Up, Down, and All Around Transformations of Lines WARM UP Identif whether the equation represents a proportional or non-proportional relationship. Then state whether the graph of the line will increase or decrease from left to right. 1. 5 9. 5 3 3. 5 3 NP NP p It decrease intase ase LEARNING GOALS Translate linear graphs horizontall and verticall. Use transformations to graph linear relationships. Determine the slopes of parallel lines. Identif parallel lines. Eplore transformations of parallel lines. You have learned how the coordinates of an image are affected when a pre-image is translated, reflected, rotated, or dilated. How can ou use knowledge about geometric transformations to transform the graphs and equations of linear relationships? LESSON : Up, Down, and All Around M-53
Getting Started Transformation Station Consider nabc with coordinates A (, ), B (, ), and C (, ) shown on the coordinate plane. 1. Suppose the triangle is translated in a single direction. In general, how does this affect the coordinates of the figure? A 0 C B (, ) New Coordinates Units Up Units Down L t k Units Left Units Right Rt. Suppose the triangle is reflected across an ais. How does this affect the coordinates of the figure? (, ) -Ais -Ais New Coordinates L C M-5 TOPIC 1: From Proportions to Linear Relationships
3. Suppose the triangle is rotated through an angle with the origin as the center of rotation. How does this affect the coordinates of the figure? (, ) New Coordinates 90 Counterclockwise 10 70 Counterclockwise G I E C f 901 Clockwise. Suppose the triangle is dilated b a factor of m with a center of dilation at the origin. How does this affect the coordinates of the figure? f CES (, ) Dilation,10 New Coordinates Cm m Dilate SF origin 5. How do ou think translations, reflections, rotations, and dilations affect lines? The transformations will operate on lines just as theoperate on geometricfigures LESSON : Up, Down, and All Around M-55
ACTIVITY.1 Translating Lines 1 = In this activit, ou will investigate how the equation of a line changes as ou translate the line up and down the -ais. 3 fin 3 1 l m I 7 1 0 1 3 1 Consider the graph of the basic linear equation 5, which is of the form 5 m. The line represents a proportional relationship with a rate of change, or slope, of 1. 1. Trace the aes and the line 5 on a sheet of patt paper. 3 L 1. Keep the -ais on our patt paper on top of the corresponding -ais of the coordinate plane. Slide the line 5 up and down the -ais. a. How does the slope of the line change as ou move it up and down the -ais? The slope doesn't change b. How do the coordinates of the line change as ou move it up and down the -ais? The X values sta the same and the values change to match the distance moved M-5 TOPIC 1: From Proportions to Linear Relationships
3. Translate the line 5 up units. a. Graph and label the line with its equation. b. Compare the equation of 5 to the equation of its translation up units. What do ou notice? k same slope of 1 I. Translate the line 5 down units. a. Graph and label the line with its equation. Now has t since it was translated up Be sure to use a straightedge as ou draw lines throughout this lesson. b. Compare the graph and equation of 5 to the graph and equation of its translation down units. What do ou notice? 5. For an -value, how does the -value change when ou translate 5 up or down? K same slope 1 K equation now has mtb The b. Are the translated lines proportional or non-proportional relationships? Eplain our reasoning. translated down value changes based on how much I translate the line Non proportional Doesn't go throughorigin LESSON : Up, Down, and All Around M-57
