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Pre-Algebra: 6.8 - Dilations

Added Apr 21, 2021 •

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Transformation #3 - Dilation 1) Find new points for Image 2) Graph Image 3) Find Scale Factor of Image 4) If given Image and SF, find Original Points

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Malaysian is actually the third transformation remember Transformations let's go so dilation reflection there are four but first let's go transformation trans4orm a shotguns third one fourth or not really going to deal with it so you guys are going to deal with the later on so we had translation where it shipped we had an object and we shifted it so that's the original and was it the original yes the original and where the new one is let's say we had the image right here so it shifted it translated it slid over its lit over based on FX and then why so we always go to the X and then why so there was a conversion the translation was we did something 7 XY X1 y1 X2 to Y to we a distracted multiply divide who knows but we added or subtracted to something you know some number and we got a new X and I knew what that was translation we had a reflection and there were two Reflections and we had a reflection on x axis on x axis here's the x-axis let's say we had this triangle right here and we flipped it it would be a mirror image on the x-axis right here so you need it and here's the thing we we only did the basic Reflections you're going to learn other Reflections where they're not on the X & Y axis there on different taxis or that's going to be in algebra so we flipped it this with the ex didn't run the extension the ex didn't change the only thing that changed was the why that's why it helps to make it my simple sketch so this right here your ex-wife that's why it's called a transformation your x y coordinate change based on this constant set up the only thing that change was that Ry notice it slipped sitar wise Flintstone X is positive Y is negative and then reflection on the y-axis I say we did the same thing we have some triangle right here and we flipped it to the y-axis we flipped it over it looks like that right there this one right here noticed it didn't go up or down so why stay the same the only thing that stayed change that the x is negative or positive switched so there are different transmitter so there is another one there is another one where you reflected on an equation so the axis is named after an equation you're going to learn that eventually took too much on it but you're going to learn it went another so they're not just I'm just going to show you where I had you don't need to know right now if we have like an X Y axis the axis is called the axis of the axis of symmetry is notice right here this line right here your x-axis is the acids it means it's exactly the same but a mirror image it's symmetrical set an axis of symmetry it looks like that you flip it on this diagonal or I can flip it on this diagonal it doesn't even have to be a diagonal it could be nothing could be a straight line right there there's your axes of symmetry so these are just the basic things we're doing there's translation and then they were doing dilation and there is a fourth one Which ion I personally I admit that something rotation when I told that you had to think if you if you have this as a flag right here if you had a flag are just a minute I'm trying to think just imagine I wrote sudeste there's rotation imma give it to you right now rotation about the origin imagine you have like a little flag connected to whatever shape you had and you're going to rotate it x amount of degrees all you're doing is rotating it around and it's going to change so if it's right up here it will probably look like this but if it's over here it would look like this so no. it just keeps on turning so it's like five different anyone it's down here it's flip the other way pinwheel that's weird that I always get mixed up so you're not that you'll eventually and all the hardest one man when you're not rotating on your hand what if you rotate out here yeah so that's what you're going to get in few but we're not doing though so we learned translation which was super easy all we did was add or subtract the X & Y just had to figure out what that pad and then Reflections we do bees on the x-axis and the y-axis you just got to remember which is which that's why you draw sketches does got to learn to do that draw a quick sketch to visualize that will always help you so today we are doing dilation 6-8 dilation so we've already did Transformations translations and Reflections on the coordinate plane chapter 2 grass. violations on it so you're going to have to learn how to graph and find the scale factor of a dilation that's why we save this one to know cuz I'm like a reflection and a translation where the object stays same size the image stays the same size in the dilation it shrinks or grows dilation so let's go mesolimbic Cavalier where we need to worry about supposed she and larges photo to 8 by 10 compare and contrast the original photo and the enlargement so he had to think quick sketch we know it's a rectangle 4 by 5 2 eight-by-ten if you're comparing it so just look at it matching size for 285 to 10 what happened so do we go bigger or smaller reduction or enlarged and if you choose one but how much so what's an enlargement thank you 2 x 2 x 2 x 2 and you can literally see that you go here to here here to here and there's a lot of different ways but remember when proportions were comparing things you could go 5 to 4 and 10 to 8 but when you go scale factor you need to go from one scale factor