1. Calculus Whatever you think you can or think you can’t – you are right. Henry Ford

2. Precalculus with Limits A Graphing Approach

3. LINES IN THE PLANE Section 1.1 3

4. Starter Factor the following trinomials. 1. x 2 - 6x – 27 2. x 2 - 11x + 28 3. 2x 2 + 11x – 40 4. 3x 2 - 16x +5 4

5. Starter - Answers Factor the following trinomials. 1. x 2 – 6x – 27 (x – 9)(x + 3) 2. x 2 – 11x + 28 (x – 7)(x – 4) 3. 2x 2 + 11x – 40 (2x – 5)(x + 8) 4. 3x 2 – 16x + 5 (3x – 1)(x – 5) 5

6. What are we going to learn today? 1. Find the slope of lines. 2. Write linear equations given points on lines and their slopes. 3. Use slope-intercept forms of linear equations to sketch lines. 4. Use slope to identify parallel and perpendicular lines. 6

7. Lines What is the minimum number of items that you must know in order to determine a unique line? Answer to follow. 7

8. Distance How far is it from A to B? 10 0 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 A B 7 8

9. Distance How far is it from C to D? 10 0 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 CD 9 9

10. Distance How far is it from E to F? 10 0 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 E F 8 10

11. Distance How did you find the distance? 1. Counted between points. 2.F – E = -1 – (-9) = -1+9 = 8 3. |F – E| = |-1- -9| = |-1+9| = |8| = 8 10 0 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 E F 8 (Right – Left) Also notice that |F-E| = |E-F|= |-9-(-1)|=|-9+1|= 8 11

12. Distance 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. (-7, -3). (4, 5) How far is it between the points? 11 8 c 2 = a 2 + b 2 = 8 2 + 11 2 = 64 + 121 = 185 4 – (-7) = 4+7 = 11 5-(-3) = 5+3 = 8 12

13. Distance 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. (-7, -3) Could you use the distance formula? c 2 = a 2 + b 2 = 8 2 + 11 2 = 64 + 121 = 185. (4, 5) 5-(-3) = 8 4 – (-7) = 11 13

14. Slope 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7 By definition, slope = m =. (-8, 2) m is positive, y-values increase from left to right.. (5, 6) where (x 1, y 1 ) and (x 2, y 2 ) are any two nonvertical points. 14

15. Slope 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7 By definition, slope = m = where (x 1, y 1 ) and (x 2, y 2 ) are any two nonvertical points.. (-8, 2). (5, 6) 15

16. Slope 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7 By definition, slope = m =. (-6, 5). (4, -3) m is negative, y-values decrease from left to right. where (x 1, y 1 ) and (x 2, y 2 ) are any two nonvertical points. 16

17. Slope 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7 By definition, slope = m =. (-5, 3). (4, 3) m is zero, y-values are constant from left to right. All horizontal lines have m = 0. where (x 1, y 1 ) and (x 2, y 2 ) are any two nonvertical points. 17

18. Slope 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7 By definition, slope = m =. (-9, -3). (-9, 5) m is undefined, x-values are equal. All vertical lines have m = undefined. where (x 1, y 1 ) and (x 2, y 2 ) are any two nonvertical points. 18

19. The Slope of a Line 1.A line with positive slope (m > 0) rises from left to right. 2.A line with negative slope (m < 0) falls from left to right. 3.A line with zero slope (m = 0) is horizontal. 4.A line with undefined slope is vertical. 19

20. Slope Since, then 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. (x 2, y 2 ). (x 1, y 1 ) Keep (x 1, y 1 ) fixed and let (x 2,y 2 ) be a variable point on the line.. (x, y) (x, y).. (x, y) Now our equation becomes or 20

21. Equations of a Line This is the point-slope form of the equation of a line. slope part of the point skip variables part of the point 21

22. Equations of a Line Every linear function crosses the y-axis. x y (0, a) (0, b) (0, c) Every y-intercept has the form (0, y-intercept). 22

23. Equations of a Line The point-slope form was y - y 1 =m(x – x 1 ). Now we can use one of the intercepts such as (o, b) as a point on the line. Then y - y 1 =m(x – x 1 ) becomes y - b=m(x – 0) or y – b = mx and then y = mx + b. This is the slope-intercept form of a line. 23

24. Equations of a Line Using the slope-intercept form y = mx + b, we can replace m with to get To clear the fraction, multiply both sides by d to get Rearrange terms to get Or, in general form Ax + By + C = 0 where A > 0. 24

25. Equations of a Line 1. General Form: 2. Vertical line: 3. Horizontal line: 4. Slope-intercept form: 5. Point-slope form: where A > 0 25

26. Linear Models 1. Use the information provided to find the equation of a line. 2. The line can be used to estimate values between two given points. (interpolation) 3. Be cautious using the line outside the given points. (extrapolation) 26

27. Linear Models My car gets 26 miles per gallon (mpg) when I drive 70 mph. However, if I only go 45 mph, I get 38 mpg. How many miles per gallon would I get if I went 60 mph? 27

28. Linear Models 50 60 70 80 90 20 30 40 10 -5 40 50 60 0 10 20 -10 30.. miles per hour miles per gallon Does this mean that at 5 mph, I would get over 55 mpg? 28

29. Miles per gallon 29

30. Sketching Graphs of Lines 1. Rearrange terms to put in slope-intercept form, y = mx + b. 2. Put a dot on the y-intercept, b. 3. Convert the slope, m, into a fraction, n/d. 4. From the y-intercept, count up (n positive) or down (n negative) n units. 5. From this point, count right (d positive) or left (d negative) d units. Place a dot at the point. 6. Connect the two points with a straight line. 30

31. Sketching Graphs of Lines Sketch. 1. b = -4 2. rise = 2 3. run = 3 4. Connect the dots with a line. 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. 31..

32. Sketching Graphs of Lines Sketch. 1. b = -4 2. rise = -1 3. run = 3 4. Connect the dots with a line. 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. 32..

33. Sketching Graphs of Lines Sketch again. 1. b = -4 2. rise = 1 3. run = -3 4. Connect the dots with a line. 10 1 2 4 3 5 6 7 9 8 -5 -4 -3 -2 -9 -8 -6 -7 -10 2 4 3 5 6 7 -5 -4 -3 -2 -6 -7. 33..

34. Parallel Lines Two lines l 1 and l 2 are parallel if and only if (iff) their slopes are equal, m 1 = m 2. y = m 1 x + b (0, b) y = m 1 x + c (0, c) l1l1 l2l2 Lines l 1 and l 2 are parallel. 34

35. Perpendicular Lines Two lines l 1 and l 2 are perpendicular if and only if (iff) (m 1 ) (m 2 ) = -1 or m 1 = -1 / m 2. y = m 1 x + b (0, b) y = m 2 x + c (0, c) l1l1 l2l2 Lines l 1 and l 2 are perpendicular. The slopes are negative reciprocals. 35

36. Lines What is the minimum number of items that you must know in order to determine a unique line? 1. Two points. Find the slope, use point-slope form. 2. A point and the slope. Use point-slope form. 3. Slope and y-intercept. Use slope-intercept form. 36

37. 37

38. What are we going to learn today? 1. Find the slope of lines. 2. Write linear equations given points on lines and their slopes. 3. Use slope-intercept forms of linear equations to sketch lines. 4. Use slope to identify parallel and perpendicular lines. 38

39. Homework Section / page / problems 1.1 / 11 / 5 – 85 by 5s 39

40. End of Lesson 40