Understanding Function Transformations

In this video, we're going to talk about functions, transformations, and things like that. So, let's say if we have the function f of x. Let's say we add 2 to it. What type of shift do we have here? If you add 2 to it, this is known as a vertical shift. The graph is going to move up 2 units. Likewise, let's say you have x minus 3. That's a vertical shift down 3 units. Now what about f minus 4? This is a horizontal shift and it shifts... Do you think it shifts 4 units to the left or to the right? It turns out that this shift is 4 units to the right. If you set x minus 4 equal to 0, x will equal 4. So it doesn't shift to the left 4 units, but it shifts to the right. Now let's say if we have f of x plus 3. This would shift to the left 3 units. If you set the inside equal to 0, you'll get x is equal to negative 3. Now, there's some other ones that you need to know as well. If you have a negative on the outside, it reflects over the x-axis. Now, if you have a negative on the inside, it reflects over the y-axis. And if you have a negative on the outside and on the inside, it reflects over the origin. Now, let's say if we put a 2 in front of f of x. This is known as a vertical stretch. And let's say if we put a fraction in front of f of x, it's going to shrink vertically. Now, if we put the 2 on the inside, it's going to be a horizontal, not a stretch, but a horizontal shrink. So, be careful with that one. And, if we put a 1 half on the inside, this is going to be a horizontal stretch. So those are some things that you want to keep in mind in terms of transformations. So let's go over some examples. Let's say if we have the parent function f of x is equal to x squared. And basically, that's a parabola that opens upward like this. Now, let's say if we wish to graph this function x squared plus 3. This is going to be... a vertical shift up three units. So the graph is going to start at 0, 3, and it's going to look the same. Now, for example, let's say if we wanted a graph x squared minus 2. We're going to have the same type of graph, but it's going to shift down two units, and so it's going to look like that. Now what about this one? Let's say that y is equal to the absolute value of x. So here's the parent function. Now what if we put, let's say, x plus 2. How's the graph going to look like now? In this case, it's going to shift 2 units to the left. So it's going to look like this. So this point was at 0, and now this point is at negative 2. Now what about this one, the absolute value of x-3? So we're going to have the same type of function, but it's going to shift 3 units to the right. So it's going to look like that. Now what about this one? Let's say that y is equal to the square root of x. And the parent function looks like this. Go ahead and graph y is equal to negative square root x, and then square root of negative x. You can put it here, and then we'll do one more. So this one reflects over the x-axis, therefore it's going to look like this. This one reflects over the y-axis, and here's the y-axis, and so it's going to go that way. And if we have a negative on the inside and the outside, it's going to reflect over the origin. And so it's going to go towards quadrant 3. Now, a good way to, like, remember which direction it goes is to look at the signs. We have positive x and positive y. x is positive in quadrant 1 and 4, towards the right, and y is positive in 1 and 2. So when x is positive, you need to go to the right, and when y is positive, you go up, so this is going to go towards quadrant 1. Now for the next one, x is positive, y is negative. So we're going to go to the right, and then y is negative as you go down. So that takes us to quadrant 4. For this one, x is negative, but y is positive. negative on the left y is positive as you go up so that's towards quadrant 2 and for the last one X is negative and Y is negative so it's going to go towards a quadrant 4 not quadrant 4 but quadrant 3 this is quadrant 4 this example Now let's graph these three functions using points. The absolute value of x, and then 2 times the absolute value of x, and then 1 half. So we're going to have the point 0, 0, 1, 1, 2, 2. Negative 1, positive 1, negative 2, positive 2. And so that's the graph for the absolute value of x. Now for 2 times the absolute value of x, it's going to be as follows. We're still going to have the point , but when x is 1, y is going to be 2. When x is 2, y is 4. And then it's symmetric about the y-axis, so the right side and the left side will look the same. So here we have a vertical stretch. Notice that the y values were doubled. And so that's the effect of putting a 2 in front of the function. For the last example, we're going to have a vertical shrink. So, at 1, it's going to be a half. At 2, it's going to be 1. And so this is a vertical shrink. The y values were cut in half. They're half of what they were compared to that graph. So now you can visually see how a vertical stretch appears and a vertical shrink appears as well. So for a vertical stretch, the y values are increased. For a vertical shrink, the y values were decreased. So now let's go back to this graph, y equals x squared. And I want you to graph, let's say... Actually, let's use a different example. Let's use the square root of x, and let's use the square root of 2x, and also the square root of 1 half x. Hopefully I can fit all of them here. So I only need the right side of the graph. So when x is 0, y is 0. When x is 1, y is 1. The square root of 4 is 2. So when x is 4, y is 2. So that's the parent function. It looks something like that. Now for this one, when x is 0, y is 0. When x is a half, y is going to be 1. And when x is 2, y is going to be 2. So notice that the x values were decreased by 2. Here, it was 4, and it had the same y value of 2. But for the same y value, the x value is now 2. Now granted, this graph will continue to grow. But I want to stop at the point where the y value is the same. So I'm going to stop at this point here. So you can see the effect that the graph has on x. We said when x is 1 half, y will be 1. And when x is 2, y is 4. So for the same y value of 2, x is no longer 4, but is 2. So as you can see, the x value was reduced by 2. And that's why this is known as a horizontal shrink. It shrinks the x values by a factor of 2. Now let's look at the last example. Thank Now, when x is 0, y is 0. But when x is 2, y will be 1. And when x is 8, half of 8 is 4, the