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[SOUND]. Lets look at Graph Transformations.

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

Â Transformations. Now we already saw in the part 1 lesson

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

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

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

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

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

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

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

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

Â stretching and shrinking. Whereas these last two down here

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â This is 1, 2. And this is 1, 2.

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

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

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

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

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

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

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

Â it. All right.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â Alright. And this is how we work with graph

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

Â [SOUND]

Â