Sometimes function are described as input/output machines. The example to the right shows the input/output machine of

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1 G.. NTES Transformations 1 NEPT 1 Describe transformations as functions that take points in the plane as inputs and give other points as outpoints. QUIK LGER REVIEW In algebra we studied functions extensively and during that time we learned how to solve for variables using function notation. Let me show you a few examples to remind you. (1) What is the value of the function f( x) = x 3when x = -? () What is the value of the function f( x) = x+ 7when x =? (3) What is the value of x when f( x ) = 11? f( x) = x 3 f ( ) = ( ) 3 f ( ) = 3 = 1 f ( ) = 1 f( x) = x+ 7 f () = + 7 f () = 1 f( x) = x = x = x = x = x When x = -, then y = 1 (-,1) When x =, then y = 1 (,1) When y = 11, then x = (,11) In the first two examples we were given the x value and then asked to solve for the y value (the value of the function). In the third example we were given with the y value (the value of the function) and then asked to work backwards to determine the x value that would have produced that result. Sometimes function are described as input/output machines. The example to the right shows the input/output machine of f( x) = x+. When we input x = - 1, the function machine produces the output of. NNETIN T GEMETRY In geometry we have a similar input/output process when we determine how shapes are altered or moved. Geometric objects can be moved in the coordinate plane using a coordinate rule. These rules can alter the shape in many different ways. Some rules will translate the shape, some will rotate or reflect the shape, some will stretch or distort the shape, some will increase the size of the shape, etc lots of different things can happen. oordinate rules come in the following form: English Translation: oordinate Rule T maps all points (x,y) to (x + 3, y 6) So rule T will move or map all points (x,y) by adding 3 to the x value of each point, and subtracting 6 from each y value of each point. The name of the rule is a capital letter before the (x,y). The point (x,y) represents that all points in the plane will be effected by this rule. The arrow represents the geometric term of map or mapping, which is a geometric way of saying moves. Finally the coordinate description after the mapping arrow symbol represents the change produced by the rule. an you see the connection to functions? You will input values into the

2 G.. NTES Transformations coordinate rule and it will output new values, just as function did. change is that we refer to the input value as the pre- image, the original location of the point, and the output value as the image, the new location of the point. To help clarify the difference between a pre- image and an image, we use special notation. If the pre- image is point, then its image will be (said prime). The prime notation tells us that it is an image. Let me do a few examples to help you see the correlation between algebra and geometry. (1) Given coordinate rule T (x, y) > (x + 3, y 6), determine the image of (- 1, )? () Given coordinate rule G (x, y) > (x, y + 1), determine the image of (,)? Pre- Image (- 1,) Image (,- 1) Pre- Image (,) Image (10,3) oth of these examples provided the pre- image (input) values and then asked you to solve for the image (output value). We can work backwards through the process; just as in we did with the function example #3. Let us look at two where we know the image and we work backwards through the coordinate rule to solve for the pre- image. (3) Given coordinate rule T (x, y) > (x + 3, y 6), determine the pre- image of (3, )? () Given coordinate rule G (x, y) > (x, y + 1), determine the pre- image of D (-, 11)? Pre- Image (0,8) Image (3,) Pre- Image D (-,10) Image D (-,11)

3 G.. NTES Transformations 3 NEPT 3 ompare transformations that preserve distance and angle (i.e. rigid motions) to those that do not (e.g. translation vs. horizontal stretch). n ISMETRI TRNSFRMTIN (RIGID MTIN) is a transformation that preserves the distances and/or angles between the pre- image and image. D Example #1 Example # Example #3 E F ' F' E' D' I H J Rotate (Turn) Example #1 Translate (Slide) Example # Reflection (Flip) - Example #3 H' I' J' K L M M' L' K' NN- ISMETRI TRNSFRMTIN (NN- RIGID MTIN) is a transformation that does not preserve the distances and angles between the pre- image and image. Example #1 Example # Example #3 F L M D D' E G F' K L' N M' ' E' G' K' N' Stretch Where one dimension s scale factor is different than the other dimension s scale factor. Examples # and #3 represent stretches. stretch definitely distorts the shape making it a NN- ISMETRI transformation. Dilation Where both dimension s scale factors is the same. The shape is proportional, not identical. Dilation changes the size of the shape making it a NN- ISMETRI transformation. There are many ways to distort a shape but these two are the most popular.