NOTES a Translate up lt b Translatep c 7. The lines on the graph are translations of the line represented b 5. 0 kt Il c Translate downs 1 5 d translate down 7 K 7 a. Describe each translation in terms of a translation up or down. Then write the equation. b. Identif the slope of each line. m The lines drawn on the coordinate plane in Question 7 represent parallel lines. Remember that parallel lines are lines that lie in the same plane and do not intersect no matter how far the etend. Parallel lines are alwas equidistant.. Analze the graph of each line and its corresponding equation. a. How can ou verif that the lines graphed are equidistant? b. How can ou tell b looking at the set of equations that the lines are parallel? 9. Based on our investigation, complete the sentence: The line 5 1 b is a l count distance that maps the point (0, 0) onto the point point (1, 1) onto the point (1, ). i a b c d 7 ongraph If lines havethe same slopethe are vertical translation O b Itb of the line 5 and maps the paralle M-5 TOPIC 1: From Proportions to Linear Relationships
ACTIVITY. Dilating Linear Equations The graph of the basic linear equation 5 is shown on the coordinate plane. 3 1 = 3 1 0 1 1 3 3 Let's investigate how the line = changes when the rate of change, or slope, changes. 1. Use a thin piece of pasta to eplore how the characteristics of the line change as ou dilate the line 5 to create the lines with equation 5 and 5 1. Then complete the table based on our investigation. 5 5 5 1 5 m 0 1 LESSON : Up, Down, and All Around M-59
. Based on our investigation, complete the sentence: NOTES The line 5 m is a of the line 5 that maps the point (0, 0) onto the point and maps the point (1, 1) onto the point (1, ). 3. Consider the equation 5 3. Use transformations to complete the table of values. Eplain our strateg. 5 5 3 1 0 1. The equation 5 is a transformation of 5. a. How are the equations similar? How are the different? 0 M-0 TOPIC 1: From Proportions to Linear Relationships
b. Graph both equations to determine the transformation. c. Based on our investigation, complete the sentence: The line 5 is a of the line 5 that If ou graph 5, does our transformation still work to create the line 5? maps the point (0, 0) onto the point and maps the point (1, 1) onto the point (1, ). LESSON : Up, Down, and All Around M-1
ACTIVITY.3 Using Transformations to Graph Lines TCH t You have eplored how the basic linear equation 5 is translated to create the equation 5 1 b or dilated to create the equation 5 m. In this activit, ou will combine both dilations and translations to graph equations of the form 5 m 1 b. 1. Consider the set of equations. 5 5 1 3 5 5 5 1 5 a. What do all of the equations have in common? same slope m toe is it g 157 is 5 p a OY 0 a b. Use transformations to graph each equation on the coordinate plane. c. Describe the relationship among the lines. Parallel same slope M- TOPIC 1: From Proportions to Linear Relationships
. Consider the set of equations. 5 3 5 3 Conventionall, 5 is considered a reflection of 5 across the -ais. 5 3 1 5 5 3 0 a. What do all of the equations have in common? b. Use transformations to graph each equation on the coordinate plane. c. Describe the relationship among the lines. d. Describe and use a strateg for verifing the relationship among the lines. LESSON : Up, Down, and All Around M-3
3. Consider these equations. 5 1 5 1 1 5 1 3 5 1 a. Without graphing, describe the graphical relationship among the lines. parallel lines M- TOPIC 1: From Proportions to Linear Relationships b. Eplain how ou determined the relationship.. Determine if the quadrilateral formed b joining the points A (3, 1), B (, 1), C (10, 5), and D (5, 5) in alphabetical order is a parallelogram. r 10 9 7 5 3 1 0 0 all equations have slope 1 3 5 7 9 10 of C A 93 AT 1113T same slope z ne ABT 11 DT same slope m O
Now that ou understand linear equations in terms of transformations, ou can use transformations to graph lines. WORKED EXAMPLE Graph 5 3 using transformations of the basic linear equation 5. First, graph the basic equation, 5, and consider at least sets of ordered pairs on the line, for eample (0, 0), (1, 1), and (, ). Then dilate the -values b 3. Finall, translate all -values down units. 0 = Could ou have translated the line 5 down units first, and then dilated the -values b 3? = 3 = 3 LESSON : Up, Down, and All Around M-5
Tr using even numbers for the -values. 5. Graph each equation using transformations. Specif which transformations ou use. a. 5 1 1 5 0 b. 5 3 3 0 M- TOPIC 1: From Proportions to Linear Relationships