square root of 4 is 2. So when x is 8, y will be 2. So for the same y value of 2, x has increased to 8. So going from 4 to 8, we can see that x was increased by a factor of 2. So this is a horizontal stretch. make sure that the y values are the same otherwise the graph may look like a vertical shrink for instance if I were to compare This graph versus this graph. If you stretched out this might appear as if it's a vertical stretch compared to this one because this looks higher. However, you need to compare the x values for the same y value. If you do that you can clearly see that this is a horizontal shrink and not a vertical stretch. In this one you can see that it's a horizontal stretch not a vertical shrink. Now let's move on. Let's work on some other examples. So make sure you're aware of the parent functions. Let me just run through them real quick. So this is the graph for y equals x. The next one you need to be familiar with, as we mentioned earlier in this video, is y equals x squared. and here's another one this is equal to this is y equals x cube and you've seen this one already y equals square root x and then we have the cube root of x and the absolute value of x Now there's some other functions, but these are the main ones that we're going to go over for now. So how would you graph this function? x minus 2 squared plus 3. So graph it using transformations. So the parent function is x squared, which looks like this. However, we can see that we have a horizontal shift, two units to the right, and a vertical shift, up three. So I'm going to shift it two units to the right and up three. And then you can just draw a rough sketch. And that's it for that example. Now what about this one? Let's say if we have 3 minus x plus 2 squared. So once again we have a parabola. And this time, I'm going to plot it more accurately, instead of using a rough sketch. Now, this is the vertical shift of 3. If you want to, you can rewrite it this way. This is y is equal to negative x plus... plus 2 squared plus 3. So we have a vertical shift of 3. Now we also have a horizontal shift left 2. So the center, or the vertex, of the parabola is going to be at negative 2, 3, which is here. Now, does a parabola open upward or downward? Normally, it would open upward. However, we do have a negative sign, so it's going to open downward. But let's get some points. Let's graph it accurately. Now, keep in mind the parent function is x squared, so 1 squared is 1. That means that if you travel one unit to the right from your vertex, the next point will be down 1. And one unit to the left, it's also going to be down one. So if you plug in 1 into x, the y value should be 2. I mean, not 1, but negative 1, because this is negative 1 here. So for instance... Negative 1 plus 2 is 1. 1 squared is 1. So 3 minus 1 is 2, and that gives us that point. Now, 2 squared is equal to 4. So as we travel 2 units away from the vertex, we need to go down 4. So the next point is going to be 0, negative 1, and also negative 4, negative 1, due to the symmetry of the graph. And that should be enough to get a decent sketch. Now, if you want to test it, let's use negative 4. Let's see if we get negative 1. So 3 minus negative 4 plus 2 squared. So negative 4 plus 2, that's negative 2. Negative 2 squared is positive. 4 3 minus 4 is negative 1 so it does gives us that point if you try 0 you get the same thing 0 plus 2 is 2 2 squared is 4 3 minus 4 is negative 1 so this technique works if you don't want to make a table and if you want to draw an accurate sketch Let's try this one. Let's say that y is equal to 4 minus the square root of 3 minus x. Let's draw an accurate sketch. Now, let's rewrite it first. You need to see it like this. So there's a vertical shift up 4 units. And there's a horizontal shift, but it looks a little different. Is it 3 units to the right or 3 units to the left? If you're ever unsure, set the inside equal to 0 and solve for x. I'm going to take this term and move it to this side. It's going to switch from negative x to positive x. And so x is equal to positive 3. So that indicates that we have a horizontal shift to the right of 3 units. So the starting point is going to be at 3, 4. Now the parent function is the square root of x. And we have a negative on the outside and a negative in front of the x. So will the graph shift towards quadrant 1, towards quadrant 2, towards quadrant 3, or towards quadrant 4? Well, x is negative towards the left, and y is negative as you go down. So it's going to go towards quadrant 3. So now that we know the direction in which this graph is going to go, keep in mind... Having this negative sign here, it reflects over the x-axis, and having it here, it reflects over the y-axis. Originally, this is the square root of x. If you reflect it over the x-axis, it looks like this. If you reflect it over the y-axis, it looks like that. When it's reflected both about the x-axis and the y-axis, it's equivalent to reflecting about the origin. So we can clearly see that it's going to go in this direction. Now let's get some other points. So the parent function is the square root of x. The square root of 1 is 1. That means that as we move one unit to the left, we need to go down one. So that's going to give us the point 2, 3. So if you plug in 2, you should get a y value of 3. So 4 minus the square root of 3 minus 2. 3 minus 2 is 1. And the square root of 1 is 1. 4 minus 1 is 3. Which we do get now the next best point to use is for the square root of 4 is 2 So as we travel 4 units to the left we need to go down to So four units to the left will take us to the x value of negative one And we need to go down to so the y value will be two And let's do it one more time the square root of nine is three So this is at three so if we go nine units to the left three minus nine is negative six That's going to take us to this point, and we need to go down three, so we're starting at four four minus three is one So we're going to have the point negative 6, 1. So if you plug in negative 6, that should give you 1. So 4 minus square root 3 minus negative 6. 3 minus negative 6 is the same as 3 plus 6, which is 9. And the square root of 9 is 3, and 4 minus 3 is 1. So we do get this point. So now we can plot it. And so, let me see if I can do that a little better. It should be something like that. My graph is not perfect, but you get the point. It has this general shape to it. And that's how you can graph that particular function.

