Point-Slope Form and Writing Linear Equations

Transcription

1 Point-Slope Form and Writing Linear Equations California Content Standards 6.0 Graph a linear equation. Develop 7.0 Derive linear equations by using the point-slope formula. Develop What You'll Learn To graph and write linear equations using point-slope form To write a linear equation using data... And Why To write an equation relating altitude and the boiling point of water, as in Example Check Skills You'll Need (ru) for Help Lessons s-1 and 1-7 Find the rate of change of the data in each table. 1. I v I II I 2. I v I II y y I 3. X "' y "' Simplify each expression (x - 5) 5. 5(x + 2) 6. -~(x - 6) II>~ New Vocabulary point-slope form., ,._,_ ~ , ~ ~ Using Po1nt-Siope Form You can use the definition of slope to find another form of a linear equation called the point-slope form. y2- yl x2- xl = m Use the definition of slope. Suppose you know that a line passes through the point (3, 4) and has slope 2. y-4 x-3= 2 Substitute (3, 4} for (x 1, y 1 } and substitute (x, y} for (x 2, y 2 }. Substitute 2 for m. ~ = j (x - 3) = 2(x - 3) Multiply each side by x- 3. y - 4 = 2(x - 3) Simplify the left side of the equation. The equation y - 4 = 2(x - 3) is in point-slope form. y - 4 = 2(x - 3) /' t ' y-coordinate slope x-coordinate Definition Point-Slope Form of a Linear Equation The point-slope form of the equation of a nonverticalline that passes through the point (x 1, y 1 ) and has slope m is y - Y1 = m(x - x 1 ) 252 Chapter 5 Linear Equations and Their Graphs

2 Graphing Using Point-Slope Form Graph the equation y - 5 =! (x - 2). The equation shows that the line passes through (2, 5) and has a slope i. Start at (2, 5). Using the slope, go up 1 unit and right 2 units to (4, 6). Draw a line through the two castandardscheck J) Graph the equationy- 5 = -j(x + 2). Writing an Equation in Point-Slope Form Write the equation of the line that has slope -3 that passes through the point ( -1, 7). y - Yl = m(x - x 1 ) y - 7 = -3[x- (-1)] y - 7 = -3(x + 1) Use the point-slope form. Substitute (-1, 7) for (x 1, y 1 ) and -3 form. Simplify inside the grouping CA Standards Check ~ Write the equation of the line that has slope~ that passes through the point (10, -8). If you know two points on a line, first use them to find the slope. Then you can write an equation using either point. Using Two Points to Write an Equation nline active~ I I ~ L---.,... :: : -- o=-...,..,... u -.:.--.j For: Point-Slope Activity Use: Interactive Textbook, 5-4 Write equations for the line in point-slope form and in slope-intercept form. Step 1 Find the slope. Y2- Y1 = m X2 - Xl The slope IS 3. Step 2 Use either point to write the equation in point-slope form. Use (2, 3). y - Yl = m(x - x 1 ) y - 3 = ~ (x - 2) ~ ol / 2 Step 3 Rewrite the equation from Step 2 in slopeintercept form. y - 3 = ~(x - 2) y- 3 = ~x- 5! y =~X- 2! CA Sta~dards Check J) a. Write an equation for the line in Example 3 in point-slope form using the point ( -1, -5). b. Write the equation you found in part (a) in slope-intercept form. Lesson 5-4 Point-Slope Form and Writing Linear Equations 253

3 r-writing Ltnear Equations Using Data You can write a linear equation to model data in tables. Two sets of data have a linear relationship if the rate of change between consecutive pairs of data is the same. For data that have a linear relationship, the rate of change is the slope. Writing an Equation Using a Table Is the relationship shown by the data linear? If so, model the data with an equation. Step 1 Find the rate of change for consecutive ordered pairs. 4( X y Step 2 Use the slope and a point to write an equation. y - Yl = m (x - x 1 ) Substitute (5, 7) for 2 (x 1 4 =2, Y1) 3 6 and i form ( 1 2 =2 5 7 y - 7 =!ex - 5) 6( =2 (i/ CA Standards Check Is the relationship shown by the data at the right linear? If so, model the data with an equation. Application X -11 y Is the relationship shown by the data linear? If so, model the data with an equation Boiling Point of Water Altitude Temperature (1000 ft) (Of) At 5280 feet above sea level, it takes 17 minutes to hardboil an egg. To cook the same egg at sea level takes more than 40% longer. Step 1 Find the rates of change for consecutive ordered pairs. _il l12_ The relationship is linear. The rate of change is -1.8 degrees Fahrenheit per 1000 ft of altitude. Step 2 Use the slope and a point to write an equation. y - Yl = m (x - x 1 ) Use the point-slope form. y = -1.8(x - 3) Substitute (3, 206.6) for (x 1, y 1 ) and -1.8 for m. The equation y = -1.8(x - 3) relates altitude in thousands of feet x to the boiling point temperature in degrees Fahrenheit. 254 Chapter 5 Linear Equations and Their Graphs

