transformations - Online Tutor, Practice Problems & Exam Prep

Transformations of functions include reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, changing its sign, while shifts move the function horizontally or vertically, represented by $f(x-h+k)$. Stretches and shrinks alter the graph's shape based on constants multiplied inside or outside the function. Understanding these transformations is crucial for analyzing the domain and range of functions, as they can significantly change these properties.

Introduction to Transformations

Hey, everyone, and welcome back. Up to this point, we've spent a lot of time talking about functions. In this video, we're going to be taking a look at the transformation of functions. This topic can seem a bit complicated at first because there are 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 3 basic transformations, which we're going to cover. 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. A transformation occurs when a function is manipulated such that it changes position or shape. There are 3 main examples of transformation: reflections, shifts, and stretches.

For a reflection, this occurs when a function is folded over a certain axis. If we were to take this function that we see 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. Another type of transformation is a shift. The shift occurs when you move a function. If we were to take this function, which is currently at position zero zero (the origin), and move it to some new location, the graph would look different. Notice how we literally took the function and moved it somewhere else—that's a shift's transformation.

The last transformation we're going to look at is a stretch, which occurs when you imagine squeezing a function together. So if you imagine taking this function and stretching it vertically, it squeezes the function together—that's the idea of a stretch. These are the 3 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. It's also important to know how the function notation is going to change in these certain situations.

A reflection is going to become negative when you reflect over the x-axis. We started with f(x), and this became -f(x). A shift is going to turn into this function, f(x-h+k), where h represents the horizontal shift, and k represents the vertical shift.

For a stretch transformation, the function is going to look like this, where you have some constant multiplied by the function f(x): c∙f(x). 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. We're given the function f(x)=|x| and the corresponding graph. We need to match the following functions P of X, Q of X, and R of X to the correct corresponding graph, because all of these functions are transformations of our original function the absolute value of x. For P(x)=|x-3|+2, this looks most like a shift transformation. Thus, graph 2 matches with function A.

For Q(x)=-|x|, this seems like a reflection. Graph 1, which has just been flipped, matches with option B. By elimination, for function C, this matches with graph number 3, but notice we have a negative sign and constant in our function, causing both a vertical stretch and a reflection transformation. So, overall, these are the three functions that match with the three graphs below. That is the basic idea of a transformation of function. Hopefully, you found this helpful. Let me know if you have any questions, and thanks for watching.

Reflections of Functions

Welcome back, everyone. In the last video, we talked about transformations of functions. We discussed how there are 3 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 creasing 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 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 one, one, 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. We could also say that our function goes from \( f(x) \) to \( -f(x) \) 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 one, one, which we have on this left side, and we reflected over to positive one, one. So notice that our y value stayed the same, but our x values changed 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 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 \( f(x) = x + 2 \) and we're told to let \( h(x) \) be the function \( f(x) \) with a reflection over the x-axis. We're asked to sketch the graph of \( h(x) \) and determine which of the functions below matches the new function \( h(x) \). So what I'm first going to do is see if I can sketch the reflection \( h(x) \) on this graph over here. So this is the original function that we see \( f(x) \) sketched for us and I see in our example that it's given we're going to have a reflection over the x-axis. You can imagine taking this graph, 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 2 and finished at a y value of negative 2 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 \( h(x) \) from our original function \( f(x) \), but we're also asked to figure out what the new equation is going to look like based on the 4 options below.

Well, our function \( h(x) \) is simply \( f(x) \) 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 \( -f(x) \). So to draw this out, we're given that \( f(x) = x + 2 \), so this whole thing is going to be \( -x + 2 \), and if I go ahead and distribute this negative sign into the parenthesis, we're going to end up with \( -x - 2 \). So this is the equation for our reflected function \( h(x) \) and if I look at the 4 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.