In this video, we're going to talk about functions, transformations, and things like that. So, let's say if we have the function f of x. Let's say we add 2 to it. What type of shift do we have here?

If you add 2 to it, this is known as a vertical shift. The graph is going to move up 2 units. Likewise, let's say you have x minus 3. That's a vertical shift down 3 units.

Now what about f minus 4? This is a horizontal shift and it shifts... Do you think it shifts 4 units to the left or to the right?

It turns out that this shift is 4 units to the right. If you set x minus 4 equal to 0, x will equal 4. So it doesn't shift to the left 4 units, but it shifts to the right. Now let's say if we have f of x plus 3. This would shift to the left 3 units.

If you set the inside equal to 0, you'll get x is equal to negative 3. Now, there's some other ones that you need to know as well. If you have a negative on the outside, it reflects over the x-axis. Now, if you have a negative on the inside, it reflects over the y-axis.

And if you have a negative on the outside and on the inside, it reflects over the origin. Now, let's say if we put a 2 in front of f of x. This is known as a vertical stretch. And let's say if we put a fraction in front of f of x, it's going to shrink vertically. Now, if we put the 2 on the inside, it's going to be a horizontal, not a stretch, but a horizontal shrink.

So, be careful with that one. And, if we put a 1 half on the inside, this is going to be a horizontal stretch. So those are some things that you want to keep in mind in terms of transformations. So let's go over some examples.

Let's say if we have the parent function f of x is equal to x squared. And basically, that's a parabola that opens upward like this. Now, let's say if we wish to graph this function x squared plus 3. This is going to be...

a vertical shift up three units. So the graph is going to start at 0, 3, and it's going to look the same. Now, for example, let's say if we wanted a graph x squared minus 2. We're going to have the same type of graph, but it's going to shift down two units, and so it's going to look like that. Now what about this one? Let's say that y is equal to the absolute value of x.

So here's the parent function. Now what if we put, let's say, x plus 2. How's the graph going to look like now? In this case, it's going to shift 2 units to the left.

So it's going to look like this. So this point was at 0, and now this point is at negative 2. Now what about this one, the absolute value of x-3? So we're going to have the same type of function, but it's going to shift 3 units to the right. So it's going to look like that. Now what about this one?

