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Transformations Video Notes

Added Apr 20, 2017 •

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Use this video to understand the formulas used to translate, rotate, reflect and dilate a point or an image

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going to be talking about translation reflection rotation dilation Oregon's in motion so polygons in motion soap you been talking predominantly about quadrilaterals and laugh lesson we finished out with called the quad family tree if you look here is it something you're going to have to memorize is all the different properties that these quad and we had two primary branches of the Quad family tree the parallel side and then a non parallelogram side which consists of solely a kite and then your generic and isosceles trapezoid my name in the unit is polygons and motion so this lesson is going to be predominately talking about the concept of coordinate motion motion before we do that when I do a quick little review  reusing a brain and hand reading techniques which were going to finger track read under draw label list cross out  something to track and read with me put your finger here and track with me the damn has a median 360 ft on the top of the down three times larger than the bottom of the damn find the length of the top and bottom of the damn so let's underline are keywords here so the median 360 feet the top of the dam is 3 times larger than the bottom of the dam and our goal is to find the top bottom here  find the length of both the top and the bottom so if you look over here at the picture of a damn it down is roughly a trapezoid so we're going to go ahead and use this trapezoid shape is our guy and we have our faces and comedian which connects the midpoint of both sides and our goal is to ultimately in the scenario find these lengths so let's write our equation out remember equation has two sides with an equal sign always write your equations with two sides and an equal sign that here's your equation median equals what's base / 2 so are median we're actually told is 360 so when are equation here we're going to have 360 equals now we're not told the length of the base is but we do know that the top what is 3 times bigger than the bottom the one day is going to be three times bigger than the other day so we're going to have a 3 relationship  and that's going to be over to  now we simply Sovereign cross multiply one  request we got 350 * 2 and we're going to go ahead and add these is borax divide both sides by 4 notice I didn't actually Mall play 360 x 2 cuz I knew ahead of time that I'm going to have to factor out anyway so I know to goes into 4 twice and then the two goes into 36180 so my ex is 180 is this an accent 3x relationship so one day is going to be 180 let's check for our unit if you have units in y storyline your answer must have unit 380 feet  20 - 3 x 180 which is 540 so once again make sure when you use all equations that you have two sides with an equal sign now let's talk about motion on a coordinate plane motion on a coordinate plane your keywords that you need make the mill you're with his the word translation reflection rotation dilation K-factor clockwise and counterclockwise all of these emotions and these emotions are often used in computer graphics CG effects recap translation just means to fly you translate something all we're doing is we're sliding the right the word fly and when you have coordinates when you translate but they you have y coordinate of XY coordinate and we Translate buy an additional set of coordinates will call at PQ no let that looks use H for horizontal V4 vertical then all you do is you add you take the X and you add the h turn it which is the horizontal component you take the Y and you add the vertical component the easiest way to simplify this is say this out loud to slide to the left add to the right slide add to the left add to the right to slide add to the left and to the right show you how this actually works we're going to take a triangle ABC and translate it by 6 units to the left and two down 6 units to the left and 2 units down so first let's get our coordinates of a and a sitting there at 4 three that's your coordinate for an 3D is sitting at seven and -1 7 + -1 + C is that positive to negative 2 positive to negative two I'm going to slide it into steps here to slide add to the right we're going to go six units to the left the technically that's going to be when something goes left or down that's going to be a negative number so when am I sex to the left and then we're going to go to unit down which is also a negative number can simply bunnyhop this if you'd like it's a little bit more cumbersome so let me show you the bunny hoppy method fix to the left 10 2 3 4 5 6  at 6 to the left the technically it's -6 and then two units down one two  can you play that's negative to so our new coordinate if you will is going to be for a week all that sub AA with a little hat smart you take a look play that new coordinate is going to be negative 2  positive one let's bunny hop but you could just do it this way again it's add to the left add to the right so we take for  and we add our which is -6 the 4-6 is -2 then you take your y component we add which is going to add -2 - 10 - 2 - 3 - 2 is  pawsitive one does that you can do this numerically we're going to do the other two numerically here  so here I'm going to take sick 10 minute - 7 - 6 + a course I get positive one  and then -1 - 2 is -3  tell my hubby coordinate is going to be positive one -3 it's going to be right there that's RB coordinate sub d positive one negative three and our last one here we're going to go to -6 and of course going to give up negative for + -2 - 2 gives us another negative for sore last coordinate is -4 -4  if you look very closely  what we end up is the exact thing trying it's simply slid six to the left and two down that's how you translate once again say this out loud with me too I'd add to the left to the right that's translation now we have reflection and there's two play reflection is reflection over the y-axis and Its Reflection over the x axis  to reflect over the y-axis dry yourself a little baby that we were reflecting like this across the y-axis when you have whatever your X and Y is to reflect tell me the why you simply opposite the x coordinate to flip over