1. A(-2, 2), B(4, -2) 3 EXAMPLE. Graph the line y = Move up 3 units. Quick Check See left.

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1 -. Plan Objectives To graph lines given their equations To write equations of lines Eamples Graphing Lines in Slope- Intercept Form Graphing Lines Using Intercepts Transforming to Slope- Intercept Form Using Point-Slope Form Writing an Equation of a Line Given Two Points Equations of Horizontal and Vertical Lines - What You ll Learn To graph lines given their equations To write equations of lines... And Wh To determine whether a wheelchair ramp complies with the law, as in Eercise Part Graphing Lines Lines in the Coordinate Plane Check Skills You ll Need GO for Help ALGEBRA page Find the slope of the line that contains each pair of points.. A(-, ), B(, -). P(, 0), X(0, -) undefined or. R(-, -), S(, -) 0. K(-, ), T(-, ) no slope. C(0, ), D(, ). E(-, ), F(, -) 7. G(-8, -9), H(-, -) 8. L(7, -0), M(, -) New Vocabular slope-intercept form standard form of a linear equation point-slope form Math Background René Descartes published his philosophical treatise Discours de la méthode in 7. Its appendi Géométrie contained the first published record of methods of analtic geometr, permanentl sealing the partnership between geometr and algebra. Within 0 ears, the methods of analtic geometr would lead to the invention of calculus. More Math Background: p. D Lesson Planning and Resources See p. E for a list of the resources that support this lesson. Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Slope Algebra Review, page. Vocabular Tip The -intercept is the -coordinate of the point where a line crosses the -ais. The -intercept is the -coordinate of the point where a line crosses the -ais. O Chapter Parallel and Perpendicular Lines In algebra, ou learned that the graph of a linear equation is a line. The slope-intercept form of a linear equation is = m + b, where m is the slope of the line and b is the -intercept. Each line at the right has slope, but the lines have -intercepts of, -, and -. B Postulate - (two points determine a line), ou need onl two points to graph a line. The -intercept gives ou one point. You can use the slope to plot another. Graph the line = +. Graph the line = Graphing Lines in Slope-Intercept Form 8 Move up units O Move right units Start at (0, ) -. See left. = + slope -intercept O Special Needs L For Eample, help students see that another point on the line can be found b moving three units up and four units right from (, ), or b moving three units down and four units left from (0, ). learning stle: visual Below Level L Encourage students to plot three points when graphing a linear equation. Because algebra errors produce noncollinear points, students can check their work. learning stle: visual

2 . Vocabular Tip In the standard form, A, B, and C are constants. In Eample, + = is in standard form with A =, B =, and C =. O The standard form of a linear equation is A + B = C, where A, B, and C are real numbers and A and B are not both zero. To graph an equation written in standard form, ou can readil find two points for the graph b finding the - and -intercepts. Algebra Graph + =. Graphing Lines Using Intercepts Step To find the -intercept, Step To find the -intercept, substitute 0 for ; solve for. substitute 0 for ; solve for. Step + = + = (0) + = + (0) = = = = = The -intercept is. The -intercept is. A point on the line is (0, ). A point on the line is (, 0). Plot (0, ) and (, 0). Draw the line containing the two points. Graph - + =-8. As an alternative, ou can graph an equation in standard form b transforming it into slope-intercept form. Knowing the slope and -intercept beforehand can give ou a good mental image of what the graph should look like. (0, ) (, 0) O Transforming to Slope-Intercept Form Algebra Graph - = 9. Step Transform the equation to Step Use the -intercept and the slope slope-intercept form. to plot two points and draw the line containing them. - = 9 - =- + 9 = + 9 O = - 9 The -intercept is and the slope is. Graph - + =-. See left.. O. Teach Guided Instruction Math Tip For the standard form of a linear equation, review the meaning of real numbers, and consider the case A = B = 0, where there is no line. Students are accustomed to solving for and adding to each side of the equation. Remind them that the slope-intercept form isolates. Additional Eamples Use the slope and -intercept to graph the line = Check that points (0, 9) and (, 7) are on students graphs. Use the -intercept and -intercept to graph - = 0. Check that points (, 0) and (0, ) are on students graphs. Transform the equation - + = to slope-intercept form, and then graph the resulting equation. ± ; check that points (, ) and (0, ) are on students graphs. Lesson - Lines in the Coordinate Plane 7 Advanced Learners L Have students find the slope and intercept of the line in standard form a + b = c. learning stle: verbal English Language Learners ELL Write the general forms of the point-slope, slopeintercept, and standard form of a linear equation on the board. Point out that the names relate to what information the respective form provides. learning stle: visual 7