4 (iff CA Standards Check <a) Is the relationship shown by the data in the table linear? If it is, model the data with an equation. Working Outdoors Temperature Calories Burned per Day. 68 F F F F 3330 In Example 5 you could rewrite y = -1.8(x - 3) as y = -1.8x They-intercept of 212 F is the boiling point of water at sea level. For more help with the three forms of a linear equation, see page 258. Summary Slope-Intercept Form y=mx+b m is the slope and b is they-intercept. Examples Y - _2x Linear Equations Standard Form Ax+ By= C A andb are not both 0. 2x + 3y = 5 Point-Slope Form (y - Yt) = m(x - x 1 ) (xv y 1 ) lies on the graph of the equation, and m is the slope. y - 1 = -~(x - 1) For more exercises, see Extra Skills and Word Problem Practice. L -u:a_ac::a_i Practice by Example for Help Example 1 (page 253) Example 2 (page 253) Graph each equation. 1. y - 2 = (x - 3) 4. y + 5 = - (X - 2) 7. y + 3 = -2(x - 1) 2.y- 2 = 2(x- 3) 5.y + 1 = ~(x + 4) 8.y- 4 = (x- 5) 3. y - 2 = - ~ (X - 3) 6. y - 1 = - 3(x + 2) 9.y- 2 = 3(x + 2) Write an equation in point-slope form for the line through the given point that has the given slope. 10. (3, -4); m = (4, 2); m = -~ 12. ( 0, 2); m = ~ 13. ( -2, -7); m = -~ 16. ( -5, 2); m = ( 4, 0); m = (1, -8); m = -! 15. (5, -8);m = ( -6, 1); m = ~ Example 3 (page 253) A line passes through the given points. Write an equation for the line in pointslope form. Then rewrite the equation in slope-intercept form. 19. ( -1, 0), (1, 2) 20. (3, 5), (0, 0) 21. (4, -2), (9, -8) 22. (6, -4), ( -3, 5) 23. ( -1, -5), ( -7, -6) 24. ( -3, -4), (3, -2) 25. (2, 7), (1, -4) 26. ( -2, 6), (5, 1) 27. (3, -8), ( -2, 5) 28. ( 1, 1), (3, 2) 29. (1, 2 ), ( -~, 4) 30. (0.2, 1.1), (7, 3) Lesson 5-4 Point-Slope Form and Writing Linear Equations 255

5 Example 4 (page 254) Is the relationship shown by the data linear? If so, model the data with an equation. 31. X y 32. X y X y I I I I 49 Example 5 (page 254) 34. Speed Over Posted Fine 35. Volume Weight Speed Limit (mi/h) ($) (gal) (I b) Apply Your Skills Write an equation of each line in point-slope form X -2 Ot 2 Write one equation of the line through the given points in point-slope form and one in standard form using integers. 39. (1, 4), ( -1, 1) 48. ( -8, 4), ( -4, -2) 51. (0, 1), (-3, 0) 40. (6, -3), (-2, -3) 43. (-6, 6), (3, 3) 46. (2, 2), (-1, 7) 49. (2, 4), (-3, -6) 52. (-2, 4), (0, -5) 41. (0, 0), (-1, -2) 44. (2, 3), (-1, 5) 47. (-7, 1), (5, -1) 50. (5, 3), ( 4, 5) 53. (6, 2), (1, -1) 54. At the surface of the ocean, pressure is 1 atmosphere. At 66 ft below sea level, the pressure is 3 atmospheres. The relationship of pressure and depth is linear. a. Write an equation for the data. b. Predict the pressure at 100ft below sea level. 55. Worldwide carbon monoxide emissions are decreasing about 2.6 million metric tons each year. In 1991, carbon monoxide emissions were 79 million metric tons. Use a linear equation to model the relationship between carbon monoxide emissions and time. Let x = 91 correspond to a. Write an equation in point-slope form that contains the point ( -4, -6). Explain your steps. b. How many equations could you write in part (a)? Explain. 57. Critical Thinking How would the graph of y - 12 = 8(x - 2) change if all of the subtraction signs were changed to addition signs? 58. Reasoning Is y - 5 = 2(x - 1) an equation of a line through (4, 11)? Explain. 256 Chapter 5 Linear Equations and Their Graphs

6 a Challenge 59. Use the graph at the right. a. Write an equation to model the data. b. What is the speed of sound at 15 C? c. Predict the speed of sound at 60 C. Write an equation in slope-intercept form of each line described below. 60. The line contains the point ( - 3, - 5) and has the same slope as y + 2 = 7(x + 3). ~ 350 s :;; 340 v v 0.. V) The line contains the point (1, 3) and has the same y-intercept as y- 5 = 2(x - 1). Effect of Air Temperature on Speed of Sound o 40 Temperature ( C) Visit: PHSchool.com Web Code: bae The line contains the point (2, -2) and has the same x-intercept as y + 9 = 3(x - 4). 63. The table shows data that you can model using a linear function. a. Find the value of y when x = 6. b. Find the value of y when x = 120. c. Find the value of x when y = 11. d. Find the value of x when y = 50. X y ,~\-MiilfipleCiioicePracfice Alg1 7.0 For California Standards Tutorials, visit PHSchool.com. Web Code: baq A line has a slope of - 3 and passes through the point (1, - 5). What is the equation for the line? ClD Y = -3x- 2 y = - 3x + 16 CD 2y = - 3x - 14 y = -3x + 8 Alg When y - 1 = - ~(x - 3) is written in standard form using positive integers, what is the smallest possible coefficient of x? C!D - 4 -~ CD 4 5 Alg Which pair of numbers are reciprocals? , 8 5,7 CD2> -2 3 _4 4' 3 Mixeil Review for Help Lesson 5-3 Graph each line x + 7 y = x + 9y = Sx - 4y = x - 8y = Sx + 18y = x + 4y = - 21 Lesson 4-7 Use inductive reasoning to find the next two numbers in each pattern ,-7, - 2, ' 6' 6' , 2.52, 2.59, , , -3.3, , 36, 72, , - 3, 1,... nline Lesson Quiz Visit: PHSchool.com, Web Code: baa

