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, altering its position without changing its shape. Stretches and shrinks adjust the graph's height or width based on a constant multiplier. Understanding these transformations is crucial for analyzing the domain and range of functions, as they can significantly alter these properties. For example, a vertical shift can change the range while leaving the domain unchanged.

Intro to Transformations

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. This topic can seem a bit complicated at first because there are many 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 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 now.

A transformation occurs when a function is manipulated such that it changes position or shape. An example of this, and actually, there are three main examples, are the three examples we have listed down here. 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. Another type of transformation is a shift. A shift occurs when you move a function. So if we were to take this function, which is currently at the position zero zero, 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 shift transformation. 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. 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. 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(x), and this became negative f(x). A shift is going to turn into this function: fx-h+k. In this notation, the h represents the horizontal shift, and the k represents the vertical shift. For a stretch transformation, the function is going to look like this: c⋅fx, where 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. In this example, we're given the function fx=|x|, and we are also given the corresponding graph. What we're asked to do is match the following functions: P of X, Q 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.

Now for this first function that I see, px is equal to absolute value of x minus 3 plus 2, 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 3 plus 2. So what I'm going to do is look for whichever one of these graphs appears to be a shift, and if I look at all of these, number 2 looks a lot like a shifted version of our original graph, because we started here at the origin, and then we finished somewhere up here. So I'm going to say that graph 2 matches with function a.

Now let's take a look at function b. We have that qx is equal to the negative absolute value of x. Now we first need to figur

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 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 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 negative 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 1,1, which we have on this left side, and we reflected over to positive 1,1. 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 \( f(x) = x + 2 \) 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 gonna have a reflection over the x-axis. You can imagine taking this graph, creasing 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 negative \( f(x) \). So to draw this out, we're given that \( f(x) = x + 2 \), so this whole thing is going to be negative \( x + 2 \), and if I go ahead and distribute this negative sign into the parenthesis, we're going to end up with negative \( x - 2 \). 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.

Reflections of Functions Example 1

So let's see if we can solve this problem. Below, we have the function f(x), which is the curve you see, represented by 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 that we have been learning about. In this situation, what's going to 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-axis and folding it down. 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. 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 is 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. $g\left(x\right)=\sqrt{x}-2$

Shifts of 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 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. Let's get right into this. A shift occurs when a function is moved either vertically or horizontally from its original position. It's important to note 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.

The general form for shifts is when you have your original function f of x that goes to fx-h+k after you've shifted the function. In this notation, the h represents the horizontal shift, whereas the k represents the vertical shift. Now, to understand this concept a bit better, let's look at the strictly vertical shift. This means that you're only going to move your graph vertically. For example, if we took this parabola, this green curve, and we shifted it upwards so it looked as it has moved from a position of 0 to a position of 2. The x values stayed the same, but the Y value changed by 2. So for the vertical shift, it's the Y values that change.

Now let's look at the horizontal shift. An example of a horizontal shift is whenever your graph moves strictly horizontally. For instance, we have our original curve here at 0, the green curve, and then it 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. Also, it's important to know how the function behaves when dealing with the shift transformation. Notice in both cases we have the same general form that we saw at the start here, but for the vertical shift, our h value is always 0, so we can ignore this h value in the equation. So our function starts as fx, and then this gets transformed to fx+k, where k is the vertical shift, and since we shifted up by 2, we would say that k is 2. Notice how whenever we have a plus 2, the graph shifts up. In this situation, if you ever see fx+k, you're always going to shift up, whereas, if you see fx-k, the graph is going to shift down.

Now, for the horizontal shift in this situation, the k value is 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 fx just like before, but then we went to fx-h, and because h represents the horizontal shift, and we ended up at an x value of 3, we would say we have fx-3, and that's the horizontal shift. Whenever you see fx-h, this means your graph is going to shift to the right, whereas if you see fx+h your graph shifts to the left. 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 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 fx become fx--2, but we know that two negative signs are going to cancel each other, so this would actually become fx+2. 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. In this example, we're given the function fx, and we're asked to sketch the transformation fx-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 also see that we have a k value of positive 3. 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 2, 3. 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.

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 & 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. So 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 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 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. So, we know that g(x) is a transformation of our original function f(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(x) is going to become negative 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. So we were originally going to be at this position after the reflection, but now this graph got shifted 2 units to the left. So we started here at 0, the origin, and we finished centered here at negative 20. Now whenever we have a shift transformation, you're going to have that your function becomes f(x minus the horizontal shift) + 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 2. So that means we're going to have our function negative f(x - (-2)), but notice that these negative signs will just cancel, so all we're going to end up having is negative 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 transformed function g(x), g(x) is going to be the negative absolute value of x + 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.

Graphs of Shifted & Reflected Functions Example 1

See if we can solve this problem. In this problem, we have a function \( h(x) \), which is a transformation of the original function \( f(x) = x^3 \). We're told in this transformation, the function is reflected over the x-axis and then shifted down two units. We're asked to write an equation for \( h(x) \) and then sketch a graph of the function \( h(x) \). So, let's begin. Now 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. Now we can also see here we have a shift down 2 units, and 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. 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 \( f(x) = x^3 \). 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 \( x^3 \) is going to become \(-x^3 \). 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 2 units. So since we're shifting down 2 units, this \( k \) represents the vertical shift. And since we're shifted down, that means \( k \) is going to be negative 2 for the downward shift. So if we add \( k \) to this equation that we have here, our transformation is going to become \(-x^3 \) and we're going to have minus 2 because it's a negative 2 that we plug in for \( k \). So, this right here is going to be our transformed function \( h(x) \), 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 \( h(x) \), and that's part a of this problem.