Example 1

So let's see if we can solve this problem. Below, we have the function \( f(x) \), which is represented by the curve you see or the green dotted lines. We are asked to sketch a graph of \( g(x) \) where \( g(x) \) is the function \( f(x) \) after it has been reflected over the x-axis. This is a reflection transformation, as we have been learning about, and in this situation, what will happen is that we will have our graph reflected over the x-axis. When this happens, recall that reflecting over the x-axis is like taking the 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 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 a 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 are creasing this like a piece of paper, so this graph is going to look something like that. Once we have reflected over the x-axis, this is what the new function \( g(x) \) is going to look like with respect to our original function \( f(x) \), this green dotted curve. This is how you can solve the problem. Hope you found this helpful.

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

Shifting A Functions

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 video 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 a 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(x) that goes to f(x−h +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 see here, this green curve, and we shifted it up here, so it looked something like this. Notice how we went from a position of 0 to a position of 2. The x-values stayed the same, but the y-value changed by 2. 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 0, the green curve, and then let's say that this gets shifted over to the right. So our graph went from 0 to this position of 3, 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 0, so we can just ignore this h value that we see in the equation. So our function starts as f(x), and then this gets transformed tof(x +k), where k is the vertical shift, and since we shifted up by 2 we would say that k is 2. And notice how whenever we have a plus 2 the graph shifts up. So in this situation if you ever see f(x) +k you're always going to shift up, whereas if you see f(x) −k, the graph is going to shift down. Since we had a positive 2, we went up by 2. 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 0. 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(x) just like before, but then we went to f(x−h), and because h represents the horizontal shift, and we ended up at an x-value of 3, we would say we have f(x−3) and that's the horizontal shift. Now whenever you see f(x) −h, this means your graph is going to shift to the right, whereas if you see f(x) +h your graph shifts to the left, and we can see that because f(x−3) caused our graph to shift to the right by 3.

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 2, our graph would look something like this, and that means that the h value would be negative 2, so we would end up having our original function f(x) become f(x−−2), but we know that two negative signs are going to cancel each other, so this would actually become f(x) +2). 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 2 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(x), and we're asked to sketch the transformation f(x−2 +3). 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 2 that we have right there, and I can see that there's a minus sign in front of both which means we're going to shift to the right. So I can keep this as positive 2 for our h value, I can see also that we have a k value of positive 3, So what this means is since we have positive 2 for our H, and positive 3 for our K, that means we're going to shift 2 units to the right from our original position, and we're going to shift 3 units up. So we started here at 0, 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.

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

Graphs of Shifted and Reflected Functions

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. 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.

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. We started here at 0 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 2 transformations combined. 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 became 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 3 and our vertical shift of 2. And notice for the combined reflection and shift, we had this negative sign which showed up for the reflection, but then we had this 3 and this 2 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 combine the multiple transformations, but let's actually see if we can do an example of this ourselves.

This example says if \( g(x) \) is a transformation of the function \( f(x) \) is the absolute value of \( x \), write the equation for \( g(x) \). Now notice on this graph we have the original function \( f(x) \), and we also have the transformation \( g(x) \). So based on this graph, we're trying to figure out how \( g(x) \) is going to change \( f(x) \) or basically what the final equation for \( g(x) \) is going to look like. And to solve this, well, to figure out \( g(x) \), we need to figure out how \( f(x) \) has changed. 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 \( f(x) \) is going to become \( -f(x) \) 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. We were originally going to be at this position after the reflection, but now this graph got shifted 2 units to the left. We started here at 0, the origin, and we finished centered over here at negative 20. Now whenever we have a shift transformation, your function becomes \( f(x - \text{horizontal shift} + \text{vertical shift}) \). 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 2. So that means we're going to have our function \( -f(x - (-2)) \), 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 \( -f(x + 2) \). 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 \( g(x) \), \( g(x) \) is going to be the negative absolute value of \( x \) plus 2, because notice how we had \( x + 2 \) 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 transformed 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.

Example 2

See if we can solve this problem. In this problem, we have a function h(x), which is a transformation of the original function f(x) = x³. We're told in this transformation, the function is reflected over the x-axis and then shifted down two units. We are asked to write an equation for h(x) and sketch a graph of the function h(x). So let's begin.