7 There are three forms of a linear equation that you have studied in this chapter: slope-intercept form standard form point-slope form To understand and remember these forms, it may help you to connect the common meaning of the words with their specialized meanings in mathematics. Word Common Meaning Mathematical Meaning Slope An inclined surface The rate of change that gives the steepness of a line: (for example, the slope of a hill) vertical change rise slope = horizontal change = run Intercept To cut off from a path The values of the points at which a line intersects (for example, to intercept a (or cuts) the x-axis or y-axis football) Standard Generally accepted A general form of an equation Point A dot or speck (noun) A fixed location on a coordinate plane; every point has a unique x-value andy-value Slope-intercept form, standard form, and point-slope form all give clues about the lines they will graph. Slope-intercept form tells the slope of the line and its y-intercept. Example y = 3x - 1 The graph of this equation has a slope of 3 and a y-intercept of -1. Standard form is the general form for linear equations. You can model many real-world situations using the standard form. Also, it is easy to find the x- and y-intercepts of the graph of an equation in this form. Example 5x - 2y = 100 The graph of this equation intersects the y-axis at ~f = - 50 and intersects the x-axis at 1 ~ 0 = 20. Point-slope form tells a point on the line and the slope of the line. Example y - 2 = 3(x - 5) The graph of this equation passes through the point EXERCISES {5, 2), and its slope is 3. For each situation, which form of a linear equation is easiest to write? 1. A submarine started at sea level and submerged at a steady rate of 8 feet per minute. 2. A plant growing at a steady rate measured 2 in. on day 5 and 13 in. on day 21. Find the common meaning and the mathematical meaning of each word. 3. function 4. relation 5. range 6. domain 7. variable 8. base 9. element 10. term 258 Vocabulary Builder Understanding Vocabulary

8 Parallel and Perpendicular Lines ~ California Content Standards 7.0 Derive linear equations by using the point-slope formula. Master 8.0 Understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Find the equation of a line perpendicular to a given line that passes through a given point. Develop, Master What You'll Learn To determine whether lines are parallel To determine whether lines are perpendicular... And Why To use parallel and perpendicular lines to plan a bike path, as in Example Check Skills You'll Need for Help Lessons 1-6 and 5-2 What is the reciprocal of each fraction? _2 5 What are the slope and y-intercept of each equation? 5. y = ~X y = ~X y = 6x.. >):» New Vocabulary parallel lines perpendicular lines negative reciprocal 4. _1 5 S. y = 6x + 2, Parallel Lines In the graph at the right, the red and blue lines are parallel. Parallel lines are lines in the same plane that never intersect. The equation of the red line is y = 1 x + ~. The equation of the blue line is y = 1 x - 1. You can use slope-intercept form to determine whether the lines are parallel. Property Slopes of Parallel Lines Nonverticallines are parallel if they have the same slope and different y-intercepts. Any two vertical lines are parallel. Example The equations y = ~ x + 1 andy = ~ x - 3 have the same slope,~, and different y-intercepts. The graphs of the two equations are parallel. Determining Whether Lines Are Parallel Are the graphs of y = - ~ x + 5 and 2x + 6y = 12 parallel? Explain. Write 2x + 6y = 12 in slope-intercept form. Then compare withy = -~x y = - 2x + 12 Subtract 2x from each side. 6 f = -Zx Divide each side by 6. The lanes for competitive swimming are CA Standards Check y = -~x + 2 Simplify. The lines are parallel. They have the same slope, -~, and different y-intercepts. 1) Are the graphs of -6x + 8y = -24 andy = ~x - 7 parallel? Explain. Lesson 5-5 Parallel and Perpendicular Lines 259

