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[SOUND] Let's look at graph transformations.

[SOUND] For example, lets sketch y equals to the square root x plus two plus one by

using graph transformation. Now, there are different types of graph

transformations. There are transformations that we call

translations, which are rigid movements of graphs and then there's also

transformations called stretching, shrinking, and reflecting, which are

rigid movements of graphs. And in this lesson, we're going to be

looking at the translations or rigid movements of graphs and we have the

following. [SOUND] That if f is a function and a and

b are positive real numbers, then to obtain the graph of f(x) - a,

we shift the graph of f to the right a units. And to obtain the graph of y = f(x

+ a), we shift the graph of f to the left a units. But to obtain the graph of f(x +

b, we shift the graph of f up b units. And to obtain the graph of y = f(x) - b,

we shift the graph of f down b units. So to sketch graphs using graph

transformations or these translations here,

what we do is we start with what's called a base function.

And looking here, we can start by sketching y equal to the square root of

x. So let's say that this is the y-axis and

this is the x-axis. The square root of x looks like this,

where this is the origin, (0,0). What are some other points on this graph?

Well, we know when x = 1, that y will also be equal to one,

which means, we know we have the point (1,1) on our graph.

Now what other point lies on our graph? Well, we wouldn't want to put, for

example, x = 2 in here, because then the y value would be square root of two.

So what's the next x value that we know the square root of after one here?

Well, that would be four, wouldn't it? Because we know that the square root of

four is two. So that's another point we have in our

graph. So this is four and this is two here.

So this is the point (4,2). Alright. So what is this + 2 here do,

under the square root, to this graph? Well, looking down here, we're in the

second case, aren't we? We're adding a positive number to the x value within the

function, which means we're going to shift this

graph to the left a units, or in this case, two units.

So, we're going to shift this graph two units to the left.

When we do that, what's going to happen to this point here (0,0)? The y

coordinate is not going to change, but the x coordinate is going to have two

subtracted from it, which means it's going to get translated

to the point (-2,0). And what about this point here, (1,1)?

This is going to go to (1-2,1) or (-1,1). And what about this point here? This is

going to go to (4-2,2) or (2,2). All right. So let's sketch y equal to the

square root of x + 2. So let's say this is the y-axis and this

is the x-axis. So we just found that the origin at point

(0,0) was moving to (-2,0). So we have that point on our graph here,

(-2,0). We also found that the point (1,1) was

moving to (-1,1), so we also have that point on our graph

(-1,1). And then, we found that the point (4,2)

was moving to (2,2). So we also have that point on our graph,

(2,2). So this shape is exactly the same, it's

just moved two units to the left. Alright.

And looking back up here, what does this + 1 do to the second graph here?

Well, we're in this third case down here. We are now adding a positive number to

the y value here on the second graph, which means we shift this second graph

upward b units or upward one unit. So when we do that,

what happens to this point here at (-2,0)? This point (-2,0) goes to

(-2,0+1) or 1. And what about this point here, (-1,1)?

This point (-1,1) is going to go to -1 and then 1 + 1 or 2.

And this last point (2,2) is going to go to 2 and then 2 + 1 or 3.

That is we have the following graph of y is equal to the square root of x plus two

plus one. So, here is the y-axis and this is the

x-axis. The point (-2,0) is moving to (-2,1).

So, it's here, (-2,1). And the point -11 is moving to -12.

So here is the point -12, and finally, the point 22 is moving to 23.

So here is the point 23. So our graph will look like this.

Notice that we're just rigidly moving to the left and then up.

Alright, let's look at another example. Let's sketch this graph here using graph

transformations. Again, we're going to use just

translations here in this lesson. So looking at our function up here, what

is our base function? It is y is equal to the absolute value of x.

So let's sketch this. So let's say this is the y-axis and this

is the x-axis, the absolute value looks like this.

We have the point (0,0) on our graph, as well as (1,1) and (-1,1).

So looking back up here, what does this -3 do to this graph here? Well, we're in

this first case here, aren't we? We are subtracting a positive number from the x

coordinate inside the function, which means we can shift this graph to the

right a units or three units. So what's going to happen to this point,

(0,0)? It's going to get moved to (0+3) or

(3,0). In this point, what's going to happen to

this point, (1,1)? It's going to move to 1 + 3, or (4,1).

And what about this point, (-1,1)? It's going to move to (1+3) or (4,1).

And what about this point, (-1,1)? It's going to move to (-1+3) or (2,1).

That is, we have the following graph of y is equal to the absolute value of x - 3.

We have the y-axis here and the x-axis. So, (0,0) is moving to (3,0), so one,

two, three, zero. And (1,1) is moving to (4,1), so here is