Introduction A young woman uses her reflection in a mirror to give herself a facial.

Transcription

1 Algebra/Geometry Blend Unit #2: Transformations Lesson 2: Reflections Introduction A young woman uses her reflection in a mirror to give herself a facial. [page 1] Name Period Date Have you ever mimicked your siblings by mirroring their actions? Have you ever been in a car and seen an emergency vehicle in a rearview mirror? Have you ever held a magazine or book up to a mirror? In all of these situations, what you see or do is being done in reverse. When you are standing face to face with your brother and he moves his right hand, you must move your left hand in order to mimic him. The ambulance in the rearview mirror will have "AMBULANCE" written in reverse on the hood. The magazine or book will look backwards in the mirror image. In this lesson, you will learn how reflecting a figure on the coordinate plane, which is a rigid motion, creates a new geometric figure that is congruent to the original. Learn [page 2] Reflections A reflection is a where the mirror image of a figure is shown directly its line of. Notice that both figures are the same and. The segment lengths and angle measures have not changed. Therefore, quadrilateral ABCD is to quadrilateral A'B'C'D'. When a figure is moved to a different location without altering its shape or size (including reflecting the figure), this is known as a. You will witness this phenomenon more closely in the coming pages.

2 When a figure is reflected, all points along each segment are reflected. In other words, a point lying on AB in the pre-image will lie on A'B' in the image. If point E is the of AB, then point E' will be the of A'B'. Therefore, a reflected figure preserves and of points, just like with translated figures. Also, if a line of symmetry exists in the pre-image, it will exist in the image. However, notice that the of the two figures is different. The pre-image is named quadrilateral ABCD. The points are listed The image must be named quadrilateral A'B'C'D' to keep the corresponding sides and angles in the proper order. In this case, the points are listed. Therefore, the orientation of reflected images is. Learn [page 3] Discovering the Rules of Reflection Reflections X-Axis If a pre-image is reflected across the x-axis, how do you find the image? Another way to say "reflect across the x-axis" is to say "reflect across the line " since the line created by graphing y=0 is the same as the x-axis. An image that is a reflection across the x-axis, or across the line y=0, will have the same x-coordinates as the pre-image but opposite y-coordinates. Therefore, the rule for reflecting an image across the x-axis can be described as (x, y) (x, y). When a figure reflects to a new position on the coordinate grid, the reflected image can be expressed in a form like a function. Remember that a function is like a machine. A number is placed into the machine and, after it is processed, a new result emerges. The x- and y-coordinates can be thought of as two separate function machines working together to give the coordinate of the new position. Imagine a figure where point A is located at ( 5, 3). The rule for reflecting an image across the x-axis is to take the original point (x,y) and change it to (x, y). If you take the x- coordinate of 5 and apply the rule, is made to the x-coordinate at all. So when x is put into our function machine, it comes out as x, or unchanged. This means the x-coordinate when reflecting will still be 5. The y-coordinate must change to y. If you take 3 and place it into the function machine, the machine asks you to take the opposite of that value to get your output. The opposite of 3 is. This means the function machine helped us take ( 5, 3) as an input and generate ( 5, 3) as the output for point A'.

3 Reflections Y-Axis If a pre-image is reflected across the y-axis, how do you find the image? Another way to say "reflect across the y-axis" is to say "reflect across the line " since the line created by graphing x=0 is the same as the y-axis. An image that is a reflection across the y-axis, or across the line x=0, will have opposite x-coordinates from the pre-image but identical y-coordinates. Therefore, the rule for reflecting an image across the y-axis can be described as (x, y) ( x, y). Just like in the previous example, the idea of a function machine is used to help find the new coordinates for this reflection. Point A is located at ( 5, 3). The rule to follow for reflecting a figure across the y-axis is to take the original coordinate (x, y) and change it to ( x, y). If you take the x-coordinate of 5 and place it into the function machine, the machine asks you to take the opposite of that value to get your output. The opposite of 5 is. If you take the y-coordinate of 3 and apply the rule, is made to the y-coordinate at all. So when y is put into our function machine, it comes out as y, or unchanged. This means the function machine helped take ( 5, 3) as an input and generate (5, 3) as the output for point A'. Reflections Line y = x If a pre-image is reflected across the line y = x, how do you find the image? To find an image that has been reflected across the line y = x, the x- and y-coordinates. Therefore, the rule for reflecting an image across the line y = x can be described as (x, y) (y, x). Just like in the previous two examples, the idea of a function machine is used to help find the new coordinates for this reflection. Point A is located at ( 4, 3).The rule to follow for reflecting a figure across the line y=x is to take the original coordinate (x,y) and change it to (y,x). In other words, the position of the x- and y-coordinates are going to. If you take the x-coordinate of 4 and place it into the function machine, the machine asks you to the position of the x-coordinate to the y-coordinate position to get your output. If you take the y-coordinate of 3 and apply the rule, the machine asks you to switch the position of the y-coordinate to the x-coordinate. This means the function machine helped take ( 4,3) as an input and generate (3, 4) as the output for point A'.

