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Lesson 2-3 Objective The student will be able to:

Learn how to write equations using slope-intercept form and identify slope and y-intercept from an equation.

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Lesson 2-3 Objective The student will be able to:

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  1. Lesson 2-3 ObjectiveThe student will be able to: • write equations using slope-intercept form. • identify slope and y-intercept from an equation

  2. Important!!! This is one of the big concepts in algebra. You need to thoroughly understand this! Slope – Intercept Form y = mx + b m represents the slope b represents the y-intercept

  3. Writing Equations When asked to write an equation, you need to know two things – slope (m) and y-intercept (b). There are three types of problems you will face.

  4. Writing Equations – Type #1 2 6 Write an equation in slope-intercept form of the line that has a slope of 2 and a y-intercept of 6. To write an equation, you need two things: slope (m) = y – intercept (b) = We have both!! Plug them into slope-intercept form y = mx + b y = 2x + 6

  5. Write the equation of a line that has a y-intercept of -3 and a slope of -4. • y = -3x – 4 • y = -4x – 3 • y = -3x + 4 • y = -4x + 3

  6. Writing Equations – Type #2 3 ??? Write an equation of the line that has a slope of 3 and goes through the point (2,1). To write an equation, you need two things: slope (m) = y – intercept (b) = We have to find the y-intercept!! Plug in the slope and ordered pair into y = mx + b 1 = 3(2) + b

  7. Writing Equations – Type #2 1 = 3(2) + b Solve the equation for b 1 = 6 + b -6 -6 -5 = b To write an equation, you need two things: slope (m) = y – intercept (b) = y = 3x - 5 3 -5

  8. Writing Equations – Type #3 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = We need both!! First, we have to find the slope. Plug the points into the slope formula. Simplify ??? ???

  9. Writing Equations – Type #3 Write an equation of the line that goes through the points (-2, 1) and (4, 2). To write an equation, you need two things: slope (m) = y – intercept (b) = It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use? I’m using (4, 2) because both numbers are positive. 2 = (4) + b ???

  10. Writing Equations – Type #3 2 = (4) + b Solve the equation for b 2 = + b To write an equation, you need two things: slope (m) = y – intercept (b) =

  11. Write an equation of the line that goes through the points (0, 1) and (1, 4). • y = 3x + 4 • y = 3x + 1 • y = -3x + 4 • y = -3x + 1

  12. To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b. Find the slope and y-intercept. • y = 3x – 7 y = mx + b m = 3, b = -7

  13. Find the slope and y-intercept. m = b = 0 2) y = x y = mx + b y = x + 0 3) y = 5 y = mx + b y = 0x + 5 m = 0 b = 5

  14. Find the slope and y-intercept.4) 5x - 3y = 6 Write it in slope-intercept form. (y = mx + b) 5x – 3y = 6 -3y = -5x + 6 y = x - 2 -3 -3 -3 m = b = -2

  15. Find the slope and y-intercept. 5) 2y + 2 = 4x Write it in slope-intercept form. (y = mx + b) 2y + 2 = 4x 2y = 4x - 2 y = 2x - 1 2 2 2 m = 2 b = -1

  16. Find the slope and y-intercept of y = -2x + 4 • m = 2; b = 4 • m = 4; b = 2 • m = -2; b = 4 • m = 4; b = -2

  17. WarPPPPPpm-Up BELLWORK – Wedn. 10-22-14 Write an equation of the line in slope-intercept form. 1. passes through (3, 4), m = 3 ANSWER y = 3x– 5 2. passes through (–2, 2) and (1, 8) ANSWER y = 2x + 6

  18. Wedn 10-22-14 Additional Bellwork

  19. x y Graphing a Line Given a Point & Slope • Graph a line though the point (2, -6) with m=2/3 • Graph (2, -6) • Count up 2 for the rise, and to the right 3 for the run • Plot the point, repeat, then connect

  20. x y Graphing Lines • m = - ½ b = 3 • Plot y-intercept (b) • Use the slope to find two more points • Connect

  21. (3, -3) m = undefined x y Graph the Line

  22. Graph using slope intercept form

  23. POINT-SLOPE FORM • Chapter 2-3

  24. Example 1 Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2. Write point-slope form. y –y1= m(x –x1) y +3=2(x–4) Substitute 2form, 4 for x1, and–3 fory1.

  25. 1. Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2. ANSWER y – 4 = –2(x + 1) Guided Practice

  26. 2 3 y+2 = (x – 3). SOLUTION Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2). 2 3 Example 2 Graph the equation Plot the point (3, –2). Find a second point on the line using the slope. Draw a line through both points.

  27. – Graph the equation 2. ANSWER Guided Practice y–1 = (x – 2).

  28. y2 – y1 = m x2 – x1 3 – 1 2 = = = –1 –1 – 1 –2 Example 3 Write an equation in point-slope form of the line shown. SOLUTION STEP 1 Find the slope of the line.