But we're also asked to sketch a graph of \( h(x) \) 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. Recall 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 to go into full 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 2 units. So this graph is not going to be centered at the origin, it's actually going to be centered down 1, 2 units right here at negative 2. So, our final graph is going to look like this for the transformed function. So this is our transformation \( h(x) \) 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.

Stretches & Shrinks of Functions

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 keep 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(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 that 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 inside of the function. And when it comes to the vertical case, whenever you have a situation where the constant that you have is between 0 and 1, the graph is going to vertically shrink. You're going to see a vertical shrink if this happens, whereas if you see your constant is greater than 1, then this means the graph is going to stretch. Now in the case for the horizontal stretch or compression, if you see that your constant is between 0 and 1, then your graph is actually going to do a horizontal stretch, whereas if the constant is greater than 1, 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(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 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. So 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), right? Because we're vertically stretching. Likewise, at this point (-1,1), we would stretch up here to (-1,2), And then at this point (-2, -2), we would stretch down here to (-2, -4) as a y-value. 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 1/2 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 1/2, rather than being at this point (2,2), we're going to be at (2,1) because we're shrinking our graph. And rather than being here at (1, -1) we're actually going to be at (1, -0.5). Likewise we’d be at a y-value of positive (1, 0.5) there, and we’d be at a y-value of negative (2, -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 1/2 multiplied inside the function. So in this case 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. 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 1/2 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 negative 1, we would be here at negative 2, and then rather than being at negative 2 we would be over here at negative 4. 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.

Stretches & Shrinks of Functions Example 1

Okay. So let's give this problem a try. Here we have the function f of x, which is equal to c times x squared, and we're asked to graph our function f of x when c is equal to 2 and when c is equal to 1 half. Now, what we have on the graph already here is a curve that represents our function f of x is equal to x squared. 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 begin. For the case where our constant here is equal to 2, this means that fx=2∙x2. This is what the function is going to look like if we plug 2 into the constant.

Now, I'm going to try a bunch of different x values. First, I'll try an x value of 0. If this happens, we would have 2 times 0 squared, and 2 times 0 is 0; 0 squared is just 0. So, this means we're going to have a point at 0. Next, I'll plot a value of -1. If I do this, we're going to get 2 times negative one squared, just replacing this x with a -1. In this case, 2 times -1 is -2, so we'll have -2 squared, and -2 squared is actually 4. Thus, at a value of -1, we're going to be at a y-value of 4. Lastly, I'll try a point of 1. So, in this case, we'll have 2∙x2, where x = 1, giving a value of 4. So our graph is going to look something like this when we replace this constant with a 2. This makes sense because we discussed previously that having a constant multiplied inside of your function causes a horizontal shrink if the constant is greater than 1.

But now let's try the situation where our constant is 1 half. This means our constant is between 0 and 1, so let's see what happens. In this case, our function f(x) is going to become 12∙x2. Well, let's begin by plugging in an x value of 0. In this case, we have 1 half times 0 squared, and anything multiplied by 0 is just 0, so we have the same origin point at 0. Now, I'll try an x value of 2. In this case, we get 1 half times 2 squared. The 1 half and 2 will cancel each other out, giving you 1 squared, which is just 1. So at an x value of 2, you would be at a y value of 1, and similarly, for -2, you would again be at a y value of 1. This results in a horizontal stretch because our constant is between 0 and 1, which differs from the horizontal shrink we had previously.

It's interesting to note that when we had a horizontal shrink, it appeared as if there was a vertical stretch. Likewise, when we had a horizontal stretch, it seemed as if there was a vertical compression or shrink. These appearances are often the case in symmetric graphs like parabolas, suggesting a correlation between horizontal and vertical transformations.

So, this is how you solve the problem and understand the effects of constants on the graph of a quadratic function. Hopefully, you found this explanation helpful.

Domain & Range of Transformed Functions

Welcome back, everyone. So, up to this point, we've talked about the 3 main types of transformations: 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 when 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. 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. 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, 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 12 units from the position 0. By looking at this graph, if I go ahead and try to 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.

To really solidify this concept, let's try an example to see how we do. 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) = (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. Here, the h corresponds to this 3 because we have x - h within the function, and inside the square function, we have x - 3. Our h is 3. Also, our k value, which is positive 2, modifies the original position. Since the h was positive, the graph will shift right and since the k is positive, the graph will shift up. So our new parabola is going to go 3 units to the right and 2 units up.

If I look at the domain and range of this new parabola, I can see that the domain is going to be all real numbers because this parabola expands in all directions to the left and right. So, our domain goes from negative infinity to positive infinity for this new function g(x). But what about the range? Well, originally, the range of the initial parabola goes from 0 to infinity because we can see that on the y-axis, it goes from 0 and continuously goes up. But after our transformation, our range goes from 2 to infinity, meaning our range is from 2 to infinity and includes this value. This demonstrates how the domain stays the same, but the range changes when we shift 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: $\left[2,4\right]$