4 G.. NTES Transformations THE ISMETRI TRNSFRMTINS (1) THE REFLETIN EXPLNTIN reflection over a line m (notation R m ) is an isometric transformation in which each point of the original figure (pre- image) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line. ecause a reflection is an isometry, the image does not change size or shape. The line of reflection is the perpendicular bisector of the segment joining every point and its image. m ' Rm ( Δ ) =Δ ' DEFINITIN reflection in a line m is a isometric transformation that maps every point P in the plane to a point P, so that the following properties are true; 1. If point P is NT on line m, then line m is the perpendicular bisector of PP '. m. If point P is N line m, then P = P m P P' P = P' PRPERTIES ISMETRI PRPERTIES - ecause a reflection is an isometric transformation the following properties are preserved between the pre- image and its image: Distance (lengths of segments are the same) ngle measure (angles stay the same) Parallelism (things that were parallel are still parallel) ollinearity (points on a line, remain on the line) fter a reflection, the pre- image and image are identical.

5 G.. NTES Transformations TRNSFRMTIN PRPERTIES ecause a reflection is a transformation that maps all points the perpendicular distance on the opposite side of the line of reflection the following properties are present. m DISTNES RE DIFFERENT - - Points in the plane move different distances, depending on their distance from the line of reflection. Points farther away from the line of reflection move a greater distance than those closer to the line of reflection. Notice how is greater than. Notice that because line m is the perpendicular to ', ' and ' they are all parallel to each other. RIENTTIN IS REVERSED The pre- image has a reversed orientation than its image. rientation is the order of the points about the shape. In Δ the points in a clockwise direction come in the order of but in the image the points in a clockwise direction come in the order of. This occurs because a reflection creates the mirror image. SPEIL PINTS Points on the line of reflection do not move at all under the reflection. The pre- image (D) = image (D ) when the point is on the line of reflection. ' D = D'

6 G.. NTES Transformations 6 REFLETIN N THE RDINTE GRID Reflection over the y axis. When we reflect over the y axis, the y values are unchanged and the x values are negated (opposite). (-3,) (-1,3) (-,-1) ' (1,3) (,-1) (3,) (-x,y) ' (x,y) RULE FR REFLETIN VER THE Y XIS R ( x, y) = ( x, y) y axis T(x,y) ( x, y) Reflection over the x axis. (-,3) (x,y) When we reflect over the x axis, the x values are unchanged and the y values are negated (opposite). (-,) (-,-) (-1,1) (-1,-1)' ' (-,-3) ' (x,-y) RULE FR REFLETIN VER THE X XIS R ( x, y) = ( x, y) x axis T(x,y) (x, y) Reflection over the y = x line. When we reflect over the y = x line, the x and y values are reversed. 6 D (0,) E (,6) F (3,) F' (,3) E' (6,) F (x,y) F' (y,x) D' (,0) RULE FR REFLETIN VER THE Y = X LINE R ( x, y) ( y, x) y = x = T(x,y) (y, x)

7 G.. NTES Transformations 7 () THE RTTIN EXPLNTIN rotation is an isometric transformation that turns a figure about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation (notation R center, degree ). n object and its rotation are the same shape and size, but the figures may be turned in different directions. ( ) R, Δ =Δ ' ' ' ' ' DEFINITIN rotation about a Point through Ɵ degrees is an isometric transformation that maps every point P in the plane to a point P, so that the following properties are true; 1. If point P is NT point, then P = P and m PP = Ɵ. P'. If point P IS point, then P = P. The center of rotation is the NLY point in the plane that is unaffected by a rotation. P = P = P'

8 G.. NTES Transformations 8 RTTIN DIRETIN ne full rotation is 360, this would return all points in the plane to their original location. ecause a rotation can go in two directions along the same arc we need to define positive and negative rotation values. UNTERLKWISE IS PSITIVE DIRETIN, and clockwise is a negative direction. Students often want to know why this backwards relationship happens.. my belief is that the rotation direction comes from the coordinate plane, In the coordinate plane, the initial arm of the angle is the positive x axis, and the terminal arm opens in a counterclockwise direction from there. PRPERTIES ISMETRI PRPERTIES - ecause a rotation is an isometric transformation the following properties are preserved between the pre- image and its image: Distance (lengths of segments are the same) ngle measure (angles stay the same) Parallelism (things that were parallel are still parallel) ollinearity (points on a line, remain on the line) fter a rotation, the pre- image and image are identical. TRNSFRMTIN PRPERTIES ecause a rotation is a transformation that maps all points along an arc the following properties are present. DISTNES RE DIFFERENT - - Points in the plane move ' different distances, depending on their distance from the center of rotation. point farther away from the center of rotation maps a greater distance than those points closer to the center of rotation. Notice that different from a reflection, ', ' and ' are NT parallel to each other. RIENTTIN IS THE SME The pre- image has the same orientation as its image. rientation is the order of the points about the shape. In Δ the points in a clockwise direction come in the order of and in the image the points in a clockwise direction come in the order of. They are the same!! (Sometimes students want to say that a rotation changes the orientation of the shape because the shape has been turned to create a different look of the shape. this is NT an orientation change!) SPEIL PINTS The center of rotation is the only point in the plane that is unchanged, =.

