 And if you look at your worksheet and this can also be found on page 202 of our Coburn textbook but you have function families and if you can figure out what function family comes from then it will be easier to be able to write the equation of a function that's been transformed or even just to figure out how it was transformed. So we have linear quadratic absolute value looks like a V square root if you remember starts at zero zero and goes up from there. We've got this cubic and then we've got a cube root which almost looks like a cubic that you just kind of turned on its side and flipped over. So we want to reference that as we go. So here we want to practice. So determine the parent function and describe how the graph shifted and write the new function. H is the way it shifted in the x direction and K is the way it shifted in the y direction. So the parent function here is going to be quadratic. I'll take the vertex with how it moved. So it looks like it moved over three. So H would be three and it moved down two and down would be a negative two. And so now we have a quadratic with H and K. So quadratics look like y is equal to x minus H quantity squared plus K. So plugging in what we know, y is equal to x minus three quantity squared minus two since it was a negative two for the K. Okay, so now we're going to talk about reflections. And in reflections we're going to call this a vertical reflection. If you look at your paper it says the y is equal to the opposite of f of x. So that means that here we had our other graph. This was the parent function. The parent function here is the square root. When I had this point over here it was four two. Now I'm going to have that same x but I'm going to have the opposite y. So we now have the point x negative y. That's what we're going to be graphing for what we have which is exactly what we have. And so it says the opposite of f of x so the square root of x was our function. We want the opposite of that. This one is going to be a horizontal reflection. What a horizontal reflection says is that y is equal to f of the opposite of x. The original graph looks something like this and it won't be perfect but we'll see what we're talking about. Okay so on my function that was graphed I have the point one negative one. Now according to this I should be able to find on that original graph the opposite so negative one but the y stays the same. Negative one. So I should have this negative one negative one on my red graph. So let's look and see if we do. Negative one negative one there would be right there. So you can see that they reflected across this way. This point right here looks like maybe it's we'll say five negative two that's on my function and then we should have negative five negative two on the parent function. So here we look over here and five negative two sure enough there it is. If we wanted to see one up here we would have the point one one on my that's my parent function. The red one is my parent function and I should then on my function have negative one one. The black. So negative one one sure enough there they are. Find a point look to see what changed. Did my x change or did my y change? You'll know which reflection you have. Okay now we're going to try to talk about smushes and stretches. If we smush something basically your y's get multiplied by whatever happens to your graph and that's going to be by your a value. Let's think back to the quadratic when we talked about it was x minus h it's really a times x minus h quantity squared plus k. We didn't put the a in there before it was just a one but now we're going to have it's a and different things happen with this a. If a is greater than one then this that means this thing is going to happen faster. So it's going to be something like this and if we have a is between zero and one then it's going to take even longer to do it so it's going to be something that looks something like this. Okay you've got two examples on your paper. So let's talk about parent function here. Parent function here is the cubic. This is a stretch and this is a smush. That's what I call it anyway. So we have look at my red one that's probably actually what a normal cubic looks like so it's been elongated it takes a little bit longer before it turns so we would call that a smush. That means that we have an a that is going to be between zero and one. It's going to be a fraction. So if we write our equation we can write any fraction we want because it's just a possible one. Let's say it's half the rate. So half of and this is a cubic so it would be half x cubed. Now is it still at the origin so as it moved up and down left and right it just smushed. Let's put it all together. Helpful if you follow a particular sequence then nice things happen and the basic rule is apply the transformation closest to the x and then work your way out and the constant is always going to be less. When we look at this one we have an h and a k and that's all we have going on. So the parent function here is the absolute value and h remember this is x minus h so it was a negative four and k is going to be two and the a here is just a one. So we want to graph we know that the x of that absolute value graph would be on zero zero. Now we have to go left four and then up two and that would be right here and now my absolute value graph is going to sketch something like this. Now we have a reflection going here so our parent family here is going to be the square root and h is going to be three because x minus h a is going to be negative one and that's the reflection and remember if it's out like that then we have a vertical reflection or in other words we're going to have x and negative y and then we have k to be a positive three as well. So here's how we do it. Let's take this point right here zero zero and the first thing we see closest to x is that minus three so we go three units to the right and then we have this reflection so if I could I would move down but I don't. Anything on the x-axis stays on the x-axis. Let's try and make sure that we can do another point. Origin is still sitting right here and then I'm going to have go over three one two three and then reflect that so it's going to be four negative one. This is just one and we've got two parts done. We're not done yet so I'm going to put an x there. Now we have to shift everything up three. So from here from the x-axis I'm going to go up one two three so there is my minimum point and this next one I'm going to go up one two three so it's this point right here and then I would draw in my graph. In fact let's make it a different color just so that we can see the difference