3 Guided Instruction Part Writing Equations of Lines Visual Learners Have students use coordinate graphs to see that slopes of -, -, and - are the same. A third form for an equation of a line is point-slope form. The point-slope form for a nonvertical line through point (, ) with slope m is - = m( - ). Using Point-Slope Form Error Prevention Students ma think the equation of a horizontal line begins with = because the -ais is horizontal. Point out that when the value of is constant, the value of changes to form a vertical line like the -ais, and when the value of is constant, the value of changes to form a horizontal line like the -ais. Algebra Write an equation of the line through point P(-, ) with slope. - = m( - ) Use point-slope form. - = [ - (-)] Substitute for m and (, ) for (, ). - = ( + ) Simplif. Write an equation of the line with slope - that contains point P(, -). ( ) B Postulate -, ou need onl two points to write an equation of a line. Writing an Equation of a Line Given Two Points Additional Eamples Write an equation in pointslope form of the line with slope -8 that contains P(, ). ± 8( ) Write an equation in pointslope form of the line that contains the points G(, -9) and H(-, ). ± 9 ( ) or ( ± ) Write equations for the horizontal line and the vertical line that contain A(-7, -). horizontal line: ; vertical line: 7 Resources Dail Notetaking Guide - L Dail Notetaking Guide - Adapted Instruction L Algebra Write an equation of the line through A(-, ) and B(, -). Step Step Find the slope. m = m = () Substitute (, ) for (, ) and (, ) for (, ). m = Simplif. Select one of the points. Write an equation in point-slope form. - = m( - ) - = f ()g Substitute (-, ) for (, ) and for the slope. - = ( ) Simplif. Write an equation of the line that contains the points P(, 0) and Q(7, -). 0 ( ) or ( 7) Recall that the slope of a horizontal line is 0 and the slope of a vertical line is undefined. Thus, horizontal and vertical lines have easil recognized equations. Equations of Horizontal and Vertical Lines Write equations for the horizontal line and the vertical line that contain P(, ). O P (, ) Closure Write the equation of the line containing the points (-, ) and (, ) in point-slope form, slopeintercept form, and standard form. ( ± ) or ( ); ± 8; 8 8 Chapter Parallel and Perpendicular Lines 8. Equations ma var from the pt. chosen. Samples are given. Ever point on the horizontal line through P(, ) has a -coordinate of. The equation of the line is =. It crosses the -ais at (0, ). Ever point on the vertical line through P(, ) has an -coordinate of. The equation of the line is =. It crosses the -ais at (, 0). Write equations of the horizontal and vertical lines that contain the point P(, -). ;. ( 0). ( ). ( ). ( ) 7. 0 ( ) 8. 0 ( 8) 8

EXERCISES For more eercises, see Etra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A GO Practice b Eample for Help Eamples, (pages, 7) Eample (page 7) Eample (page 8).