9 To write the equation of a line parallel to a given line, you use the slope of the given line and the point-slope form of a linear equation. Writing Equations of Parallel Lines Video Tutor Help Visit: PHSchool.com Web Code: bae-0775 Write an equation for the line that contains (5, 1) and is parallel toy = ~x - 4. Step 1 Identify the slope of the given line. Y - ~ X- 4-5 t slope Step 2 Write the equation of the line through (5, 1) using slope-intercept form. y - Y1 = m(x - x 1 ) point-slope form y - 1 = ~ (x - 5) Substitute (5, 1} for (x 1,y 1 } and i form. y - 1 = ~ x - ~ ( 5) Use the Distributive Property. y - 1 = ~x - 3 Simplify. y = ~x - 2 Add 1 to each side. (ij CA Standards Check Write an equation for the line that contains (2, -6) and is parallel toy = 3x + 9. r-per=p-endiculariines " The lines at the right are perpendicular. Perpendicular lines are lines that intersect to form right angles. The equation of the red line is y = -lx - 1. The equation of the blue line is y = 4x + 2. Property Slopes of Perpendicular Lines Two lines are perpendicular if the product of their slopes is -1. A vertical and a horizontal line are also perpendicular. Example The slope of y = -lx - 1 is -l. The slope of y = 4x + 2 is 4. Since -l 4 = -1, the graphs of the two equations are perpendicular. The product of two numbers is -1 if one number is the negative reciprocal of the other. Here is how to find the negative reciprocal of a number. Start with Find its Write the a fraction: -l ---+ reciprocal: -~ negative reciprocal: ~ Since -~ ~ = - 1, ~ is the negative reciprocal of-~- Start with Find its Write its an integer: reciprocal: l negative reciprocal: -l- Since 4( -l) = -1, -l is the negative reciprocal of Chapter 5 Linear Equations and Their Graphs

10 You can use the negative reciprocal of the slope of a given line to write an equation of a' line perpendicular to that line. Writing Equations for Perpendicular Lines Find the equation of the line that contains (0, - 2) and is perpendicular toy = 5x + 3. After finding the slopes of the perpendicular lines, multiply the two slopes as a check. 5( -i) = - 1 Step 1 Identify the slope of the given line. y = 5x + 3 t slope Step 2 Find the negative reciprocal of the slope. Step 3 Use the slope-intercept form to write an equation. y= mx+ b y = - ~ x + (-2) y = -~x- 2 The equation is y = -~ x - 2. The negative reciprocal of 5 is - ~. Substitute -! for m, and - 2 for b. Simplify. {i! CA Standards Check ~ Write an equation of the line that contains (1, 8) and is perpendicular to y = i x + 1. You can use equations of parallel and perpendicular lines to solve some real-world problems. Application A jogging path for a new city park will connect the park entrance to Park Road. The path will be perpendicular to Park Road. Write an equation for the line representing the bike path. y-intercept (4, 5) new path r I : IJ I... 1"1 I I perpendicular to Park Road (2, 1) Beverly Gardens Park in Beverly Hills is a 1.9-mile-long linear park with a gravel jogging path. Step 1 Find the slope m of Park Road. m = ;~ =: ~~ = ~ = ~ = ~ = 2 Points (2, 1} and (4, 5) are on Park Road. Step 2 Find the negative reciprocal of the slope. The negative reciprocal of 2 is -~.So the slope of the bike path is -~. They-intercept is 4. The equation for the bike path is y = - ~ x + 4. {i! CA Standards A second jogging path is planned. It will be parallel to Park Road and will also contain the park entrance. Write an equation for the line representing this jogging path. Lesson 5-5 Parallel and Perpendicular Lines 261

11 For more exercises, see Extra Skills and Word Problem Practice. Practice by Example Example 1 for (page 259) Help Find the slope of a line parallel to the graph of each equation. 1. y =! X y = -~X y = X 4. y = x + 4y = x - y = 5 Are the graphs of the lines in each pair parallel? Explain. 7. y = 4x y = -~x y = ix + 3-4x + 3y = 21 3x + 2y = 8 X- 3y = y = -!x + ~ 5x- l Oy= y = - 3x 12. y = ~ X- 2 2lx + 7y = 14-3x + 4y = 8 Example 2 (page 260) Write an equation for the line that is parallel to the given line and that passes through the given point. 13. y = 6x - 2; (0, 0) 15. y = -2x + 3; ( -3, 5) 17. y = 0.5x - 8; (8, - 5) 14. y = - 3x; (3, 0) 16. y = - ~ x + 6; (- 4, - 6) 18. y = -~ x + 12; (5, - 3) Example 3 (page 261) Find the slope of a line perpendicular to the graph of each equation. 19. y = 2x 20. y = - 3x 21. y = ~ x y = - ~ x + 3y = y = - 8 Write an equation for the line that is perpendicular to the given line and that passes through the given point. 25. y = 2x + 7; (0, 0) 27. y = -ix + 2;(4,2) lox + 8y = 3; (15, 12) 26. y = X - 3; ( 4, 6) 28. 3x + 5y = 7; ( - 1, 2) 30. 4x - 2y = 9; (8, -2) Example 4 (page 261) Apply Your Skills 31. A city's civil engineer is planning a new parking garage and a new street. The new street will go from the entrance of the parking garage to Handel St. It will be perpendicular to H andel St. What is the equation of the line representing ~he new street? Tell whether the lines for each pair of equations are parallel, perpendicular, or neither. 32. y = 4x + ~, y = -l x y = -X + 5, y = X y = ~ - 4, y = i X x + y = 2,2x + y = y = ~X - 6, y = ~ X y = 5x, y = - 5x x = 2,y = x- 5y = 3, -5x + 3y = Write an equation for a line parallel to the graph of 4x - y = Chapter 5 Linear Equations and Their Graphs