Let's say that y is equal to the square root of x. And the parent function looks like this. Go ahead and graph y is equal to negative square root x, and then square root of negative x.

You can put it here, and then we'll do one more. So this one reflects over the x-axis, therefore it's going to look like this. This one reflects over the y-axis, and here's the y-axis, and so it's going to go that way. And if we have a negative on the inside and the outside, it's going to reflect over the origin.

And so it's going to go towards quadrant 3. Now, a good way to, like, remember which direction it goes is to look at the signs. We have positive x and positive y. x is positive in quadrant 1 and 4, towards the right, and y is positive in 1 and 2. So when x is positive, you need to go to the right, and when y is positive, you go up, so this is going to go towards quadrant 1. Now for the next one, x is positive, y is negative. So we're going to go to the right, and then y is negative as you go down.

So that takes us to quadrant 4. For this one, x is negative, but y is positive. negative on the left y is positive as you go up so that's towards quadrant 2 and for the last one X is negative and Y is negative so it's going to go towards a quadrant 4 not quadrant 4 but quadrant 3 this is quadrant 4 this example Now let's graph these three functions using points. The absolute value of x, and then 2 times the absolute value of x, and then 1 half. So we're going to have the point 0, 0, 1, 1, 2, 2. Negative 1, positive 1, negative 2, positive 2. And so that's the graph for the absolute value of x. Now for 2 times the absolute value of x, it's going to be as follows.

We're still going to have the point , but when x is 1, y is going to be 2. When x is 2, y is 4. And then it's symmetric about the y-axis, so the right side and the left side will look the same. So here we have a vertical stretch. Notice that the y values were doubled.

And so that's the effect of putting a 2 in front of the function. For the last example, we're going to have a vertical shrink. So, at 1, it's going to be a half. At 2, it's going to be 1. And so this is a vertical shrink.

The y values were cut in half. They're half of what they were compared to that graph. So now you can visually see how a vertical stretch appears and a vertical shrink appears as well. So for a vertical stretch, the y values are increased.

For a vertical shrink, the y values were decreased. So now let's go back to this graph, y equals x squared. And I want you to graph, let's say... Actually, let's use a different example. Let's use the square root of x, and let's use the square root of 2x, and also the square root of 1 half x.

Hopefully I can fit all of them here. So I only need the right side of the graph. So when x is 0, y is 0. When x is 1, y is 1. The square root of 4 is 2. So when x is 4, y is 2. So that's the parent function.

It looks something like that. Now for this one, when x is 0, y is 0. When x is a half, y is going to be 1. And when x is 2, y is going to be 2. So notice that the x values were decreased by 2. Here, it was 4, and it had the same y value of 2. But for the same y value, the x value is now 2. Now granted, this graph will continue to grow. But I want to stop at the point where the y value is the same. So I'm going to stop at this point here.

So you can see the effect that the graph has on x. We said when x is 1 half, y will be 1. And when x is 2, y is 4. So for the same y value of 2, x is no longer 4, but is 2. So as you can see, the x value was reduced by 2. And that's why this is known as a horizontal shrink. It shrinks the x values by a factor of 2. Now let's look at the last example.

Thank Now, when x is 0, y is 0. But when x is 2, y will be 1. And when x is 8, half of 8 is 4, the square root of 4 is 2. So when x is 8, y will be 2. So for the same y value of 2, x has increased to 8. So going from 4 to 8, we can see that x was increased by a factor of 2. So this is a horizontal stretch. make sure that the y values are the same otherwise the graph may look like a vertical shrink for instance if I were to compare This graph versus this graph. If you stretched out this might appear as if it's a vertical stretch compared to this one because this looks higher. However, you need to compare the x values for the same y value. If you do that you can clearly see that this is a horizontal shrink and not a vertical stretch.

In this one you can see that it's a horizontal stretch not a vertical shrink. Now let's move on. Let's work on some other examples. So make sure you're aware of the parent functions.

Let me just run through them real quick. So this is the graph for y equals x. The next one you need to be familiar with, as we mentioned earlier in this video, is y equals x squared.

and here's another one this is equal to this is y equals x cube and you've seen this one already y equals square root x and then we have the cube root of x and the absolute value of x Now there's some other functions, but these are the main ones that we're going to go over for now. So how would you graph this function? x minus 2 squared plus 3. So graph it using transformations.

So the parent function is x squared, which looks like this. However, we can see that we have a horizontal shift, two units to the right, and a vertical shift, up three. So I'm going to shift it two units to the right and up three.