4 Reflections Horizontal and Vertical Lines If a pre-image is reflected across a vertical or horizontal line, how do you find the image? While determining the relationship between a pre-image and its reflection across a vertical or horizontal line is not as straightforward as the other methods, it is still possible to find. Notice that each point is the same distance away from the line of reflection as its corresponding point. For example, if point A is 4 units away from the line of reflection, point A' is 4 units away from the line of reflection but on the other side. Realize, however, that point A and point B in the same figure will necessarily be the away from the line of reflection. So you will need to find the distance for in the polygon before finding each corresponding point. Learn [page 4] Examples of Reflections Once you have learned how to reflect an image across the coordinate plane, you are ready to put it to use in some examples and problems. Take a look! Example One Pentagon HOUSE is shown on the coordinate plane. Reflect pentagon HOUSE across the: x-axis (y = 0) y-axis (x = 0) line y = x line y = 2 line x = 1

5 Reflection Across the X-Axis (y = 0) When a figure is reflected across the x-axis, the x-coordinates of the image will stay the same, while the y-coordinates will be the opposite of the pre-image. In other words, (x, y) (x, y). HOUSE H O U S E H( 4, 5) H (, ) O( 2, 3) O (, ) U( 3, 1) U (, ) S( 5, 1) S (, ) E( 6, 3) E (, ) [Graph the new points] Reflection Across the Y-Axis (x = 0) When a figure is reflected across the y-axis, the x-coordinates of the image will be the opposite of the pre-image, while the y-coordinates will be identical. In other words, (x, y) ( x, y). HOUSE H O U S E H( 4, 5) H (, ) O( 2, 3) O (, ) U( 3, 1) U (, ) S( 5, 1) S (, ) E( 6, 3) E (, ) [Graph the new points]

6 Reflection over the line y = x When a figure is reflected across the line y = x, the x-coordinates and y-coordinates of the pre-image will be reversed for the image. In other words, (x, y) (y, x). HOUSE H O U S E H( 4, 5) H (, ) O( 2, 3) O (, ) U( 3, 1) U (, ) S( 5, 1) S (, ) E( 6, 3) E (, ) [Graph the new points] Reflection over the line y = -2 When a figure is reflected across a horizontal or vertical line, you need to the distance between each point in the figure and the line of reflection. Point H is points away, point O is points away, point U is points away, point S is points away, and point E is points away. Now you must count this same distance from the line of reflection to find each corresponding point. Point H' will be points away, point O' will be points away, point U' will be points away, point S' will be points away, and point E' will be points away.

7 Now connect the points with segments, and you have the reflected image! [here. I ll help] H O U S E H (-4, -9) O (-2, -7) U (-3, -5) S (-5, -5) E (-6, -7) Reflection over the line x = 1 When a figure is reflected across a horizontal or vertical line, you need to count the distance between each point in the figure and the line of reflection. Point H is points away, point O is points away, point U is points away, point S is points away, and point E is points away. Now you must count this same distance away from the line of reflection to find each corresponding point. Point H' will be points away, point O' will be points away, point U' will be points away, point S' will be points away, and point E' will be points away.

8 Now connect the points with segments, and you have the reflected image! [here. I ll help] H O U S E H (6, 5) O (4, 3) U (5, 1) S (7, 1) E (8, 3) Example Two Isosceles trapezoid TRAP and its transformation T'R'A'P' are shown on the coordinate plane. Which two transformations are applied to trapezoid TRAP to create T'R'A'P'? Because the trapezoid image is, you can determine that the preimage was across the x-axis. If it had been reflected across the y- axis, the image would not look much different from the original. So reflect trapezoid TRAP across the x-axis by changing the sign on each y-coordinate.

9 Once you have the reflected image, trapezoid T1R1A1P1, you can see that this image needs to be to arrive at trapezoid T'R'A'P'. Find the translation rule by counting up or down and left or right from P1 to P'. In order to reach P' from P1, you must count units and units. Therefore, the translation rule is (x, y) > (, ). So the two transformations applied to trapezoid TRAP to create T'R'A'P' is a over the and then a 8 units left and 2 units up. Note that this is not the only series of transformations that would create trapezoid T'R'A'P'. You could also have reflected TRAP across the line y = 1 and translated 8 units left. Try it yourself! [don t!]

