good morning everyone the next type of transformation that we are going to talk about is a reflection today specifically we're going to talk about reflections over the X or Y axis so first let's start with a definition a reflection is a transformation in which a figure is flipped over a line of symmetry so if you look at that image to the left you can if you start with the blue and you go over to the green one you can tell that it is flipped and the line that it flipped over is the y-axis so that yellow would be the line of symmetry the next part says each point and it's image are the same distance from the line of reflection so looking at that image I'm gonna zoom in let's start let's say right here this was a which means that up here would be a prime so if I count to the yellow line this goes 1 2 3 units so on the other side a prime 1 2 3 is also 3 units and it should do that with every point so say this one was B it's only 1 away over here B prime also only one away this is C makes this C Prime 1 2 3 4 so on this side it should also be 1 2 3 4 okay so let's look at some examples so let's start with what if I wanted to reflect over the x-axis so let's start by filling in this table with the pre image our pre image is there in blue X Y Z X is at the point negative 3 positive 1 why is that the point one for and Z is at the point three two okay so up there in the title it says reflect over the x-axis so my x-axis you see how they're labeled X is the one that goes side to side so I want to flip this blue figure across that yellow line so if we follow the rule that it was before we know that every single point in the image is the same distance as it was in the preimage okay so let's do it like this let's start with X so point X was only 1 away from the line so I need to do that same thing here let me do this way only one square away from the line so now I need to do that but on the other side one square way so X prime should go right there that's X prime and I'm going to go ahead and write it in that's the point negative 3 negative 1 then for y Prime we're gonna do the same thing so looking at the point why it's one two three four squares away from the yellow line so I need to do four squares the other direction one two three four so that is where Y Prime should land and then same thing with Z these one two squares away so we're gonna do the same thing one two squares away Z prime should be right there so let me connect these points and that would be my triangle reflected over the x-axis now let me finish writing in the points for the image so Y prime is that positive one negative four and Z prime is that positive 3 negative two so that's one way that you can do reflections by actually counting the squares on the graph but with every transformation that we have we're going to have a coordinate rule that we can follow so I want to know if you can figure out what the rule should be here so let's take a look at what it says on our table if you notice on our preimage x was at negative 3 positive 1 but on the image the point is on negative 3 negative 1 so if you look at those numbers the only thing that really changed is that the 1 is negative now hey then on the Y it was one for but then it changed to one negative four so same thing the Y value turned negative and for z3 two turned into three negative two so again it's only the Y value that changed and the only thing that changed is that now it's negative so that is actually the rule that you have to follow when you're doing reflections over the x axis the X will stay the same nothing happened with it but with the Y value it turns negative so negative Y let's do another example like this okay so let's reflect over the x-axis and as you can see I went ahead and I wrote down what the rule is that way we wouldn't forget and I already went through and wrote down all the coordinates for each of these points de and F so instead of counting squares what I want to do is use this rule so for point D negative three negative three if I were to follow this rule the only thing that's gonna change is that the Y value has a negative sign but it's already negative here so I'm gonna write a side note what happens when you have a double negative but let me just write it over here so if it was negative three but then I'm making it negative again hopefully we remember the rule that when there's two negatives it basically is just positive three okay so let me go back to our slide our x value is going to stay the same but our y-value since its double negative is going to become positive so D prime should be at negative 3 positive 3 which is up here then for a prime the same thing the x value is going to stay the same but the Y value we're going to change the sign so since it was negative 2 we're gonna change that to a positive 2 so 3 comma 2 that's our Y prime and then lastly our F the x value stays the same - but we change the sign of the Y value but notice on this one our Y value is 0 F is on the x axis so there is no such thing as positive 0 or negative 0 0 just always stays the same so 0 stays the same so that point F prime actually lands right on top of regular F so let's draw this triangle and see how you can tell that it's reflecting over the x axis so that's two different ways of doing it either you can count from this graph or you could use the rule honestly it's probably gonna be easiest to just use the rule when you're reflecting over the x axis you just gotta make the Y value negative okay so let's try reflecting over the other axis reflecting the y axis so I'm gonna go ahead and fill in the points for P Q and R P is at negative 4 negative 3 Q is at negative 3 positive 3 and R is at negative 1 positive 2 so let's write those in okay so remember that this time I want to reflect over the y-axis so our y-axis is the one that goes up and down so we're going to count from that axis for P it's one two three four squares away so we needed to do that same amount on the other side one two three four that is where P prime should be 4q it's one two three away so again three-way one two three that is where Q prime should be and in last R is only one square away from the yellow line so we're gonna go one square the other direction that's where our prime should be you know it's draw our triangle and that's how I would look reflected over the y-axis let's write the coordinates for each of those points so P prime is that positive 4 negative 3 Q prime is a positive 3 positive 3 and R prime is a positive 1 positive 2 so can we figure out what the rule should be when we're reflecting over the y-axis well it's actually pretty simple so again just take a look at the coordinates here negative 4 negative 3 change to positive 4 negative 3 negative 3 positive 3 change to positive 3 positive 3 so the numbers are the same the only thing that changed is look at those first numbers negative 4 change to positive negative 3 change to positive negative 1 change to positive so in this case it's the x-value that changes so we're gonna put that as negative x but the y-value stays exactly the same okay so let's use this rule to do the last example there's my image LMN or I'm sorry the preimage I already wrote the coordinates and I wrote down our rule so that we wouldn't forget but now let's write where our image should be so if I'm following this rule Y is gonna stay exactly the same but X is gonna change signs so on point L this negative 2 is gonna change to a positive but this 4 is gonna stay the same okay with the next point positive - is going to change to negative but the y-value stays the same and the last point that negative 4 is going to change to positive 4 but the y-value stays the same so let's go and graph those points so we can see what our image looks like - 4 that's all prime negative to positive 1 that's M Prime and 4 negative 3 is n Prime so these triangles overlap a little bit but that's fine because you can still clearly see that it's reflecting over the y-axis like we need it to