4 EXERCISES For more eercises, see Etra Skill, Word Problem, and Proof Practice. Practice and Problem Solving A GO Practice b Eample for Help Eamples, (pages, 7) Eample (page 7) Eample (page 8). ( 0) or Eample (page 8) Algebra Graph each line.. See back of book.. = +. = +. = -. = + Algebra Graph each line using intercepts. 0. See back of book.. + =. + = 7. - = = = =. Algebra Write each equation in slope-intercept form and graph the line.. See back of book.. = +. - =. + =. 8 + =. + =. - = 8 Algebra Write an equation in point-slope form of the line that contains the given point and has the given slope. ( ) ( ) ( ) 7. P(, ), slope 8. X(, -), slope 9. R(-, ), slope - 0. A(-, -), slope -. V(, ), slope. C(0, ), slope ( ) ( ) See left. Write an equation in point-slope form of the line that contains the given points. 8. See margin p. 8.. D(0, ), E(, 8). F(, ), G(, ). H(, ), K(-, ). A(-, ), B(, 0) 7. L(-, 0), M(-, -) 8. P(8, 0), Q(-, ). Practice Assignment Guide A B -, -8,,, A B 7-, 9-, 7-, - C Challenge 7- Test Prep -9 Mied Review Homework To check students understanding of ke skills and concepts, go over Eercises 8, 0, 8,,. Eercise 8 Ask: In which quadrant of the coordinate plane is the graph of this equation relevant? Quadrant B Real-World Eample (page 8) Appl Your Skills Connection NASA s Advanced Communications Technolog Satellite has a capacit for 0,000 phone calls. Write equations for (a) the horizontal line and (b) the vertical line that contain the given point. a A(, 7) b. a. 0. Y(, -) b. a.. N(0, -) b. 0 a.. E(, ) b. Graph each line. 7. See back of book.. =. =-. = 9. = 7. = 8. Telephone Rates The equation C = $.0m + $.9 represents the cost (C) of a long distance telephone call of m minutes. a. What is the slope of the line? 0.0 b. What does the slope represent in this situation? the cost per minute c. What is the -intercept (C-intercept)?.9 the initial charge for d. What does the -intercept represent in this situation? a call 9. Error Analsis A classmate claims that having no slope and having a slope of 0 are the same. Is our classmate correct? Eplain. No; a line with no slope is a vertical line. 0 slope is a horizontal line. 0. a. What is the slope of the -ais? Eplain. m 0; it is a horizontal line. b. Write an equation for the -ais. 0. a. What is the slope of the -ais? Eplain. Undefined; it is a vertical line. b. Write an equation for the -ais. 0 Identif the form of each equation. To graph the line, would ou use the given form or change to another form? Eplain.. See margin =. = =-( - ) Lesson - Lines in the Coordinate Plane 9 GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Name Class Date Practice - Are the lines parallel, perpendicular, or neither? Eplain. Slopes of Parallel and Perpendicular Lines. =. = +. + =. - - =- = + - = 8 + = = 7. =. + = 0 7. = 8. + = = 0 + = = 8 + = Are lines l and l parallel, perpendicular, or neither? Eplain * ) * ) ) ) XY * ) * ) * ) ) *XY *XY Write an XY equation for the line perpendicular to that contains point Z.. : + =-, Z(, ). : = +, Z(, 8) 7. : - + = 0, Z(-, -) 8. : = 0, Z(-, 9. XY : =-, Z(0, 0) 0. *XY : = +, Z(, -) Write an XY equation for the line parallel to that contains point Z. XY ). Aviation Two planes are fling side b side at the same altitude. It is important that their paths do not intersect. One plane is fling along the path given b the line - = 0. What is the slope-intercept form of the line that must be the path of another plane passing through the point L(-, -) so that the planes do not collide? Graph the paths of the two planes. L L L L L Pearson Education, Inc. All rights reserved.. Answers ma var. Samples are given.. The eq. is in standard form; change to slopeintercept form, because it is eas to graph the eq. from that form.. The eq. is in slope-int. form; use slope-int. form, because the eq. is alread in that form.. The eq. is in point-slope form; use point-slope form, because the eq. is alread in that form. 9

. Assess & Reteach Lesson Quiz. Find the -intercept and the -intercept of the line + =-80. -intercept:, -intercept: 0 Three points are on a coordinate plane: A(, ), B(-, -), and C(, -).