12 Visit: PHSchool.com Web Code: bae-0505 Find the equation for each line r t-: I "' ~ y ' X X _L.., t Use the map below for Exercises What is the slope of New Hampshire Avenue? 48. Show that the parts of Pennsylvania Avenue and Massachusetts Avenue near New Hampshire Avenue are parallel. 49. Show that New Hampshire Avenue is not perpendicular to Pennsylvania Avenue. 50. Which pair of equations graph as perpendicular lines?.y=4x+! C[) y= ~x+l (O y~-~x-9 y = -4x + 3 y = 3x - 1 y = ~x + 8 y = ~x Are the graphs of 2x + 7y = 6 and 7y = 2x + 6 parallel? Explain. 52. Are the graphs of 8x + 3y = 6 and 8x - 3y = 6 perpendicular? Explain. 53. Writing Are all horizontal lines parallel? Explain. lesson 5-5 Parallel and Perpendicular Lines 263

13 54. Critical Thinking Explain how you can tell that the graphs of 7x- 3y = 5 and 7x - 3y = 8 are parallel without finding their slopes. Math Reasoning Tell whether each statement is true or false. Explain your choice. 55. Two lines with positive slopes can be perpendicular. For Exercises 55-57, sketch a graph to help you understand the statement in each exercise. 56. Two lines with positive slopes can be parallel. 57. The graphs of two different direct variations can be parallel. A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Use slopes to determine whether each figure is a parallelogram. 58. B 59. J X 0 D M R A quadrilateral with four right angles is a rectangle. Use slopes to determine whether each figure is a re~tangle 'y R X Challenge Tell whether the lines in each pair are parallel, perpendicular, or neither. 64. ax - by = 5; -ax + by = ax+ by= 8;bx- ay = 1 Assume the two lines are perpendicular. Find an equation for each line For what value of k are the graphs of 3x + 12y = 8 and 6y = kx - 5 parallel? Perpendicular? 69. A quadrilateral with two pairs of parallel sides and with diagonals that are perpendi~ular is a rhombus. Quadrilateral ABCD has vertices A( -2, 2), B(1, 6), C(6, 6), and D(3, 2). Show that ABCD is a rhombus. 70. A triangle with two sides that are perpendicular to each other is a right triangle. Triangle PQR has vertices P(3, 3), Q(2, -2), and R(O, 1). Determine whether PQR is a right triangle. Explain. 264 Chapter 5 Linear Equations and Their Graphs

14 ~-MiiltipleCiioiCi!Pracfice For California Standards Tutorials, visit PHSchool.com. Web Code: baq-9045 Alg Which equation represents a line that is parallel toy = -2x + 1? y = -2x - 1 CD y = -!x + 1 CD y = 2x - 1 y=!x+1 Alg Which line has a slope of 2 and a y-intercept of -1? CD CD 2 'y X Alg Which equation represents a line that contains the points (-3, 4) and (1, -4)? y = -!x + 4 CD y = 2x - 4 CD y = -2x - 2 y =!x + 3 Mixeil Review for Help Lesson S r- T r r r ~y 76. T. f ' 1 iy I ~ ~ ~! I X!d:_ t 01 2 X Lesson 4-2 Determine whether each relation is a function. 77. {(1, 1), (2, 2), (3, 3)} 78. {(1, 3), (2, 5), (3, 5)} 79. {( 5' 1)' ( 5' 2)' ( 4' 3)} 80. {(1, 3), (2, 2), (3, 1)} r o -che'ckpoint Quiz2 [ essons s:3-throl.lglfs:s 1. Write an equation for the line through (3, 4) that has a slope of -l Find the x- and y-intercepts of each equation. Then write an equation of the line that is parallel to the given line and that passes through the given point. 2. X + y = 3; ( 5, 4) 3. 3x + 2y = 1; ( -2, 6) Write an equation of the line that is perpendicular to the given line and that passes through the given point. 4. y = - 4x + 2; (0, 2) 5. y =~X+ 6;(-6,2) nline Lesson Quiz Visit: PHSchool.com, Web Code: baa

15 For some problems, it may help to draw a diagram of the given information if one is not provided. The points R, S, and T lie on a line in order such that the length of ST is twice the length of RS. The length of RT is 5 em more than the length of ST. Find the length of RS and ST. Draw RT. Since ST is twice as long as RS, let RS = x and ST = 2x. Since the length of RT is 5 em more than the length of ST, let RT = 5 + 2x. 2x X R s T You can see from your diagram that 2x + x = 5 + 2x. Solve this equation, and you find that x = 5. The length of RS is 5 em and the length of ST is 10 em. The points A( -8, 1), B( -2, 7) and C(4, -11) are the vertices of a triangle. Is Triangle ABC a right triangle? Two lines form a right angle if the product of their slopes is -1. When you draw a diagram, you can see that the right angle cannot be at point C. You need to see if the product of the slopes of AC and AB or of BC and AB is -1. The slope of AB is 1, and the slope of AC is -1. The product of 1 and -1 is -1. So Triangle ABC is a right triangle. y r ~~ B( -2, 7) I -~.:.. : \ X C(4, -11) 1. The points L( 4, 0), M(10, 0), and N(7, 5) form D.LMN. What is the sum of the slopes of the three sides of the triangle? -~ O CD 6 2. Three towns A, B, and C lie on a straight road in that order. The distance from B to C is 6 miles more than twice the distance from A to B. The distance from A to C is 2 miles more than four times the distance from A to B. What is the distance from A to B? 2 4 CD Test-Taking Strategies Drawing a Diagram