the why you offered to be at say this out loud with me to flip over the why you opposite the x2 flip over the Y office the DS let me show you how this work let's get can you park ordinance the same coordinates which is 4 and 3 * a coordinate Abby coordinate is 7 + 1 -1 and RC coordinate is do + -2  now we're going to flip over this way so just take a line and we're going to go flip over reflect Premier image over the y-axis go to flip over the why we opposite the X so Step One is I simply want you to Circle tell the x coordinate those are the ones were changing the leave the wise alone  that just write in your wife so we have three F1 -2  to flip over the why we opposite the X so we're going to do that's for this becomes negative for 7 is becomes -7 that's two thus becomes negative two now we graph play our first one is negative for positive 3 negative or positive 3 that's our sub -7 positive 1 shipping -7 negative 1 hrp corded or sub d last one is negative to negative two it's connected and it's easier to see the shapes  so you can see the image it's not applied it's a mere we neared play the Y access to this is called a reflection so to reflect over the water flip over the Yu office at The X is twice with me to flip over the Y opposite the X again to flip over the Y office at The X now you can get the next one it was we're going to flip over the X  and if delete just the opposite to flip over the X we office at the Y so if you have to flip over the horse until you have X and Y and you want to flip over the XU office it the y-coordinate so balloon egg bear a negative doesn't always mean it negative per say you simply opposite CU opposite the y-coordinate go to flip over the XU office at the Y let's get our current coordinates here we're going to make day is going to be -8 + -8 be is -6 negative 4 can see is negative two negative V dry your new coordinate so we're going to flip over the X to flip over the X we opposite belt why so Circle the y-coordinate  the exes day completely the same so don't mess what time in factors rewrite them so that their you don't mess with them  flip over the X we opposite the Y so we're going to take this opposite play we going to go to positive Bank opposite of negative for is positive for opposite of -6 is positive 6 there's your reflective coordinates so our sub a looks like this -8 how's the debate there's our new Subway -6 positive for there's are stubby and negative to positive 6 there's our subsea we connect them you see that we have a mirrored image now over the x axis let's recap this verbally to flip over the y-axis to the flip over the y u office at The X tell me the why you off at the BX  and this one is true to flip over the  you opposite the white lower the Axew office at the Y  those are both reflection so so far what we have is we have translation which is the slide actions which is the mirrored image  so we have translation reflection in the last one for now is rotation that's where you take a shape and you're going to rotate it to some degree can you take a coordinates and we're going to rotate now there's three different rotations were going to do Chris were going to rotate with called 90 degrees clockwise  90° clockwise that means we're going Around the Clock okay so to rotate to rotate 90 clockwise draw clockwise Arrow  this is what we do  we take our XY and it goes to two steps you switch and went to the right so it's switch YX  Android clock with arrow and flip the sign on the right  go to rotate 90 clockwise switch and flip to the right so we take her ex why we switch we have Wyatt and then we draw your clockwise arrow and flip the sign on the right because it's why isn't clock wife goes to the right say this out loud to rotate 90 clockwise switching flip to the right let me show you how this works let's take our Originals get our coordinates are original coordinates here of course are 4 and 3 Danby is at 7 negative one and see is at positive to negative two so we're rotating 90 clockwise which means we switch and flip the sign to the right switching flip play the right the first thing I like you to do is simply put in a blank template you going to do this in two steps we're going to switch One Step lip is a second step do not do these two steps at the same time switch then flip the deuce which then flip the sign in to step switch and flip the step one we're going to switch reorder so that's 4 3 this becomes 3/4  then here is seven- one we have negative 1 7 if two two we have negative two, or two- to we have to so that is the switch can you switch the order now can I flip the sign because it's 90 clockwise we flip the signs on the right to draw a little clockwise Arrow flip that sign clockwise Arrow flip that sign clockwise Arrow flip the sign  Knowledge Graph is new new phase 3 negative for positive 3 negative Force Waco hear positive 3 negative 4 so I knew sub-points right here pawsitive three- for that's our Sade Savvy is -1 negative 7 -1 negative 7 going to be way I'm here the last one is negative to negative two scaly -1 seven is this side that's RB and then we have negative to negative 2 cats RC subsea to beat connect those okay it's hard to kind of see the rotation here why it's 90 so there's your shape the easiest way to double-check rotation is rotating from the origin so just connect to the original point so let's go from the origin to a and then from the origin to sub and you can see that we have created a 90 degree angle and we have gone clockwise you want to see that again I'll do that with C so from the origin to see and origin to sub see you see that you play right angle and that you have moved clockwise so we have rotated clockwise here once again to rotate 90 to the right you switching flip to the right to rotate 90 clockwise which switching flip to the right now to rotate counter-clockwise we're going to switch and flip to the left 90 counterclockwise switches flip to the left so here's XY that's going to become first you switch YX and then we're going to flip to the left this time not to the right because it's counterclockwise does say that aloud with me three times rotate 90 counterclockwise switch and flip to the left  to rotate 90 counterclockwise switch left to the left draw that every time to rotate 90 counterclockwise switching flip to the left so this is 90 counterclockwise switching flip to the left do this together  here's your a