between the work and the actual. So you can see that definitely reflected and it went over three and up three. We have the quadratic functions. H is it's x minus H so H is going to be negative one. A is two and in this case A is not the reflection but it's going to be the stretch. So it's going to be skinnier and K is going to be positive two. So we do our negative one. Here's the vertex that would be one but let's also take this point here and make it become there okay. And then the second part is to multiply the y values by two. Well this one's going to stay here because it doesn't multiply but this is a y value of one and now it's going to multiply by two and become two. Maybe we should have done the third one. One one two four and then we would have gone to the left one so now it's going to be there and then we move it up but four times two is going to be eight so it's going to be way up here. So you can see this graph now is something like this. You can see it getting skinnier but we still have one more piece to take care of and that's going to be the up two units. So we go from the x-axis up two for the vertex. We go from the two up two for the second point and then this one is also going to move up. So you can see that we have a graph that looks something like this. It's skinnier than a typical quadratic graph and it's been shifted over one and up two. Now sometimes you don't have an actual function you just have a graph and then this is going to describe to me how to move all the points on that graph. So we need to think about what's happening here. Well h is going to be equal to negative one and let's do a next because that's the next thing we would do and that's negative f so that means that it's a negative one and it means that it's going to reflect x-axis. And then we have k is equal to negative four. So let's start. Let's take this point right here and we want to go over to the left one and let's take this point right here go over to the left one and we go right there. Take this one and we'll go right there and take this one and it would go right there and finally this one would go right there. So if I were to graph my graph right now it would look something like that. Okay that's step one. So the next one says reflect everything or remember anything on the x-axis stays on the x-axis. This point right here was at negative three three so I would need it to be negative three negative three. Remember you change the y. I just have to come down here and say one two three and that would be that point. That one stays. This one has to reflect. It's a negative three right now so it needs to become a positive three one two three. And then finally this point right here is going to be not negative six but positive six. So if I were to draw that one right now it would look something like this. Same basic shape. It's just shifting. And then we have our final one where we have to drop everything down four. So take this point and finally it gets to move again one two three four and I'm taking my red points and shifting them. I'm going to take this point and go down one two three four. This point was also on my graph so I'm going to go one two three four and now I have this part of my graph. And then I'm going to take this one and come down four. I'm going to take this point four now we have the three points that we connect to make this part. Okay so if I come in here and erase all the other colors we can see a little bit better what the final looks like all by itself. So that green is what we want. The blue is the original. It's been shifted over one it's reflected it's been shifted down four. Now we want to go and see if we can make an equation out of all this. Well that's not usually too bad except for this nasty little A but we can do that. We're going to have a look at this graph and we want to think about the origin. The origin was our vertex now it's my new vertex. Well my new vertex is at point negative two three and that tells me H and K. H is negative two because if we went left K is two because it moved up. And then A is going to have a negative value because it's not an upper Avala but it's reflected down so I know it's going to have a negative but I don't know what this value is. But that's why there's this nice little point right here that says negative three negative one. You can't see it very well but it's right there. And we can take what we know Y is equal to A times X minus H quantity squared plus K and I'm not even going to worry about putting a negative in for now I'm just but we can plug and chug because this is my X, this is my Y and I have H and I have K. The Y happens to be negative one is equal to some A times X from my point is negative three minus my H which is negative two so I'm going to add two that's a negative three and then plus my K which is K was three not two. Negative one is equal to, oh this is quantity squared, so negative three plus two is going to be negative one squared is going to be one A plus three, subtracting three we have negative four is equal to A. So now we have everything we need because we know A, we know H, we know K and we can say that Y is equal to negative four and it looks like it's stretched times X minus the negative two or plus two quantity squared and then plus three. Let's try one more. So we have this parent function and the parent function looks like it's going to be a cubic and so we used to have something here at the origin and it looks like it's about right here now. So what did we do? We went over one to the right so it's a positive one is my H and it looks like we went down four so that would be a negative four and then we have the point three negative one that X equal three and Y equal negative one. So now we have HKXY we just have to put it into a cubic this time. So if we do our work over here we have Y is equal to A times X minus H cubed plus K. Y is negative one is equal to A which we don't know and then X is three minus one which would be H and then we're going to cube that value plus our K value which is negative four. I put the plus before I thought about my number. Alright so let's bring the four over this time make it look nicer. So we bring the four to the other side and now we're going to have three is equal to A times two cubed and two cubed is just eight so eight A equal three and A is going to be equal to three divided by eight. It's a fraction and this looks like it did elongate a little bit take a little bit longer to get places so we write our equation in this form. So A is three eights times X minus my H so minus one but it's a cubic so I have to cube that and then I went down four so minus four.