First off, we should recognize what type of transformations are happening here. In this first case, I see that we have a reflection over the x-axis. Recall from previous videos that when we reflect over the x-axis, our function f(x) becomes -f(x). This is something we learned in the video on reflections. We can also see here we have a shift down 2 units. Since we're only shifted down, this means we're only being vertically shifted. When this happens, our original function f(x) becomes f(x) + k, where k represents the vertical shift. So these are the two situations we need to watch out for in this problem.

We’ll start off by examining the function here, which is f(x) = x³. The first transformation that happens is a reflection over the x-axis, and when this happens, the function becomes negative. So x³ is going to become -x³. This is the first transformation where we reflect over the x-axis. The next transformation is the shift down 2 units. Since we're shifting down 2 units, this k represents the vertical shift. Since we're shifted down, that means k is going to be -2 for the downward shift. Adding k to this equation, our transformation becomes -x³ - 2. This right here is our transformed function h(x), and this is what the new equation is going to look like. This is how you can determine the equation for h(x), and that's part a of this problem.

We are also asked to sketch a graph of h(x) as well. To do this, let's once again look at the transformations we have. This is the original function right here. First off, we have a reflection over the x-axis. Recall that when reflecting over the x-axis, you can imagine folding your graph over the x-axis, as if increasing it like a piece of paper. So this portion of the function is going to go down like this, and then this portion of the function is going to fold up 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 2 units. So the graph is not going to be centered at the origin, it’s actually going to be centered down, 2 units right here at -2. So our final graph is going to look like this for the transformed function.

This is our transformation h(x) as a graph, and this is the corresponding equation. This is how you can deal with multiple transformations on a single function. Hope you found this helpful.

Stretches and Shrinks of Functions

Welcome back, everyone. Up to this point, we've been talking about transformations of functions. We've looked 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. The nice thing about the stretch and shrink transformation is, unlike shifts where we deal with two numbers in our function, the stretch and shrink only involve one number. With stretches and shrinks, there are a few things you'll have to keep track of and remember for how the graph is going to behave, but in this video, we're 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. To understand this better, let’s look at these two cases down here. In this example, on the left, we have a vertical stretch or compression. I want to mention right off the bat 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.

In this example here, we have a function f(x), which is plotted on this graph; it's the 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. 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. 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 looks like this. So this is the function after you've stretched the graph horizontally, whereas if you were to perform 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.

The idea of a horizontal stretch or compression depends on where the constant is multiplied. 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 inside of the function. In the vertical case, whenever you have a constant between 0 and 1, the graph is going to vertically shrink. You're going to see a vertical shrink if this happens, whereas if your constant is greater than 1, it means the graph is going to stretch.

In the case of the horizontal stretch or compression, if your constant is between 0 and 1, then your graph is going to do a horizontal stretch, whereas if the constant is greater than 1, the graph is going to horizontally shrink. Notice how the vertical stretch or compression is the opposite of the horizontal stretch or compression when it comes to what constant stretches versus shrinks the graph. This is 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(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. Let’s first start with the case that we see on the left, which is a. For case a, we have 2 times the function, and the 2 is multiplied outside f(x). Whenever you have your constant multiplied outside of the function, this corresponds to the vertical case and, since our constant we see is 2, that's greater than 1, which means we're going to have a vertical stretch. What this means is our graph is going to vertically stretch by a factor of 2, so we can see right here that we have the point (2,2), but our y values are going to stretch vertically, so we're going to end up actually at (2,4), and at this point where we have (1, -1), we would stretch down here to (1, -2). Likewise, at this point (-1,1), we would stretch up here to (-1,2), and then at this point, we would stretch down here to (-1, -4). So our vertical stretch will 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 0.5 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 stretch or compression, but notice that the constant is now between 0 and 1, which means we're going to have a vertical shrink. So everything is going to shrink by a factor of 0.5; rather than being at this point (2,2), we're going to be at (2,1) because we're shrinking our graph. Rather than being here at one, we're actually going to be at (1, -0.5). Likewise, we'd be at a y-value of positive 0.5 there, and we'd be at a y-value of negative 1 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 0.5 multiplied inside the function. Since we're on the inside of the function this corresponds to the horizontal stretch or compression, and since the constant that we have is between 0 and 1, this corresponds to a horizontal stretch. 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 a 0.5 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 2 would cancel the 2 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 2, it would be over here at 4, and down here rather than being at an x value of 1 we'd be at an x value of 2, and rather than being at -1 we would be here at -2, and then rather than being at -2 we would be over here at -4. So in this case, our graph is going to horizontally stretch. Notice how we basically stretched it on the horizontal axis.