10 You Try! Practice One Triangle PQR is graphed on the coordinate plane with vertices at the following ordered pairs: P( 4, 6), Q( 2, 3), R(3, 3) If triangle PQR is reflected across the x-axis to create triangle P'Q'R', what are the ordered pairs of each vertex of triangle P'Q'R'? When a figure is reflected across the x-axis, or y=0, the sign of the y-coordinates, while the x-coordinates the. In other words, the figure is reflected according to the rule (x, y) (x, y). So the vertices of triangle P'Q'R' are P'(, ), Q'(, ), R'(, ). Take a look at the graph. Practice Two Hexagon JKLMNO is graphed on the coordinate plane with vertices at the following ordered pairs: J( 6, 2), K( 3, 2), L( 2, 0), M( 3, 2), N( 6, 2), O( 7, 0) If hexagon JKLMNO is reflected across the y-axis to create hexagon J'K'L'M'N'O', what are the ordered pairs of each vertex of hexagon J'K'L'M'N'O'? When a figure is reflected across the y-axis, or x=0, the sign of the x-coordinates, while the y-coordinates the. In other words, the figure is reflected according to the rule (x, y) ( x, y). So the vertices of hexagon J'K'L'M'N'O' are J'(, ), K'(, ), L'(, ), M'(, ), N'(, ), O'(, ). Take a look at the graph.

11 Practice Three Kite DEFG is graphed on the coordinate plane with vertices at the following ordered pairs: D(2, 8), E(0, 5), F(2, 3), G(4, 5) If kite DEFG is reflected across the line y = x to create kite D'E'F'G', what are the ordered pairs of each vertex of kite D'E'F'G'? When a figure is reflected across the line y = x, the numerical values of the x-and y-coordinates are reversed. In other words, the figure is reflected according to the rule (x, y) (y, x). So the vertices of kite D'E'F'G' are D'(, ) E'(, ) F'(, ) G'(, ) Take a look at the graph. Practice Four Rectangle ABCD is shown on the coordinate plane. Part 1: Reflect rectangle ABCD across the line x = 1 to create rectangle A'B'C'D'. Part 2: Reflect rectangle ABCD across the line y = 1 to create rectangle A''B''C''D''.

12 Part 1 First, graph the vertical line x = 1. Since rectangle ABCD is being reflected across a line, the y- coordinates will remain the same. To find the x-coordinates of each point, determine how far apart each point is away from the line of reflection. The reflected points will be the same distance away. From points A and B, you count units to the to reach the line of reflection. Therefore, count 3 units to the left from the line of reflection to find reflected points A' and B'. From points C and D, you count units to the to reach the line of reflection. Therefore, count 8 units to the left from the line of reflection to find reflected points C' and D'. The graph below shows the original rectangle and its reflected figure.

13 Part 2 First, graph the horizontal line y = 1. Since rectangle ABCD is being reflected across a line, the x-coordinates will remain the. To find the y-coordinates of each point, determine how far apart each point is away from the line of reflection. The reflected points will be the same distance away but on the other side. From points A and D, you count units to reach the line of reflection. Therefore, count 2 units up from the line of reflection to find reflected points A'' and D''. From points B and C, you count units to reach the line of reflection. Therefore, count 5 units up from the line of reflection to find reflected points B'' and C''. The graph below shows the original rectangle and its reflected figure.

14 Learn [page 5] Figure 1 Look at the three figures below. Each shows a hexagon reflected on the coordinate grid. Figure 1 shows the hexagon reflected using the rule (x, y) (x, y) since it is a reflection across the. This means all x-coordinates are and the y-coordinates are the of the original value. Notice that if you draw the vectors of each point as they translate, you wind up with a series of line segments that are in length. As mentioned earlier, a reflection is a. This means that after the reflection has taken place, all and are to the original figure. The size and shape of the original figure has not changed. Notice that the length from each point to its new reflected coordinate creates a series of line segments that are congruent between corresponding vertices. Figure 2 Looking at figure 2, notice that the hexagon has been reflected across the y-axis. What rule can be used to show this reflection? See if you can discover what series of steps would carry the first figure onto the second one. ANS: Since the hexagon is reflected over the y-axis, the rule (x,y) (, ) would carry the first hexagon onto the reflected one. Notice that if you draw the vectors of each point as they reflect, you wind up with a series of, just like you did for figure 1. If you move each of the points in the figure the same distance, what can you conclude about the side lengths of the original figure and the translated figure? Keep reading to discover the answer!