must go from one object do other which means we can't make like 5/4 is equal to 10 over 8 that's true it's twice as big but if you're going to make a scale factor want to go directional base and since this is the original original canoe old to new scale factor they gave us a direction to go to notice I'm drawing the smaller to bigger that's what we know what's an enlargement and all you got to do is take one set of corresponding sides hey is equal to your scale factor and all you got to do literally just go from one to the other over Skip and that's the whole thing it's new over old scale factor is new and keep it civil over the old thing and the old thing is called the original or new over Regional but that's why I got lazy I just want to work hold so what's our news and that's a woman it's up to you to decide that's what freaks people out about algebra which one do I choose their boat use both of them and they freak out and they don't do anything it's a metaphor of life just choose if you have a choice and you don't know which one well at some point you got to choose and that's the whole thing if I go my original is for 4 corresponds 284 doesn't correspond to 10 or funds to 8 so 8 / 4 but if I didn't do that it doesn't matter cuz I could do the same thing with five and ten does both of these should simplify the same thing and that's another way to check to see if it's proportional do they simplify the same now 2 is your scale factor but when you actually use it to multiply if you have fractions it's always I always keep it as a fraction just in case so I don't freak don't forget but if I can send the file stupefied and I have both to look at so it didn't matter which one so it's an enlargement bye 2 time so what is the scale factor barely got that ice literally they what the scale factor to go from what is the original photo to the enlargement what is the skin of an emergency line that's a whole thing original X if we're going original to enlargement is 1/2 that's the whole thing I don't like what they did recall that a transformation maps and original figure onto an image remember anything new is called the image and then you can have it image an image of an image of an image that right there is the algebra 2 stuff where they make you do multiple Transformations connecting you do you could do a translation into a rotation into a what shape into a dilation that's like a one-page problem so in math I always I was always happy when I had 20 problems verses for when it's for problem yes in math class you will always down when you get used to it all right I'm looking for palm tree do tonight in the that's why I told before remember that until mystery also anyways so ricasso translation and reflection are two Transformations that change the position of an object a dilation is a transformation that enlarges or reduces a figure by a scale factor so look at this translation is just shifted a reflection flipped over the puzzle pieces look translation of Base the same way reflection is faces the opposite way depending on the axes a dilation faces the same way but look at it there's the orig here's the new in this case it's the reduction it shrank down when the center of a dilation is the origin and that's another thing we're not going there yet changing stuff when the center. PlayStation is the origin you can find the coordinates of the image by multiplying each coordinate of the figure by the scale factor use the notation X Y 2 K x k one so K is a scale factor you multiply and then it either doubles triples whatever make how many times bigger it is or if you multiply it to make it a smaller number to get a reduction so always pay attention begin a lot of people picked up do I use 2/1 or 1/2 remember when you multiply by a fraction bigger than one you're enlarging the number so the numbers are going to get have to be bigger so you're making enlargement when you're multiplying by a fraction smaller one so it's going to be a reduction that's the easiest way to remember if you're doing it right so look at example one where they're now finally has vertices vertices J is 2 K is 2/6 m is 86 N is a to graph to figure an image of the polygon after a dilation with a scale factor of 1/2 so for all of these we need to find a brand new figure brand new number but you know graphing the first one but let's look at this right here let's go let's go EX Y & J that's not a j this is Jay Kay mnn so Jay is at 2 for 26868 to remember our next how do you label your image with the apostrophe the prime not a power window Prime so J Prime K Prime MP I'm in Prime stats for our next our image and we have an X Y axis for these right here and it's the second X and the second watch and what are we supposed to do if we're going to Deuce this we're drinking it down scale factor of 1/2 so 1/2 1/2 we take 1/2 1/2 x 1/2 why so just take half and sometimes they can be different but they should be different it should be the same case you need to save its constant half of 2 and 1/2 of K2 1/2 of 8484 passes 42331 and notice what I did I put them side-by-side and I made a grid you always wants what do they call it call adenine when the graphic organizer there it is we always want something visual to organize our thoughts it's like an outline it helps us keep track of the hot mess that is whatever RSA is going to be so and let's go and look at this all our numbers are positive so all I need is quadrant 1 on this right here and let's look at I'm going 128 on my x-axis I'm only going up to eight on my y-axis I'm only going up to 6 so 1 2 3 4 5 6 8 and notice I got no graph paper that's just a good estimate definitely I can see that this is longer than these