16 r vocatiulary Review linear equation (p. 239} linear function (p. 239} linear parent function (p. 239} negative reciprocal (p. 260} parallel lines (p. 259} parent function (p. 239} perpendicular lines (p. 260} point-slope form (p. 252} rate of change (p. 230} slope (p. 232}.. >):» English and Spanish Audio Online slope-intercept form (p. 240} standard form of a linear equation (p. 246} x-intercept (p. 246} y-intercept (p. 239} Choose the vocabulary term that correctly completes the sentence. 1. Two lines are _]_ if the product of their slopes is Two lines in the same plane that never intersect are _1_. ego nline PHSchool.com For: Vocabulary quiz Web Code: baj The equation y - 2 = 3(x - 5) is written in _]_form. 4. The ratio of the vertical change to the horizontal change is called the _1_. 5. They-coordinate of the point at which the graph of a line crosses the vertical axis is called the _1_. ~-SICills - a ne~-concepl:s Lesson 5-1 To find rates of change from tables and graphs (p. 230} To f ind slope (p. 232} Alg1 6.0, 7.0, 8.0 Rate of change is the relationship between two quantities that are changing. change in the dependent variable rate of change = change in the independent variable Slope is the ratio of the vertical change to the horizontal change. vertical change rise slope = horizontal change = run Find the rate of change for each situation. 6. A kitten grows from 5 oz at birth to 3 lb 5 oz at 6 months. (Hint: 1lb = 16 oz) 7. A plant measures 0.5 in. at the end of Week 1 and 14 in. at the end of Week 5. Find each rate of change. Explain what rate of change means in each situation TI!TJ 10., ;±j "!:, ~ 20 -~ --; 5 u c-::9 ':::' 200 c...c:: ~ 0.0 ~~ v -~ 10 l ~ ~ 2.5 :r:: 100 Q - (.j!.2) I l U~ Time (h) Hours of Driving Time (min) Find the slope of the line that passes through each pair of points. 11. (3, -2) and ( - 5, - 4) 12. (4.5, - 1) and (4.5, 2.6) 13. (2, 5) and (-5, -2) Chapter 5 Chapter Review 267

17 Chapter Review (continued) Lesson 5-2 To write linear equations in slope-intercept form (p. 239) To graph linear equations (p. 241) Alg1 6.0, 7.0 The graph of a linear equation is a line. The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis, and they-intercept is they-coordinate of the point where the line crosses they-axis. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is they-intercept. Write an equation of a line with the given slope andy-intercept. Then graph the equation. 14. m = 0, b = m = -7, b = ~ 16. m = ~' b = 0 Write the slope-intercept form of the equation for each line = 18. y -2 0 X 2 '1 Determine whether the ordered pair lies on the graph of the given equation. 19. (3, -10);y = -5x (8, 13);y = ~x (9,0.5);x- 2y = ( -2, 0); 4y = -8x ( -1, -3); - 6x + 3y = (0, -4); -~y =X + 2 Lesson 5-3 To graph equations using intercepts (p. 246) To write equations in standard form (p. 248} Alg1 6.0 The standard form of a linear equation is Ax + By = C, where A, B, and Care real numbers, and A and B are not both zero. Find the x- and y-intercepts. Then graph each equation x + 2y = x - 4y = 52 Write each equation in standard form. 2s.y- _3 5x y = -!x X+ 3y = y =~X- 5 Match each equation with its graph x- 3y = x + 3y = 6 A. B. 2'y 33. 2x + 3y = -6 c. For each equation, tell whether its graph is a horizontal or a vertical line. 34. y = X= y = l Chapter 5 Chapter Review

18 Lesson 5-4 To graph and write linear equations using point-slope form (p. 252) To write a linear equation using data (p. 254) Alg1 6.0, 7.0 The point-slope form of a linear equation is y - through the point (x 1, y 1 ) and has slope m. y 1 = m(x - x 1 ), which passes Use point-slope form to write an equation of a line that passes through the point (1, - 2) with slope m. 37.m = 2 38.m = i 39.m = m = 0 Use the point-slope form to write an equation of a line through the given points. 41. ( 4, 3), ( -2, 1) 42. (5, -4), (0, 2) 43. ( -1, 0), (-3, -1) Are the data linear? If so, model the data with an equation. 44. X y 45. X y 46. X y Lesson 5 5 To determine whether lines are parallel (p. 259) To determine whether lines are perpendicular (p. 260) Alg1 7.0, 8.0 Parallel lines are lines in the same plane that never intersect. Nonverticallines are parallel if they have the same slope. Two lines are perpendicular lines if they intersect to form right angles or, equivalently the product of their slopes is -1. Are the graphs of the lines in each pair parallel? Explain. 47.y =-3x-8 48.y=-~x-1 49.y=ix-2 6x + 2y = -10 2x + 5y = 20-4x + 3y = -12 Are the graphs of the lines in each pair perpendicular? 50. y = -!x y = -ix y = ~x - 4 y = 3x + 6 4x + 5y = -15 3x + 2y = 4 Write an equation for each of the following conditions. 53. parallel to y = 5x - 2, through (2, -1) 54. perpendicular toy = - 3x + 7, through (3, 5) 55. parallel toy = 9x, through (0, - 5) 56. perpendicular toy = 8x - 1, through ( 4, 10) Find the equation for each line Chapter 5 Chapter Review 269