15 Figure 3 For figure 3, another reflection was performed. What rule can be used to show this reflection? See if you can discover what series of steps would carry the first figure onto the second one. ANS: Since the hexagon is reflected across the line y=x, the rule (x,y) (, ) would carry the first hexagon onto the reflected one. Notice that if you draw the vectors of each point as they are reflected, you wind up with a series of. Since all three figures demonstrate motion, the angles and side lengths are the. This means the original hexagons and reflected hexagons are to each other. Learn [page 6] Cumulative Practice You have learned about reflections and how they are represented on the coordinate plane. It's time to check your knowledge, and see if you are ready to take the assessment. #1 Parallelogram FGHI and its reflection, parallelogram F'G'H'I', are shown below in the coordinate plane. What is the line of reflection between parallelograms FGHI and F'G'H'I'?

16 #2 Pentagon ZABCD is shown on the coordinate plane. If pentagon ZABCD is reflected across the x-axis to create pentagon Z'A'B'C'D', what is the ordered pair of point Z'? There are two ways to find the ordered pair of point Z'. First, you could remember the rule that points reflected across the x-axis will have identical x- coordinates and the sign on the y-coordinate will change. In other words, (x, y) (x, y). Since point Z is at (7, 5), its reflected point will be (, ). Or you could count the distance between point Z and the x-axis. The reflected point Z' must be the same distance away in the same direction.

17 #3 Rhombus EFGH is shown on the coordinate plane. If rhombus EFGH is reflected across the y-axis to create rhombus E'F'G'H', what is the ordered pair of point G'? There are two ways to find the ordered pair of point G'. First, you could remember the rule that points reflected across the y-axis will have the opposite sign on the x-coordinate and identical y-coordinates. In other words, (x, y) ( x, y). Since point G is at (4, 6), its reflected point will be (, ). Or you could count the distance between point G and the y-axis. The reflected point G' must be the same distance away in the same direction.

18 #4 Trapezoid IJKL is shown on the coordinate plane. If trapezoid IJKL is reflected across the line y = x to create trapezoid I'J'K'L', what are the vertices of trapezoid I'J'K'L'? When a figure is reflected across the line y = x, the x and y values are reversed. In other words, (x, y) (y, x). Vertex I Vertex J (x, y) (y, x) (x, y) (y, x) ( 6, 6) (, ) ( 3, 6) (, ) I': (, ) J': (, ) Vertex K Vertex L (x, y) (y, x) (x, y) (y, x) ( 1, 3) (, ) ( 8, 3) (, ) K': (, ) L': (, )

19 #5 Triangle ABC and its transformation, triangle A'B'C', are shown on the coordinate plane. Which two transformations are applied to triangle ABC to create A'B'C'? Because point A is not on the bottom of the triangle, triangle ABC was not across the x-axis. Since triangle ABC passes through the line y = x, it could not have been reflected across this line. If it had passed through the line, point A' would be in the bottom-right corner. Therefore, triangle ABC must have been across the. To reach triangle A'B'C' from the reflected image of triangle A1B1C1, it must be. Count the number of units and the direction between one of the points in triangle A1B1C1 to the translated point in triangle A'B'C'. Let's use point B1 and B'. Because you must count place from point B1 to point B', the rule for the x-coordinate is. Because you must count places, the rule for the y-coordinate is. The translation rule is (x, y) (, ). Triangle ABC is reflected across the and translated according to the rule (x, y) (, ) to reach triangle A'B'C'.

20 [page 7] Review GeOverview A reflection is a type of transformation that creates a mirror image of the pre-image across a line of reflection. You learned how images are created across four different lines of reflection: Across the x-axis o (x, y) (, ) Across the y-axis o (x, y) (, ) Across the line y = x o (x, y) (, ) Across a horizontal or vertical line o Count the distance between each point on the figure and the line of reflection. Then, for each point, count this from the line of reflection to find the corresponding point. Remember that a reflection can also be seen as a function, which is a type of input/output machine. If you had a point at (2, 3) and wanted to reflect it using the rule (x,y) ( x,y), you could place 2 in for x, and you would get 2 for your x-coordinate since the rule says to take the opposite of the x-coordinate. If we place 3 in for y and follow the rule, our output is 3 since the y-coordinate remains unchanged. Our new coordinate would be (, ).

21 Name Period date HW #2.02: Reflections Algebra/Geometry Blend Rules: X-axis Y-axis y=x (x, y) (x, -y) (x, y) (-x, y) (x, y) (y, x) #1 If point A (5, 2) is reflected over the x-axis, what is the location of A? #2 If the point B (4, -6) is reflected over the y-axis, what is the location of B? #3 If the point C (-3, 5) is reflected over the line y=x, what is the location of C? #4 Reflect the rectangle ABCD over the x-axis. A (, ) (, ) B (, ) (, ) C (, ) (, ) D (, ) (, ) #5 Reflect the rectangle ABCD over the line y = x. A (, ) (, ) B (, ) (, ) C (, ) (, ) D (, ) (, )