pieces but it's close enough I'm going up on 6 in the y-axis 1/2 two three four five six five men that's five so it helped us out play weird things are make it quicker so let's grab green first original later this graph the original. So 1 and 2 go over at 1 to 1 up three after 3 4 Min after 3 is 5 Sy makes it easier five years for than up to 3/4 than up to one we got a nice little rectangle and here's the thing let's label that I forgot to label the corners if they give you points you should label them so J Prime is 1/2 J Pro I'm a prime and then n is for one and Prime M Prime all right Dallisgrass our original image we go to two up to 4 that's Jay hey we go to choosing up to 6 that's okay go over to 8 then up to 6 and then n 8 mm to so you can tell them that I'll look at this this is my original this is my image and the thing about this bravado you guys are going to graphic designing one of the most stressful jobs either video games or animation is just pretty much giving a depth and you're going to have to use some of this out of dilation concept all the programs there are tikis they already have things made for you but if you look at it what song is this if you connect the points k2k Prime and then you don't want to connect to make a little. Cuz it's behind there we could see it's like Anna is zoom and you can see those lines are lines of zooming in and out they have programs that can shrink it for you but dilation stuff yes could use a bath you still out there and find most a person so that's dilation right there super one super easy but a lot of people scale factor is the hardest thing that's the next do it let's do one first so try this one out on your own except I was a bit off I was a bit off that's all right but try this one on your own will do that Checker answer so a figure has vertices rst so triangle lot easier for you guys 363-1233 and the scale factor of 1/3 so we know we're shrinking it down try that one on your own graphic dealer simple problem the simplest of problems are always the ones we miss up mess up and miss the most cuz we're two packet do I always have to do a t chart up and down are you don't that's just my preference so let's make a t chart the other way let's go x y and so we get negative 3 6 there's J 312 there's a c and then 3-3 there's tea if I'm going to do J Prime S Prime T Prime see I get in a bad habit of just making up and down so that's why I messed that all right X Prime y Prime and what are we supposed to do multiply by 1/3 you should know that * 1/3 is the same thing as dividing by 3 / 3 let's do the different color / 3 3 / 3 except I like to see left and right so I can just have a pair 111 in a same thing as no different so let's grass why goes up to well so and do I have to look at this I don't want to go 12 marks know I can count by twos but then I have an odd number on my why but that's all right and on my ones I just go one two three and I can make him bigger one two three on my way Plies I'm going to count by twos and here's the thing what is distort it distorts be when you're spacing is not equal it distorts your picture but it still puts in the same place so my wife and I don't want to make 12 * 224 610 12 cuz I can find halfway marks for each one of those so for the first one let's do this I'm counting by twos negative o- is negative negative ham I got to go -3 see I didn't pay attention I messed up with the hair illustrate all right so negative 1 and 2 hears negative 1 2 + -1 + 2 and then we got one and four one and four and then we got one and one hears one and I don't have one on the why I know why you're halfway so let's connect the dots so J Prime S Prime T Prime is my triangle jst from the image and let's craft the original -3 and 6 years negative 3246 -3 + 6 * 3 3 and 12 that's S3 and 12 is up here s J & E is at 3 and 3/3 and here's the thing I'm trying by twos Dynomite who is halfway in between right here 2214 South you don't need grass paper you guys out it doesn't need to be super perf and if you connect them legal right here t2t Prime as primed us you should see if you just cut off it's a slice that's all pretty easy let's look at example to you dial 8 every used to expand or widen Matthews to enlarge or reduce scale factor so a triangle so let's do this right here is how you would see it on my own which remind me Ren star review Ren star one more time sometime in May that'll be pretty much a something for the school to keep a record on the sea learn have you improved even a little bit so play testing is always a choice. A triangle has a vertices so we got a B and C I will write that down later find the coordinates of the triangle after a dilation with a scale factor of 2 so just by looking at the information right now you can see that are dial a is an enlargement based on our scale factor that tells us what to do with our original numbers I'm going to give you a quick sat question always pay attention what if they give you the image that's how they'll do your dirty on these tests will give you the image and I'll tell you what the scale factor is so if you're going to go backwards to find the original Point coordinates you got to take the reciprocal that's how that you do that Right Thurr it's easy work but if you're not paying attention hearts on an SAT that's actually it's at least a level 3 but might even be considered a level 4 in difficulty because they split them so watch out for the Lexile reading so why do guys are doing all right good good good let's just so you guys are give me coordinates so let me do that I'm going to do my teacher X and Y so a baby he is at one- to be is at 0 in to see is at negative to negative 1 and what are we supposed