This is the basic idea behind the stretch transformation. Hopefully, you found this video helpful, and let me know if you have any questions.

Example 3

Okay. So let's give this problem a try. Here we have the function \( f(x) = c \times x^2 \), and we're asked to graph our function \( f(x) \) when \( c = 2 \) and when \( c = 0.5 \). Now what we have on the graph already here is a curve which represents our function \( f(x) = x^2 \). This is the general case for a parabola, but we need to figure out what's going to happen if we take various constants and plug them into this function. So let's see. We're going to start for the case where our constant here is equal to 2. This means that \( f(x) = 2 \times x^2 \). This is what the function is going to look like if we plug 2 into the constant. Now what I'm going to do is try a bunch of different \( x \) values. I'll first try an \( x \) value of 0. If this were to happen, we would have \( 2 \times 0^2 \), and \( 2 \times 0 \) is 0. \( 0^2 \) is just 0. So that means we're going to have a point at 0. That's going to be one point we can plot here. Now what I'm also going to do is plot a value of -1. If I do this, we're going to get \( 2 \times (-1)^2 \), just replacing this \( x \) with -1. In this case, \( 2 \times (-1) \) is -2, so we'll have -2 squared, and -2 squared is actually positive 4. So that means that a value of -1, I mean, at -1, we're going to be at a \( y \) value of 4. So this is another point that we could put on this graph.

Now lastly, I'm going to try a point of positive 1. So in this case, we'll have \( 2 \times 1^2 \). \( 2 \times 1 \) is 2, and 2 squared is 4. So at a value of 1, we're going to or at an \( x \) value of 1, we're going to have a \( y \) value of 4. So our graph is going to look something like this when we replace this constant that we see here with a 2. 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 that it caused a horizontal shrink, and this happens whenever your constant is greater than 1. So it makes sense that we would get this situation where the graph shrinks since 2 is greater than 1. But now let's try this other situation where we have \( c = 0.5 \). This basically means our constant is between 0 and 1.

So let's see what happens if I do this. Well, in this case, our function \( f(x) \) is going to become \( \frac{1}{2} x^2 \), because now we're just gonna take this constant and replace it with the \( \frac{1}{2} \) that we have over here. Let's see how this behaves. What I'm first going to do is plug in an \( x \) value of 0 like we did before, in which case we have \( \frac{1}{2} \times 0^2 \), and anything multiplied by 0 is just 0, so we already know this whole thing will come out to 0, meaning we'll have the same origin point 0. Now next, what I'm going to do is I'm actually going to try an \( x \) value of 2. And the reason that I'm trying 2 specifically is that if I take this \( x \) and replace it with a 2, notice we're going to get \( \frac{1}{2} \times 2^2 \). And \( \frac{1}{2} \) and 2 will actually cancel each other here because taking 2 and cutting it in half will just give you 1. So really you're just getting up with 1 squared, which is just 1. So if you try an \( x \) value of positive 2, you're going to end up here at a \( y \) value of 1. And likewise, if you were to try -2, well, in this case, we would have \( \frac{1}{2} \times (-2)^2 \), in which case this 2 would cancel with that one giving us -1 squared. And -1 squared is just positive 1 because -1 times -1 will cause the negative signs to cancel. So at an \( x \) value of -2, we're going to be at a \( y \) value of positive one again, meaning our function is going to look something like this when we multiply the inside by \( \frac{1}{2} \). And notice in this case, we got a horizontal stretch because whenever our constant is between 0 and 1, we get a horizontal stretch, whereas when our constant is greater than 1, 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 likewise, 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 often 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.