And then you can just draw a rough sketch. And that's it for that example. Now what about this one?

Let's say if we have 3 minus x plus 2 squared. So once again we have a parabola. And this time, I'm going to plot it more accurately, instead of using a rough sketch. Now, this is the vertical shift of 3. If you want to, you can rewrite it this way. This is y is equal to negative x plus...

plus 2 squared plus 3. So we have a vertical shift of 3. Now we also have a horizontal shift left 2. So the center, or the vertex, of the parabola is going to be at negative 2, 3, which is here. Now, does a parabola open upward or downward? Normally, it would open upward.

However, we do have a negative sign, so it's going to open downward. But let's get some points. Let's graph it accurately.

Now, keep in mind the parent function is x squared, so 1 squared is 1. That means that if you travel one unit to the right from your vertex, the next point will be down 1. And one unit to the left, it's also going to be down one. So if you plug in 1 into x, the y value should be 2. I mean, not 1, but negative 1, because this is negative 1 here. So for instance... Negative 1 plus 2 is 1. 1 squared is 1. So 3 minus 1 is 2, and that gives us that point.

Now, 2 squared is equal to 4. So as we travel 2 units away from the vertex, we need to go down 4. So the next point is going to be 0, negative 1, and also negative 4, negative 1, due to the symmetry of the graph. And that should be enough to get a decent sketch. Now, if you want to test it, let's use negative 4. Let's see if we get negative 1. So 3 minus negative 4 plus 2 squared.

So negative 4 plus 2, that's negative 2. Negative 2 squared is positive. 4 3 minus 4 is negative 1 so it does gives us that point if you try 0 you get the same thing 0 plus 2 is 2 2 squared is 4 3 minus 4 is negative 1 so this technique works if you don't want to make a table and if you want to draw an accurate sketch Let's try this one. Let's say that y is equal to 4 minus the square root of 3 minus x. Let's draw an accurate sketch. Now, let's rewrite it first.

You need to see it like this. So there's a vertical shift up 4 units. And there's a horizontal shift, but it looks a little different.

Is it 3 units to the right or 3 units to the left? If you're ever unsure, set the inside equal to 0 and solve for x. I'm going to take this term and move it to this side.

It's going to switch from negative x to positive x. And so x is equal to positive 3. So that indicates that we have a horizontal shift to the right of 3 units. So the starting point is going to be at 3, 4. Now the parent function is the square root of x. And we have a negative on the outside and a negative in front of the x.

So will the graph shift towards quadrant 1, towards quadrant 2, towards quadrant 3, or towards quadrant 4? Well, x is negative towards the left, and y is negative as you go down. So it's going to go towards quadrant 3. So now that we know the direction in which this graph is going to go, keep in mind... Having this negative sign here, it reflects over the x-axis, and having it here, it reflects over the y-axis. Originally, this is the square root of x.

If you reflect it over the x-axis, it looks like this. If you reflect it over the y-axis, it looks like that. When it's reflected both about the x-axis and the y-axis, it's equivalent to reflecting about the origin. So we can clearly see that it's going to go in this direction.

Now let's get some other points. So the parent function is the square root of x. The square root of 1 is 1. That means that as we move one unit to the left, we need to go down one. So that's going to give us the point 2, 3. So if you plug in 2, you should get a y value of 3. So 4 minus the square root of 3 minus 2. 3 minus 2 is 1. And the square root of 1 is 1. 4 minus 1 is 3. Which we do get now the next best point to use is for the square root of 4 is 2 So as we travel 4 units to the left we need to go down to So four units to the left will take us to the x value of negative one And we need to go down to so the y value will be two And let's do it one more time the square root of nine is three So this is at three so if we go nine units to the left three minus nine is negative six That's going to take us to this point, and we need to go down three, so we're starting at four four minus three is one So we're going to have the point negative 6, 1. So if you plug in negative 6, that should give you 1. So 4 minus square root 3 minus negative 6. 3 minus negative 6 is the same as 3 plus 6, which is 9. And the square root of 9 is 3, and 4 minus 3 is 1. So we do get this point. So now we can plot it.

And so, let me see if I can do that a little better. It should be something like that. My graph is not perfect, but you get the point. It has this general shape to it.

And that's how you can graph that particular function.

Transcript for:Understanding Function Transformations

Transcript for:
Understanding Function Transformations