ACTIVITY. Reflecting and Rotating Parallel Lines You have learned what happens when a line or figure is reflected across the -ais. What happens if ou reflect a pair of lines across the -ais? 1. Line segment AB and line segment CD are shown on the coordinate plane. A C B 0 D 10 Patt paper might be helpful when reflecting across the -ais. 10 a. What is the relationship between segments AB and CD? Justif our reasoning. b. Trace line segments AB and CD onto a sheet of patt paper. Reflect the line segments across the -ais to create segments A9B9 and C9D9. c. What are the coordinates of points A9, B9, C9, and D9? d. What is the relationship between segments A9B9 and C9D9? Justif our reasoning. LESSON : Up, Down, and All Around M-7
e. Etend segments AB, CD, A9B9, and C9D9 to create lines AB, CD, A9B9, and C9D9. Draw the lines on our graph. What do ou notice about the relationship between the lines? f. Reflecting parallel lines across the same line of reflection results in lines that are. Let s eplore what happens when the segments and lines created from the points A (3, ), B (, 1), C (3, 0), and D (, 1) are rotated.. Consider the line segments AB and CD as shown on the coordinate plane. You learned that when ou rotate a point (, ) 90 degrees 10 counterclockwise about the origin, the image of the point is (, ). However, what happens when ou rotate two parallel lines? 10 0 A C B D 10 10 a. Rotate each point 90º counterclockwise to create segments A9B9 and C9D9. What are the coordinates of points A9, B9, C9, and D9? M- TOPIC 1: From Proportions to Linear Relationships
b. What is the relationship between line segments A9B9 and C9D9? Justif our reasoning. c. Rotate each original point 10º to create a new set of segments. What are the coordinates of the new points? d. What is the relationship between the segments created b rotating the original points 10º? Justif our reasoning. e. Etend line segments AB, CD, A9B9, and C9D9 to create lines AB, CD, A9B9, and C9D9. Draw the lines on graph. What do ou notice about the relationship between the lines? f. Rotating parallel lines results in lines that are. LESSON : Up, Down, and All Around M-9
NOTES TALK the TALK Are Th e Parallel? 1. Which transformations of linear graphs result in parallel lines? Eplain each response. a. dilation b a non-zero factor other than 1 b. translation up or down c. reflection across an ais d. rotation 90º counterclockwise. Create and graph four linear equations that represent lines with the same slope. Label each line with its corresponding equation. 0 M-70 TOPIC 1: From Proportions to Linear Relationships
Assignment Write Eplain how to use transformations of the basic equation 5 to graph the equation 5 m 1 b. Remember Translations, reflections, and rotations map parallel lines and line segments to corresponding parallel lines and line segments. Practice 1. Write an equation for each linear relationship after transforming 5. a. dilation b a factor of 5 b. dilation b a factor of c. reflection across the -ais d. translation down units e. dilation b a factor of, then a translation up 3 units f. reflection across the -ais, dilation b a factor of 3, and then a translation down 9 units. Use the graph of the linear relationship shown to complete each task. a. Write the equation of the line. 1 b. Write the equation of the line after a translation down units. Graph the line. c. Write the equation of the line after a translation up units. Graph the line. 1 1 1 0 1 1 1 1 Stretch Graph each given sequence of transformations. Are the equations the same? Eplain wh the equations must be the same or wh the are not the same. Use transformations to support our answer. 1. Translate 5 up units, and then dilate b a factor of.. Dilate 5 b a factor of, and then translate up units. LESSON : Up, Down, and All Around M-71
Review Draw similar triangles on the graph to determine each slope. 1.. 1 1 1 1 10 1 1 0 1 1 1 3 5 7 9 Identif the similar triangles and eplain how the triangles are similar b the Angle-Angle Similarit Theorem. 3. X. Y Q R V X Z Y Z W Solve each proportion for the unknown. 5. 3 5 3.5. 0. m = 30 M-7 TOPIC 1: From Proportions to Linear Relationships