9 G.. NTES Transformations 9 RTTIN N THE RDINTE GRID Rotation of 90 about the rigin When we rotate 90 about the origin, we see that the x and y coordinates are reversed and the new x coordinate is negated. ' (-,) (-,3) (-1,1) (1,1) (,) RULE FR RTTIN Y 90 UT THE RIGIN R (, ) (, ),90 x y = y x T(x,y) ( y, x) (3,) (-y,x) (x,y) Rotation of 180 about the rigin When we rotate by 180 about the origin, we see the x and y coordinates are negated. (-3,-) -(1,-1) ' (--,) (1,1) (,) (3,) (-x,-y) (x,y) RULE FR RTTIN Y 180 UT THE RIGIN R (, ) (, ),180 x y = x y T(x,y) ( x, y) Rotation of 70 about the rigin (,) (x,y) This is also a rotation of (1,1) (3,) When we rotate by 70 about the origin, we see that the x and y coordinates are reversed and the new y coordinate is negated. (1,-1) (,-3) ' (,-) RULE FR RTTIN Y 70 UT THE RIGIN R (, ) (, ),70 x y = y x T(x,y) (y, x) (y,-x)

10 G.. NTES Transformations 10 (3) THE TRNSLTIN EXPLNTIN translation slides an object a fixed distance in a given direction. When working in the plane this is usually represented by an arrow, the arrow provides both distance and direction of the translation. When working on the coordinate plane, a vector is used to describe the fixed distance and the given direction often denoted by <x,y>. The x value describes the effect on the x coordinates (right or left) and the y value describes the effect on the y coordinates (up or down). ' The pre- image and image have the same shape and size. ( ) T Δ xy, =Δ < > ' This is a key property to translations ll segments that are translated are parallel to each other. DEFINITIN translation is an isometric transformation that maps every two points and in the plane to points and, so that the following properties are true; 1. = (a fixed distance).. ' (a fixed direction). ' PRPERTIES ISMETRI PRPERTIES - ecause a translation is an isometric transformation the following properties are preserved between the pre- image and its image: Distance (lengths of segments are the same) ngle measure (angles stay the same) Parallelism (things that were parallel are still parallel) ollinearity (points on a line, remain on the line) fter a translation, the pre- image and image are identical.

11 G.. NTES Transformations 11 TRNSFRMTIN PRPERTIES ecause a translation is a transformation that maps all a fixed distance and in a fixed direction the following properties are present. DISTNES RE THE SME - - Points in the plane all map the exact same distance. Notice how is EQUL T. ' RIENTTIN IS THE SME The pre- image has the same orientation as its image. In Δ the points in a clockwise direction come in the order of and in the image the points in a clockwise direction come in the order of. SPEIL PINTS There are N special points, LL PINTS IN THE PLNE MVE!!! TRNSLTIN N THE RDINTE GRID translation is defined by a fixed distance and a fixed direction. This motion is usually described by an arrow or a vector. The vector describes the horizontal and vertical shifts in the plane. For example, if we slide all points 3 to the right and up we are defining a fixed distance and a direction. Translations are described in a few different way; T (x,y) > (x + 3, y + ) T ( x, y ) ( x 3, y ) < 3, > = RULE FR TRNSLTINS T(x,y) (x ± h, y ± k) h and k are real numbers T (x,y) > (x +, y - 1) T ( x, y ) ( x, y 1) <, 1 > = + T (x,y) > (x -, y + 1) T ( x, y ) ( x, y 1) <,1 > = + T (x,y) > (x -, y - ) T ( x, y ) ( x, y ) <, > =