to do with the fact we take this number and most fly to get our new coordinates to x x 2 * y a prime B Prime C Prime so we get * 2 * still is 0 * 2 * 2 * 2 easy enough to grass so let's go negative and positive so we go from 2 to -4 on the x-axis -4 mm x axis no shifting over here 1/2 1 2 3 4 I go all the way from to -4 same concept 1 2 1 2 3 4 and all we got to do is grass so let's the original one- to one negative to that's a b is at 0 I don't go anywhere 02 I've got to go up 02 there's B she is at -2 and one negative to negative one here I connect the dots and now I got enlargements so I can expand this 3 and 4 to -4 there's my a prime 04 there's B -4 + -2 air-sea Prime don't forget to Prime sorry thank you for letting me know so just connect the dots and there it is a dilation from the old not too bad but now we got a match up so look at this matchup the coordinates so multiple choice Lotto numbers to look at and easy to get you might have done the work correctly that's the thing of that that that  about multiple choice test with all these numbers and letters and symbols you did the work correctly you better match it with the correct answer so a prime should be two negative for that's wrong Crossing I don't need to bother with that is when it's done I don't need to bother so it's C or D process of elimination so both of them to -4 then I got zero and four that's go to answer C i shouldn't even look at the Finish bothering I don't need to look any further cuz these to its 04 but notice my sees are the same so be would have been the thing that differentiates little details on multiple choice guys do great work but then you throw it away but not finishing checking to see if your answers correct on mobile choice test read the test on your up on the coordinate of the image of the after dilation multiply the coordinates of each vertex by the skill Super Simple Man so saw the item we grasped it translation and transfer that right there's transformation work I will never trip and say you need to show me the transformation method way notation this is good enough that is just that's extra but some people like to show all the work can have it doing it if you're into physics I would actually start doing it so you get used to write a notation don't get into Bad Habits Like Me all right let's go look guided practice is my grandpa versus Bears it doesn't look as pretty but the colors a figure has vertices X Y and Z find the coordinates of the figure after a dilation with a scale factor of 3 so don't bother graphing just buy me the new coordinate and which one is it F G H or J y&z notice XY XY once copper with lowercase how do I know which is which that's why you got to watch out organized. -3 six 3033 and I'll put an extra they say the scale factor equals 3 so 3x 3y and I organized mypoints X Prime y Prime Z Prime what do I do at * -9 18 909 + 9 so 9 + 9 + 9 0 easiest to look for because this I look for Z Prime that's the only one with nine nine so J sometimes it's easier but if we look and start with negative 9:18 that's the only with negative 9:18 to so all the times I just got to match it up so Jay I don't even need to finish cuz nothing else bothered now here's a quick summary of this right here I already gave I didn't give you the last one I never thought about this a dilation with a factor of K will be if it's greater than 1 and remember you think about it as a treated as a fraction always treat a as a fraction is that'll help you related backed all the portion bigger than white if it's smaller than I put smaller than 1 but yeah it needs to be positive also it's a fraction less than 1 is what it's saying you have a reduction can't be negative and if it's the same if an equals one you didn't do anything with it figure it's the same thing you didn't enlarge it or reduce it it's just there they made a clone all right now next thing what if you're given a picture notice which ones so let's zoom in I was right here they did it they did a dirty they didn't tell us which is the original that's why you got to read to most of the time they would label the cornice and they would tell us which is the image which is the original to help you set it up that's why you got to go back and read the equation so a drawing have a school mascot is to be increased as shown at the right so it can be painted on the gymnasium doors what is the scale factor of the day so we know it's an enlargement so we got find the scale factor write a ratio comparing the lengths of the sides of the two images subtract the x coordinates to find the lengths let's go to that Park scale factor first so we know it's an enlargement so one thing that can help let's go with this one first let's cover that up we know our original right here XY we have one two three 2 and then we enlarged it to and then we shifted it over to it so why am why Prime we got the corner three and six 9 and 6 so the whole thing is what was the scale factor e x to get these new images what did I multiply to 1 and 2 to get 3/6 and he needs to be the same with that I multiply 232 to get 96 I was to become how does 1/3 become 3-9 thank you we multiply that by 3 so the scale factor is 3 going from the small figure to the big figure it's 3 so now we know the scale scale factor is 3 so length on dilation length of so they chose they chose to do it using length so what they did they said let's make wrecked who is this how long is this wrong compared to this 1 2 3 4 5 6 units long and remember it's the new over the original so 6/2 that's where the 6/2 came from somebody's earlier where did the nine and three come from they just decide to go blank so 9 - 3 is the best the whole thing we just kind of