5 . Assess & Reteach Lesson Quiz. Find the -intercept and the -intercept of the line + =-80. -intercept:, -intercept: 0 Three points are on a coordinate plane: A(, ), B(-, -), and C(, -).. Write an equation in pointslope form of the line with slope - that contains point C. ± ( ). Write an equation in pointslope form of the line that contains points A and B. ( ) or ± ( ± ). Write an equation of the line that contains B and C.. Graph and label the equations of the lines in Eercises above. 8 ( ) ( ) 8 O 8 8 Alternative Assessment Have students work in pairs to eplain in writing how to write the equation of a line in three different forms. The should give an eample of a question when each form would be used. Eercises 7 0 Suggest that students calculate and compare slopes. GO. 0 0., 0.08; 0 S ; it is possible onl if the ramp zigzags. Real-World nline Homework Help Visit: PHSchool.com Web Code: aue-00 C Connection To visualize a slope of, think one foot over, one inch up. c. The abs. value of the slopes is the same, but one slope is pos. and the other is neg. One -int. is at (0, 0) and the other is at (0, 0). Critical Thinking Graph three different lines having the given propert. Describe how the equations of these lines are alike and how the are different.. The lines have slope.. The lines have -intercept. See back of book. See margin. 7. Graphing Calculator Graphing calculators use slope-intercept form (rather than standard form or point-slope form) to graph lines. Choose either Eercise or Eercise and write three equations for the lines ou graphed. Use the Y= window of our graphing calculator to enter our equations. Press. Do the graphs on the screen confirm the description ou wrote previousl? Check students work. Graph each pair of lines. Then find their point of intersection. 8. See margin pp =-, = 9. = 0, = 0 0. =-, =. =, =. Building Access B law, the maimum slope of a ramp in new construction is. The plan for the new librar shows a -ft height from the ground to the main entrance. The distance from the sidewalk to the building is 0 ft. Can ou design a ramp for the librar that complies with the law? Eplain. See left.. Writing Describe the similarities of and the differences between the graphs of the equations = - and =- -. See margin p. 7.. Open-Ended Write equations for three different lines that contain the point (, ). Answers ma var. Sample:, ( ),. Critical Thinking The -intercept of a line is and the -intercept is. Use this information to write an equation for the line. See margin p. 7.. The vertices of a triangle are A(0, 0), B(, ), and C(, 0). a. 0 ( 0) GPS a. Write an equation for the line through A and B. or b. Write an equation for the line through B and C. b. ( ) or c. Compare the slopes and -intercepts of the two lines. 0 Challenge Do the three points lie on one line? Justif our answer See left. 7. Yes; the slope of AB 7. A(, ), B(, ), C(, 8) 8. D(-, -), E(, -), F(0, 0) equals the slope of BC. 9. G(, -), H(, ), I(-, 0) 0. J(-, 9), K(, -), L(, -) 8. No; the slope of DE Yes; the slope of GH equals the slope of HI. See left. does not equal the A line passes through the given points. Write an equation for the line in pointslope form. Then, rewrite the equation in standard form with integer coefficients. slope of EF. 0.Yes; the slope of JK equals the slope of KL.. R(-, ), S(0, 8). T(, ), W(7, ). X(, ), Y(, 8) ( ); ( ); ( ); 8 70 Chapter Parallel and Perpendicular Lines 70. The slopes are all different, and the -intercepts are the same. O 8. O 7 (, )

6 GO Standardized Test Prep Test Prep Multiple Choice Short Response Mied Review for Help Lesson - Lesson - Lesson -7. Which equation is equivalent to + = 0? D A. = + B. =- - C. = - D. =- +. Which pair of points A(-, ), B(-, -), C(, -), and D(7, 0), lie on the line with -intercept closest to the origin? G F. A and B G. A and C H. B and C J. B and D. What is the -intercept of the line whose equation is 9 ( )? C A. B. 9 C. - D. -9 Use the graph at the right for Eercises What is the slope of the line? J (0, ) F. - G. - H. J. 8. Which equation is the equation for the line? C (, ) A. B. C. D. O 9. The slope of line a is and its -intercept is. Line b passes through (, ) and (7, -). a. Write an equation for each line. a b. See margin. b. Graph both lines on the same coordinate plane. From the graph, what is their point of intersection? Find the sum of the measures of the angles of each polgon. 70. a nonagon 7. a pentagon 7. an -gon 7. a -gon Is each statement a good definition? If not, find a countereample. 7. A quadrilateral is a polgon with four sides. es 7. Skew lines are lines that don t intersect. No; parallel lines never intersect, but the are not skew. 7. An acute triangle is a triangle with an acute angle. No; all obtuse > have two acute ) '. Algebra For Eercises 77 80, PQ is the bisector of &MPR. Solve for a and find the missing angle measure. 77. m&mpq = a, m&qpr = a +, m&mpr = 7 a ; mlmpr m&mpq = 7a, m&qpr = a +, m&mpr = 7 a ; mlmpr 79. m&mpq = 8a - 8, m&qpr = a -, m&qpr = 7 a ; mlqpr m&mpq = a + 9, m&qpr = a -, m&mpq = 7a ; mlmpq Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, p. 9 Test-Taking Strategies, p. 88 Test-Taking Strategies with Transparencies. The -intercepts are the same, and the lines have the same steepness. One line rises from left to right while the other falls from left to right. 0. (, 0), (0, ); m 0 0 ( ), or 9. [] a. Line a: OR equivalent equation; Line b: 9 OR equivalent equation b. lesson quiz, PHSchool.com, Web Code: aue-00 Lesson - Lines in the Coordinate Plane (, ) O (0, 0) O (, ) O O point of intersection: (, 9) [] at least one correct eq. or graph 7

Parallel and Perpendicular Lines. What are the slope and y-intercept of each equation?

Parallel and Perpendicular Lines. What are the slope and y-intercept of each equation? 6 6-6 What You ll Learn To determine whether lines are parallel To determine whether lines are And Wh To use parallel and lines to plan a bike path, as in Eample Parallel Lines Parallel and Perpendicular