Domain and Range of Function Transformations

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 common 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. Therefore, 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; and 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(x). 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 3 to positive 3. 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 3 to positive 3 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 position 12 from position 0. Let's see what happens here. 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 2 to positive 4, 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 5, so our range is going to go from negative one to 5, and notice how the domain and range that we got are different 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(x) is equal to x², and we're asked to sketch a graph of the function g(x) is equal to (x − 3)² + 2 and determine its domain and range. Now, the function x² is just going to be a parabola centered at the origin, and if I look at the transformation that we're given, I notice that this looks to be in the form f(x − h) + k, which is a shift transformation. Now, I can see here that the 'h' corresponds with this 3 right here because we have x − h within the function and then inside the square function, we have x − 3. So I can tell that our 'h' is going to be 3. Now I can also see that our 'k' value is going to correspond to this positive 2. So our 'k' is positive 2. 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 1, 2, 3 units to the right and 1, 2 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. Now if I look at the domain and range of this 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 are 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(x). But what about the range? Well, originally, our range on the initial parabola goes from 0 to infinity because we can see here that on the y-axis it goes from 0 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 2 all the way to infinity, meaning our range is going to go from 2 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. 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.

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 $f\left(x\right)$.

• A. Dom: $\left[1,4\right]$ , Ran: $\left[-5,-1\right]$
• B. Dom: $\left[1,5\right]$ , Ran: $\left[-5,1\right]$
• C. Dom: $\left[-1,3\right]$ , Ran: $\left[-2,4\right]$
• D. Dom: $\left[-2,3\right]$ , Ran: $\left[2,4\right]$

Here’s what students ask on this topic:

What are the three basic types of transformations in trigonometry?

The three basic types of transformations in trigonometry are reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, changing its sign. Shifts move the function horizontally or vertically, represented by $f(x-h)+k$. Stretches and shrinks alter the graph's shape based on constants multiplied inside or outside the function. Understanding these transformations is crucial for analyzing the domain and range of functions, as they can significantly change these properties.

How does a reflection transformation affect a function's graph?

A reflection transformation affects a function's graph by flipping it over a specific axis. If the reflection is over the x-axis, the y-values of the function change signs, transforming $f\left(x\right)$ into $-f\left(x\right)$. If the reflection is over the y-axis, the x-values change signs, transforming $f\left(x\right)$ into $f(-x)$. This transformation changes the graph's orientation but not its shape.

What is the difference between a vertical and horizontal shift in function transformations?

A vertical shift moves a function's graph up or down, while a horizontal shift moves it left or right. In the function notation $f(x-h)+k$, $h$ represents the horizontal shift and $k$ represents the vertical shift. A positive $h$ shifts the graph to the right, while a negative $h$ shifts it to the left. A positive $k$ shifts the graph up, and a negative $k$ shifts it down.

How do you determine the domain and range of a transformed function?

To determine the domain and range of a transformed function, observe the new graph after the transformation. The domain is found by projecting the graph onto the x-axis, and the range is found by projecting it onto the y-axis. For example, if a function $f\left(x\right)$ is shifted to the right by 2 units and up by 3 units, the domain and range will adjust accordingly. Always consider how the transformation affects the graph's position and shape to accurately determine the new domain and range.

What is the effect of a vertical stretch or shrink on a function's graph?

A vertical stretch or shrink affects a function's graph by altering its height. If a constant $c$ is multiplied outside the function, $cf\left(x\right)$, and $c$ is greater than 1, the graph stretches vertically. If $c$ is between 0 and 1, the graph shrinks vertically. For example, multiplying $f\left(x\right)$ by 2 will double its height, while multiplying by 0.5 will halve it.