[SOUND]. Lets look at Graph Transformations.

For example, lets sketch y=3-2(x-1)^2. by using graph transformations.

Transformations. Now we already saw in the part 1 lesson

on graph transformations what this 1 and this 3 will do to the graph, but what

does this 2 and this negative do? Well we have the following. [SOUND] That if c is

greater than 1. To obtain the graph of y=cf(x), we

vertically stretch the graph we have by a factor of c.

If we divide by c, then we vertically shrink the graph by a factor of c.

And if we multiply the x inside the function by c, then we horizontally

shrink the graph of f by a factor of c. But if we divide the x by c, then it's a

horizontal stretch of this graph. So these four here determine this

stretching and shrinking. Whereas these last two down here

determine the reflecting. And what they say is that to obtain the

graph of y=-f(x), we reflect the graph about the x axis.

And if we multiply the x inside the function by a negative.

Then we reflect the graph of f, about the y axis.

So we still start with our base function; y=x^2.

So lets say that this is the y-axis, and, this is the x-axis, y=x^2 looks like

this. Where this point here is 0,0, the origin.

We also have that (1,1) lies on the graph, as well as (-1,1).

Alright, so remember, what does this -1 do inside the function? This rigidly

shifts this graph one unit to the right. Which means to each x coordinate here,

we're adding 1 while leaving the y coordinate alone.

Which means that (0-0) will move to 0+1 or (1,0). (1,1) will move to 1+1, which

is 2, and then 1. And (-1-1) will move to -1+1, which is

(0,1). Therefore y=(x-1)^2 will look like this.

Here is the y axis. Here is the x axis.

So here's 1, 2, and 1. So (0,0) moved to (1,0) which is here.

(1,1) moved to (2,1) which is here. And (-1,1) moved to (0,1) which is here.

So this graph will look like this. So this is the point (1,0).

This is the point (2,1) and this is the point (0,1).

All right. Now, let's discuss what this 2 up here

does to this graph. Well, we're in this first case down here,

aren't we? Which means that we're going to vertically stretch this graph

here by a factor of 2. Which means what? Which means for every y

coordinate that you see on the second graph, we're going to be multiplying that

y coordinate by 2. So the graph of y=2(x-1)^2 will look like

this. Say this is the y-axis, and the x-axis.

This is 1, 2. And this is 1, 2.

Now this point over here, ·(1,0), isn't really going to move, because we're

multiplying the y coordinate by 2. So, we're still that 1, and then 2,0,

which is still 1, 0, so we're here. But these other two points will move.

This is going to move 2 and then 1 * 2 which is 2.

And this is also going to move to 0 and then 1 * 2 which is 2.

So we have this point here. And this point here on our graph.

And therefore, this function will look like this, so we're vertically stretching

it. All right.

Looking back up here to the function, now what does this negative do here? Well,

looking down here, that means that we are going to reflect this graph about the x

axis. That is y=-2(x-1)^2 will look like this.

So if this is the y axis. And this is the x axis.

This is 1, 2 and this is -1, -2. We still have this same vertex here.

But now those other two points, (0,2) and (2,2) are going to be flipped down below

the x axis to (0,-2) and (2,-2). So this graph will look like this.

And finally, the graph we're looking for, this y=3-2(x-1)^2 will be a vertical

shift up three units of this graph. That is, it's going to look like this.

So here's the y axis, here's the x axis. So the point (1,0) is going to move to

(1,3), Because we need to add 3 to every y-coordinate.

And the point (0,-2) is going to move to (0,1).

And the point (2,-2) is going to move to (2,1).

It's here. So the graph we're looking for is this.

Alright. And this is how we work with graph

transformations. Thank you and we'll see you next time.

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