College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Coordinates and Graphs 2

3. Lines 2.4

4. Lines • In this section, we find equations for straight lines lying in a coordinate plane. • The equations will depend on how the line is inclined. • So, we begin by discussing the concept of slope.

5. The Slope of a Line

6. Slope of a Line • We first need a way to measure the “steepness” of a line. • This is how quickly it rises (or falls) as we move from left to right.

7. Slope of a Line • We define ‘run’to be the distance we move to the right and ‘rise’to be the corresponding distance that the line rises (or falls). • The slopeof a line is the ratio of rise to run:

8. Slope of a Line • The figure shows situations where slope is important. • Carpenters use the term pitchfor the slope of a roof or a staircase. • The term gradeis used for the slope of a road.

9. Run & Rise • If a line lies in a coordinate plane, then: • Runis the change in the x-coordinate between any two points on the line. • Riseis the corresponding change in the y-coordinate.

10. Slope of a Line—Definition • That gives us the following definition of slope. • The slopem of a nonvertical line that passes through the points A(x1, y1) and B(x2, y2) is: • The slope of a vertical line is not defined.

11. Slope of a Line • The slope is independent of which two points are chosen on the line.

12. Slope of a Line • We can see that this is true from the similar triangles in the figure:

13. Slope of a Line • The figure shows several lines labeled with their slopes.

14. Positive & Negative Slopes • Notice that: • Lines with positive slope slant upward to the right. • Lines with negative slope slant downward to the right.

15. Steep & Horizontal Lines • The steepest lines are those for which the absolute value of the slope is the largest. • A horizontal line has slope zero.

16. E.g. 1—Finding Slope of a Line Through Two Points • Find the slope of the line that passes through the points P(2, 1) and Q(8, 5) • Since any two different points determine a line, only one line passes through these two points.

17. E.g. 1—Finding Slope of a Line Through Two Points • From the definition, the slope is: • This says that, for every 3 units we move to the right, the line rises 2 units.

18. E.g. 1—Finding Slope of a Line Through Two Points • The line is drawn here.

19. Point-Slope Form of the Equations of a Line

20. Equations of Lines • Now let’s find the equation of the line that passes through a given point P(x1, y1) and has slope m.

21. Equations of Lines • A point P(x, y) with x ≠x1 lies on this line if and only if: • The slope of the line through P1 and P is equal to m, that is,

22. Equations of Lines • That equation can be rewritten in the form y – y1 = m(x – x1) • Note that the equation is also satisfied when x = x1 and y =y1. • Therefore, it is an equation of the given line.

23. Point-Slope Form of the Equation of a Line • An equation of the line that passes through the point (x1, y1) and has slope m is: y – y1 = m(x – x1)

24. E.g. 2—Equation of a Line with Given Point and Slope • Find an equation of the line through (1, –3 ) with slope –½. • Sketch the line.

25. Example (a) E.g. 2—Equation of a Line • Using the point-slope form with m = –½ , x1 = 1, y1 = –3, we obtain an equation of the line as:

26. Example (b) E.g. 2—Equation of a Line • The fact that the slope is –½ tells us that: • When we move to the right 2 units, the line drops 1 unit.

27. Example (b) E.g. 2—Equation of a Line • This enables us to sketch the line here.

28. E.g. 3—Equation of a Line Through Two Points • Find an equation of the line through the points (–1, 2) and (3, –4) • The slope of the line is:

29. E.g. 3—Equation of a Line Through Two Points • Using the point-slope form with x1 = –1 and y1 = 2, we obtain:

30. Slope-Intercept Form of the Equations of a Line

31. Equations of Lines • Suppose a nonvertical line has slope m and y-intercept b. • This means the line intersects the y-axis at the point (0, b).

32. Slope-Intercept Formof the Equation of a Line • So, the point-slope form of the equation of the line, with x = 0 and y = b, becomes: • y – b = m(x – 0) • This simplifies to y = mx + b. • Thisiscalled the slope-intercept formof the equation of a line.

33. Slope-Intercept Form of the Equation of a Line • An equation of the line that has slope m and y-intercept b is:y = mx + b

34. E.g. 4—Lines in Slope-Intercept Form • Find the equation of the line with slope 3 and y-intercept –2. • Find the slope and y-intercept of the line 3y – 2x =1.

35. Example (a) E.g. 4—Slope-Intercept Form • Since m = 3 and b = –2, from the slope- intercept form of the equation of a line, we get: y = 3x – 2

36. Example (b) E.g. 4—Slope-Intercept Form • We first write the equation in the form y =mx + b: • From the slope-intercept form of the equation of a line, we see that: the slope is m = ⅔ and the y-intercept is b = ⅓.

37. Vertical and Horizontal Lines

38. Equation of Horizontal Line • If a line is horizontal, its slope is m = 0. • So, its equation is: y = b where b is the y-intercept.

39. Equation of Vertical Line • A vertical line does not have a slope. • Still, we can write its equation as: x =a where a is the x-intercept. • This is because the x-coordinate of every point on the line is a.

40. Vertical and Horizontal Lines • An equation of the vertical line through (a, b) is: x = a • An equation of the horizontal line through (a, b) is: y = b

41. E.g. 5—Vertical and Horizontal Lines • An equation for the vertical line through(3, 5) is x = 3. • The graph of theequation x = 3 is a vertical line withx-intercept 3.

42. E.g. 5—Vertical and Horizontal Lines • The equation for the horizontal line through (8, –2) is y = –2. • The graph of the equation y = –2 is a horizontal line with y-intercept –2.

43. General Equation of a Line

44. Linear Equation • A linear equationis an equation of the formAx + By + C = 0 • where: • A, B, and C are constants. • A and B are not both 0.

45. Equation of a Line • The equation of a line is a linear equation: • A nonvertical line has the equation y =mx +b or –mx +y –b = 0 This is a linear equation with A = –m, B = 1, C = –b • A vertical line has the equation x =a or x –a = 0 This is a linear equation with A = 1, B = 0, C = –a

46. Graph of a Linear Equation • Conversely, the graph of a linear equation is a line: • If B ≠ 0, the equation becomes:This is the slope-intercept form of the equation of a line (with m = –A/B and b = –C/B). • If B = 0, the equation becomes: Ax +C = 0 or x = –C/A Thisrepresents a vertical line.

47. General Equation of a Line • We have proved the following. • The graph of every linear equationAx +By +C = 0 (A, B not both zero)is a line. • Conversely, every line is the graph of a linear equation.

48. E.g. 6—Graphing a Linear Equation • Sketch the graph of the equation 2x – 3y – 12 = 0

49. Solution 1 E.g. 6—Graphing a Linear Equation • Since the equation is linear, its graph is a line. • To draw the graph, it is enough to find any two points on the line. • The intercepts are the easiest points to find.

50. Solution 1 E.g. 6—Graphing a Linear Equation • x-intercept: • Substitute y = 0. • You get: 2x – 12 = 0. • Thus, x = 6. • y-intercept: • Substitute x = 0. • You get: –3y – 12 = 0. • Thus, y = –4.