19 ego n}ine For: Chapter Test PHSchool.com Web Code: baa-0552 Tell whether each statement is true or false. Explain. 1. A rate of change must be positive. 2. The rate of change for a vertical line is 0. Find the slope of the line that passes through each pair of points. 3. ( 4, 3), (3, 8) Graph each equation. 5. X- 4y = 8 7. y =!x ( -2, 1), (6, -1) 6. 2x + 4y = y - 1 = - 3(x - 3) Write each equation in slope-intercept form y = 8x x + 4y = 100 Find the x- andy-intercepts of each line x + 4 y = x + loy = X - 3y = x = 2y x + 2 y = X+ y = 1 Write an equation in slope-intercept form for a line that is parallel to the given line and that passes through the given point. 32. y = 5x; (2, -1) 34. y = 5x + 1; (2,-3) 36. 2x + 3 y = 9; ( -1, 4) 38. y = - 2x + 3; (-2, -1) 33. y = 5; (-3, 6) 35. y = -x - 9; (0, 5) 37. y = -!x; (3, -4) 39. y = ~X + 7; ( -1, 2) Write an equation in slope-intercept form for a line that is perpendicular to the given line and that passes through the given point. 40. (4,0);y = -2x 41. (0, 2); X = (-10,3);y = -2x (12,8);y = -3x (-1, 7);y = -ix (4, -9); -x + 2y = (2,1); -2x + 3y = (-2,3);y = -5x Write the slope-intercept form of the equation for each line below. Write an equation in point-slope form for the line with the given slope and through the given point. 17. (-2, -7);m = ~ 19. (0, 3); m = -! 21. (0,4);m = (5, -3); m = ~ 25. (-6,4);m = -~ 18. (4, -8); m = (9, 0); m = (-2,0);m = (-1, - 9); m = - ~ 26. (7, 11); m =! Determine whether the ordered pair lies on the graph of the given equation. 49. ( -2, l);y = -3x ( -1, -4); y = 8x + 2 Write an equation in point-slope form for the line through the given points. 27. (4, 9), ( -2, -6) 29. (5, -8), ( -9, -8) 28. ( -1, 0), (3, 10) 30. (0,7),(1,5) 31. Which of the following lines is not perpendicular to y = -2.5x + 13? y = 0.4x - 7 y = ~ x + 4-2x + 5y = 8 2y = 5x (6,3);y = -~x Write the equation of a line parallel to y = 0.5x (4, -7); -4x- y = You start a pet-washing service. You spend $30 on supplies. You plan to charge $5 to wash each pet. a. Write an equation to relate your profit y to the number of pets x you wash. b. Graph the equation. What are the x- and y-intercepts? 270 Chapter 5 Chapter Test

20 Cumulative Practice Some questions ask you to interpret a graph. Read the question at the right. Then follow the tips to answer the sample question. Tip 2 Look at the answer choices. The equations are written in slope-intercept form. The graph of a line is shown below. H rlj ~ X - 2 Which equation best represents the graph of the line? 1 y = 3 x y = -x ~ C0y= 3x- 1 1 y- 3x Tip 1 Think about what information you need to write the equation of a line. You can use slope, intercepts, or points on the line. Think It Through Since the answer choices are written in slope-intercept form, find the slope and y-intercept. The y-intercept is - 1. To get from the point (0, -1) to (1, 2), you move 3 units up and 1 to the right. The slope is f = 3. The equation is y = 3x - 1. The correct answer is C. As you solve problems, you must understand the meanings of mathematical terms. Choose the correct term to complete each sentence. A. The (slope, y-intercept) of a line is determined by rise run B. The (x-intercept, y-intercept) is where a line crosses the x-axis. C. The (slop e, y-intercept) of a line defines a rate of change. D. (Parallel, Perpendicular) lines have the same slope. Read each question. Then write the letter of the correct answer on your paper. 1. The graph shows lillian's distance from her house as she walks home from school. At what rate does Jillian walk? ~ 5 (Lesson 5-1) ::0 4 ~3 1 block per minute 1 minute per block CO 2 blocks per minute! block per minute 2 :~ 1 Q Time (min) 2. Which is the equation of a line with slope 3? (Lesson 5-2) y = 3x -4 (]"_)X= - 3y What is the slope of the line shown in the graph at the right? (Lesson 5-2) - 2 co~ ~ 3 4. Trisha drew a diagram with 3 squares as shown at the right. Which expression represents the perimeter ofthelargestsquare? (Lesson 2-5) z 2 CO (x + y + z) 2 4z 4x + 4y + 4z CO y=4x -3 y = - 3x- 5 X 1,Y , 5. The length of Darren's family room is 3 times the width. If the width of Darren's family room is x, which expression best represents the area? (Lesson 2-5) 3x 2 CO x + 9 9x 2 (x + 3) 2 y z California Standards Mastery Cumulative Practice 271