coordinate same a coordinate we've been using so it's positive for positive 3 happy coordinate is positive seven- one and Arcee coordinate is positive to negative two now we are Road a 90 counterclockwise there's two steps you switch the order and now weird and then you flip the sign and to the left this time that's why we draw counterclockwise Arrow  pod let you catch up here okay Step 1 make yourself a template templates  Step 2  we do this in two steps not in one step so now we just switched borders to switch the order which is going to be three four negative 1722 that's the switch check now we're going to flip the signs counterclockwise when we flip to the left hand side the dryer counterclockwise air put the sign that's positive 3 it becomes negative  counterclockwise Arrow flip the sign it's -1 it's going to be pawsitive one  counterclockwise Arrow flip the sign and it is positive now now let's try graphing these and see your results your first one is going to be the bay is -3 positive for so we go negative three positive for that's our stuff a study is positive one negative into a positive one positive 7 so that's going to be here one positive 7 is going to be way up here it's going to be RB  and then our subsea is positive to positive two so there's our study now we're going to connect them one two three  once again it's kind of hard to see rotation here unless you actually connect the line let's do that for our age so from the origin draw a line to the first day then from the or draw a line to the second a you can see that you have a 90 degree angle there and that you've gone counter-clockwise to the left if you want to check another step we can do that so start at the origin go to seed ronero the origin go to the second seed ronero you again C 90 degrees rotated counterclockwise adjusted to recap these are the tricky ones to rotate clock ways we switching flip to the right to rotate clockwise switching flip to the right switch and flip the right  to rotate 90 clockwise we switch and flip to the left switch slip to the left cats rotation so we've done three motions so far trans song reflection rotation is actually one more motion  that uh oh we have a Jimmy one more we have a reflection over the origin and rotation 180 is actually the same thing if you want to flip straight over the or you want to rotate 180 degrees it's going to be the same thing flip over the origin or one locate 180 again let me draw those arrows and show if you want to flip over the origin which is that or if you want to rotate 180 which is that see that you're going to put yourself in the same position this is an easy one if you have XY and you want to flip over the origin or rotate 180 you just negative them both are opposite both here's our Originals Again original is 43  7 negative 1 turned into -2 positive to negative two and we want to rotate 180 or Clover the origin but I recommend you do is first rear item now you flip the signs both way so that's negative for -3 negative 7 we flip the sign becomes positive one negative to positive two now we graph -4 negative 3 is here has your stud a -7 positive one  here's your stubby and YouTube positive two is your sub c  so when you connect you can see that we have flipped over the origin are we have  rotated 180 degrees which is the same thing the last one is to reflect over the origin or rotate 180 it is the same thing when did opposite your post time now the last one dilate means to change in size means to make something smaller or larger and this one is if you have XY and you dilate by what's call Factor k  all you do is you multiply at time K and my time k  set the jingle actually Rhymes Here say it out loud with me three times to turn thighs Jazz multiply again to change in size just multiply one more time to change in size add multiply  and if your Factor K is larger than one we're going to make it larger if your doctor K is smaller than one we make it smaller both are considered dilations Michelle you have this going to do one example it says we're going to draw the image of the following figure and we're going to dialate it but from the center of the origin by a scale factor of lower 3 that scale factor is your Factor k  step one we need to get our three sources of point so this point will call it a all this B will call this C so a  is at positive nine coming over here positive 9 we make this three different colors so it's easy to track will do green be is at positive thick positive 9  and see will do red is at positive 9 positive 6 so we have 3 coordinates and then we're going to dialate this by a factor of five do we have a at nine nine we have be at 6 now and then we have fee at 9 6  we're going to dilate by a factor of 5/3 in a month play all of these by 5/3 so we take line  The Chew 9 hour in a * 5/3 to get are new coordinates equals  I'm going to make this really big you can follow the math here 3 goes into 9 three times 3 * 5 is 15 and again 3 turn the 9 3 * 3 * 5 is 15 so our new coordinate our sub a coordinate it's going to be 15 15 what do the B coordinate so that's going to be fixed and nine and both of them are going to be times 5/3  time over 3 you don't need a calculator for this ready 3 goes into 6 twice  2 * 5 is 10  3 goes into 9 3 * 3 * 5 is 15  so I newbie coordinate our study is 10 + 15 C is 9 and 6 so we have 9 and 6 Time 5 / 3 trigos in the 9 3 * 3 * 5 is 15 6 goes in 3 goes into 6 twice 2 * 5 is 10 so I see coordinates subsea coordinate is going to be 1510  go back to our original playground  never going to graph these new point so are new today is 1515 so I go to 1415 follow it up to 15 which is right there there Sade  no our study is 10:15 so 10 + 15 user sub V & R Sub C is 1510 15 freaking up to 10 temperature subsea  and here's how you know you're correct we draw this obvious second image is larger than our first which makes sense because RK Shafter is 535 / 3 turn one  and so therefore it's larger than one that means we're going to have an enlargement or we're going to make the image larger when you dilate buy something over one at makes it larger dialight with something less than 1 you shrink it if you'd like a quick recap for this lesson  highly recommend you go to the last page here in your notes or second and you have a summary of all 4 translations right here on this pay

Uploaded by David Rowan.

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