  29. Example 3 STEP 2 Write the equation in point-slope form. You can use either given point. Method 1 Method 2 Use (–1, 3). Use (1, 1). y –y1= m(x –x1) y –y1=m(x – x1) y –3=–(x +1) y –1=–(x – 1) CHECK Check that the equations are equivalent by writing them in slope-intercept form. y – 3 = –x – 1 y – 1 = –x + 1 y = –x + 2 y = –x + 2

  30. 3. Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4). ANSWER 1 1 2 2 y –3 =(x – 2) or y –4 = (x – 4) Guided Practice

  31. 1. Write an equation in point-slope form of the line that passes through (6, –4) and has slope 2. ANSWER y + 4 = –2(x – 6) 2. Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10). y + 6 = 4(x + 1) ory –10 = 4(x–3) ANSWER Lesson Quiz

  32. Write Equations and Parallel and Perpendicular Lines Lesson 2-3 continued

  33. Warm-Up Are the lines parallel? Explain. 1. y– 2 = 2x, 2x + y = 7 No; one slope is 2 and the other is–2. ANSWER 2. –x = y + 4, 3x + 3y = 5 ANSWER Yes; both slopes are–1.

  34. Example 1 Write an equation of the line that passes through (–3, –5) and is parallel to the liney = 3x – 1. SOLUTION STEP1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

  35. Example 1 STEP2 Find the y-intercept. Use the slope and the given point. y=mx+b Write slope-intercept form. –5=3(–3) +b Substitute3form,3for x,and5fory. Solve forb. 4 = b STEP3 Write an equation. Usey = mx + b. Substitute3formand4forb. y = 3x + 4

  36. 1. Write an equation of the line that passes through (–2, 11) and is parallel to the liney =–x + 5. ANSWER y = –x + 9 Guided Practice

  37. SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form.The slope is 5. Example 2 Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b:x + 5y = 2 Line c:–10y – 2x = 0 Write the equations for lines band cin slope-intercept form.

  38. Lineb: ANSWER x + 5y = 2 Lines band chave slopes of – , so they are parallel. Lineahas a slope of5,the negative reciprocal of – , so it is perpendicular to lines band c. – x y = + 2 1 1 1 1 Linec: –10y – 2x = 0 5 5 5 5 5 x – y = Example 2 –10y = 2x 5y = – x + 2

  39. ANSWER parallel: b and c; perpendicular: a and b, a and c Guided Practice Determine which lines, if any, are parallel or perpendicular. Line a: 2x + 6y =–3 Line b: y = 3x – 8 Line c:–1.5y + 4.5x = 6

  40. STATE FLAG The Arizona state flag is shown in a coordinate plane. Lines aand bappear to be perpendicular. Are they? Example 3 Linea:12y = –7x + 42 Lineb:11y = 16x – 52 SOLUTION Find the slopes of the lines. Write the equations in slope-intercept form.

  41. 7 16 7 16 11 12 11 12 x y=– + 42 52 12 11 x y = – ANSWER The slope of line ais –. The slope of line bis . The two slopes are not negative reciprocals, so lines aand bare not perpendicular. Example 3 Linea:12y = –7x + 42 Lineb:11y = 16x – 52

  42. ANSWER 1 No; the slope of line ais –, the slope of line bis . The slopes are not negative reciprocals so the lines are not perpendicular. 2 3 2 Guided Practice 3. Is line a perpendicular to line b?Justify your answer using slopes. Linea:2y +x = –12 Lineb:2y = 3x – 8

  43. Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 – Example 4 Write an equation of the line that passes through(4, –5)and is perpendicular to the liney = 2x + 3. SOLUTION STEP1

  44. y= mx+b 1 – (4) +b –5= Substitute – for m, 4 for x, and –5 for y. 2 1 2 –3= b 1 2 1 y = – x – 3 Substitute – formand–3for b. 2 Example 4 STEP 2 Find the y-intercept. Use the slope and the given point. Write slope-intercept form. Solve for b. STEP3 Write an equation. y = mx + b Write slope-intercept form.

  45. Graphing Lines • Graph the line perpendicular to y = 2x + 3 that goes through the point (-2, 3). • The slope of the line is 2 so the slope of the perpendicular line is -1/2. • m = -1/2 b = 3

  46. 1 4 y = – x + 4 ANSWER Guided Practice 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

  47. 1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2. ANSWER y = 5x + 9 2. Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2. 1 ANSWER – 4 y = 4x + 3 Lesson Quiz

  48. 2. ( –1, –2), (2, 7) Warm-UpT1t Write an equation in point-slope form of the line that passes through the given points. 10-23-14 Bellwork 1. (1, 4), (6, –1) y – 4 = –(x – 1) ory + 1 = –(x – 6) ANSWER ANSWER y + 2 = 3(x + 1) ory– 7 = 3(x– 2)

  49. Writing Equations in Standard Form • MFCR Lesson 2-3

  50. Graphing Lines • Graph the line parallel to x = -1 that goes through the point (3, -3). • The slope of the line is undefined (VUX) so the slope of the parallel line is undefined.

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