21 Cumulative Practice (continued) 6. Which expression could represent the sum of 3 consecutive even integers? (Lesson 2-5) n+3 3n+6 3n+3 6n+6 7. Look at the graph at the right. If they-intercept increases by 2 and the slope stays the same, what will the x-intercept be? (Lesson 5-2) y 2 ~ 8. Which equation represents a line that is steeper than the line represented by y = ix - 1? (Lesson 5-2) y = -ix - 2 y = ~x - 1 y = 1x - 2 y = ix There is $65 in Hilo's class fund. The class is having a car wash to raise money for a trip. Hilo made the graph below to model the amount of money they will have if they charge $4 for each car washed. ~100 ::cr 80 c & 60 ~ 40 ~ 0 20 I Cars Washed How would the graph change if they charged $5 for each car washed? (Lesson 5-2) They-intercept would increase. The slope would increase. They-intercept would decrease. The slope would decrease. 10. What is the second step of the process? (Lesson 2-2) Step 1. 4(3x + 6)- 1 = -13 Step 2. Step 3. 12x + 23 = -13 Step 4. 12x = -36 Step 5. x = -3 4(3x + 5) = x = x = -13 (7x + 10)- 1 = What are the x- and y-intercepts of the equation 6x - 3y = 36? (Lesson 5-3) x = -12 andy = 6 x = 12 andy = -6 x = 6 and y = 12 x = 6 andy = What is the equation of a line through point (2, 3) with a slope of - ~? (Lesson 5-4) 5x + 7y = 11 5x + 7y = 31 5x- 7y = -11 5x -7y = What part of the statement below is the conclusion? (Activity Lab 3-3b) If 4x + 2y = 9, then 12x + 6y = 27. 8x + 4y = 18 16x + 8y = 36 4x + 2y = 9 12x + 6y = For each pair of equations, determine whether the graphs are parallel or perpendicular lines. (Lesson 5-5) (1) 2x + 3y = 9 and 4x + 6y = 27 (2) 2x + 3y = 9 and 3x - 2y = 27 (1) is parallel; (2) is perpendicular. (1) is parallel; (2) is parallel. (1) is perpendicular; (2) is perpendicular. (1) is perpendicular; (2) is parallel. 15. What is the equation of a line through point (2, 1) and perpendicular to y = ~x + 3? (Lesson 5-5) 2x- 3y = 1 3x + 2y = 8 3x + 2y = -4 3x + 2y = Suppose you earn $74.25 for working 9 hours. How much will you earn for working 15 hours? (Lesson 2-4) $120 $ $ $ Which is NOT a solution of 5x- 4 < 12? (Lesson 3-4) A line perpendicular to y = 3x - 2 passes through the point (0, 6). Which other point lies on the line? (Lesson 5-5) (9, 3) (9, -3) (-9,3) (-9, -3) 272 California Stan~ards Mastery Cumulative Practice

22 I for Help to the Lesson in green. 19. Which of the following is the solution of 6(4x - 3) = -54? {Lesson 2-2) CD Findf( -2) whenf(x) = -3x + 4. {Lesson 4-2) CD Josie found the slope of the line shown below She used the points (1, -6) and (5, 2) and wrote 2- (-6) = _ 4 = -2 to find the slope. Which of the following describes her error? {Lesson 5-1) She reversed x 1 and x 2 in the denominator. She subtracted instead of added. CD She used an x-value in the numerator. She did not simplify the negative signs correctly. 22. Which equation represents a line that passes through (2, -1) and (3, 4)? {Lesson 5-4) y = Sx -11 y=5x+7 CD y = lx y=3x Which equation represents a line that passes through (2, -1) and that is perpendicular to the line y = ~x - ~? {Lesson 5-5) y = -~x- V y = -~x + 4 CD y = 2x _ y=~x Which of the following equations has a graph that contains the ordered pairs ( - 3, 4) and (1, -4)? {Lesson 5-4) CP;) X+ 2y = 8 2x-y=4 CD 2x + y = - - 2y = Which expression is equivalent to 3(5x - 2) - 4(x + 3)? {Lesson 2-2) 19x + 6 CD 11x x x What is the range of the following relation? {(0, 3), (1, 5), (2, 3), (3, 1), (4, 0), (5, 6)} {Lesson 4-2) {0,1, 2,3,4,5} {0,1, 3,3,5,6} CD {0, 1, 3,5,6} {0, 1, 2,3,4, 5, 6} 27. What is the domain of the functionf(x) = x 2-4? {Lesson 4-2) {x:x 2: 0} CD {-2, 2} {x:x 2: -4} all real numbers 28. What part of the statement below is the hypothesis? If -2x- 5 < 7, then -2x < 12. {Activity Lab 3-3b) -2x - 5 < 7 CD X> -6 C[) -2x < X< -6 Alg1 4.0 I 10, 19, 25 - Alg1 5.0 I 4-6, 16, 17 - Alg1 6.0 I 1-3,7-9,11,21 - Alg1 7.0 I 12, 22, 24 - Alg1 8.0 I 14, 15, 18, 23 - Alg117.0 I 20,26,27 - ~ Alg I 13,28 For additional review and practice, see the California Standards Review and Practice Workbook or go online to use the For: PHSchool.com, Web Code: baq-9045 California Standards Mastery Cumulative Practice 273