this right here down so if they look right here even though the top of the head and just crossed over one to God include this one cuz it's in there so three units hi 1 2 3 4 5 6 7 at the age of subtracted the lengths 6 / 2 too much factors 3 and so what is it want us to do what's the dash which Dash the subtraction you mean what's the that's a subtraction so what they decided they decided subtract length so they went right here Nine Mile 3D just wanted to link member look at this new over original so let me go back to the picture so they just decide to do with a super long way I made it easier easier it is to compare this they decided just to look at the grass so they're 9 - 3 what is the length of the dilation so since it's not exactly on zero there's the distance problem so looking at the X's 9 -3 so that let's six units. That's where the six came from and then what how long is it the small one 3 - 1 so the distance between here to hear is to that's where they got the 6/2 that's 4 9 - 3 - 1 came from so length of the new verse over the length of the old distance formula and then they have 6 over 2 equals 3 look at how complicated they made it look how much easier it is when you make an organized brass I don't know why they don't do that the Dallas for December the scale factor is 3 all right last one right here by the steel factory Megan wants to reduce an original 8 x 10 inch photo to a 2 x 2 and 1/2 inch photo now we got a mixed number up in this what is the scale factor of the dilation and we know there's a clue right here it says it's a reduction so is it so think about how to set it up and we'll use the other one to prove it don't make this more complicated than it is cuz it's super easy scale factor someone's the remember scale factor which is your constant proportion is just new over the old or original the original deal you just need to have your Lexile to identify noodle and this doesn't have a coordinate somebody said you don't have a coordinate let's draw picture so we have eight and I'm going to compare it to a 2 by 2 and 1/2 yes yes and yes let's go check her answer so look at this we got we're trying to find a scale factor so here's a remember don't always think the bigger is a  it said we're shrinking it down so the new is to the old corresponding side is a that simplifies to 1/4 if you're paranoid and you're not sure if you did it right do it to this one set it up as 2 and 1/2 / 10 2 and 1/2 / 10 and here here's how I checked is it right here remember this is another way of writing writing division 5/2 / 10 let's call this 10/1 remember fractions and see that's why I would not like to do this one but if they say check and prove your answer you have to do this fractions another way of writing division 5/2 / 10 / 1 which. fractions the same thing as multiplying the reciprocal if you remember 1 / 10 and then all of this I can cross simplify that 5 / 5 is a 110 / I was into 1 * 112 * 2 1/4 1/4 equal one for this would have been the better one so this is one of those things you guys want to do 1/4 cuz remember we reduced it it's a reduction so if you wrote if it's four times that means you're blowing it up or time but it said we track it down so take a reciprocal so that's that's a nice affix that means you just flip the numbers around when you did the work that's okay that is totally okay to flip the numbers around but at the end you always got to ask yourself did I shrink it down or make it bigger and that'll take just take the reciprocal so right there 1/4 1/4 homework I do not remember what numbers I put it up I think it was like 152 17 big boxes of 5 to 17 and 85 to 17 and 18 so one even bro what time is it we got enough time to do one of these problems on number 1 through whatever for let's look at these right now where do you find the vertices this would have been class work if we had the regular class schedule but we don't so look at this right here find the vertices of each figure after dilation with the given scale factor k then graph the image show on if this was a you have to know how to read a graph so you got to find all the coordinates so like this one this is -2 one that means we got to divide the coordinate all the core ordinance by 1/2 / 2 * 4 / 2 this right here you get one too that means if we have 2 and 1/2 1/2 and 1/2 the coordinate beef right there so shrink it that all of this right here all of these all the coordinates need to be / 4 * 1/2 / 4 that would have been two examples 6.3 A square has vertices vertices and the coordinates of the look at this has a scale factor of 2.5 are we * * 2.5 so all of our numbers need to go bigger 2.5 * 2 * 2.5 is using a photocopier deuce that means RK has to be smaller that is reduced I posted as 8 inch by 14 inch to 5 inch by 8 and 3/4 in what is the scale of the dilation so draw a diagram and matchup corresponding sides like we did from the first session all right for you guys in class please pack up grab a wet wipe wipe down the table and your seats I know you farted on it's all right so I know he had to leave she was too nice to say hey it's alright everybody has homework it's all good and then yes you are more than welcome to did you get me what in a minute iFixit time is for you if you are in if you're in the after-school program do not forget to come to school and pick up your Makey Makey I got you for the Mega Mickey I got you guys so line up once you right down the line up right here till Sergeant Schilling instruction to move on have a good day mr. All right guys I gots all y'all you guys good

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Pre-Algebra: 6.8 - Dilations

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