1. 10/5/2015 V. J. Motto 1 Chapter 1: Linear Models V. J. Motto M110 Modeling with Elementary Functions 1.3 Two Important Questions (Revisited)

2. 10/5/2015V. J. Motto 2 Overview Housekeeping Who’s here today? Who needs materials? Homework Problems Questions Homework Two Important Questions (Revisited) Homework Assignment

3. 10/5/2015V. J. Motto 3 Example 1 --- Intercepts Where does the graph cross the x-axis? Where does the graph cross the y-axis? These points are called the x-intercept and y-intercept.

4. Problems Set 1: Let’s look at the following linear equations using inspection and our calculator to find x- and y-intercepts. 10/5/2015V. J. Motto 4

5. 10/5/2015V. J. Motto 5 Example 2 – Non-linear This is a non- linear function. X-intercepts: (-2, 0) (3, 0) Y-intercept is: (0, -6)

6. 10/5/2015V. J. Motto 6 Finding x- and y-intercepts To find the x-intercept, let y = 0 and solve for x. To find the y-intercept, let x = 0 and solve for y.

7. 10/5/2015V. J. Motto 7 Example 3: Graphing x – 3y = 6 Thus, (6, 0) is the x-intercept. Thus, (0, -2) is the y-intercept. We can use these two points to make a sketch of the graph. Sketch the graph of x – 3y = 6 by plotting the x-intercept and the y-intercept.

8. 10/5/2015V. J. Motto 8 Example 3 (continued)

9. 10/5/2015V. J. Motto 9 Special Lines Vertical Lines The graph x = c, where c is a real number, is a vertical line with x-intercept (c, 0). Horizontal Lines The graph of y = c, where c is a real number, is a horizontal line with y-intercept (0, c)

10. 10/5/2015V. J. Motto 10 Example 4 y = 1 x = 1 y = 2.5 x = 4 y = 3 x = 5.5

11. 10/5/2015V. J. Motto 11 Slope Lines can be slanted in many different ways. We use slope to measure a lines slant. The green line has a big slope, because it is slanted sharply. Because the red line is close to flat, it has a small slope. These lines have positive slope.

12. 10/5/2015V. J. Motto 12 Negative Slope Lines with negative slope point down instead of up. A line of negative slope is pictured at the right.

13. 10/5/2015V. J. Motto 13 Putting Slope in perspective We can tell whether a line’s slope is big or small, and whether the slope is positive or negative. How can we quantify this idea of slope so that we can compare slopes? Let’s start with a definition of slope.

14. 10/5/2015V. J. Motto 14 The Definition of Slope --- 1 Slope is defined as the change in the y-coordinates divided by the change in the x- coordinates. People often remember this definition as “rise/run”

15. 10/5/2015V. J. Motto 15 The Definition of Slope --- 2

16. 10/5/2015V. J. Motto 16 Example 5

17. 10/5/2015V. J. Motto 17 Problem Set 2: Find the slopes for the following pairs of points: (3, 2) and (-2, 5) (-4, 3) and ( -2, -4) (0, -4) and ( -3, -5)

18. 10/5/2015V. J. Motto 18 Slope Which line has negative slope? Which line has positive slope? Which line has the greater slope? Which line has no slope? Which line has 0 slope? Which line has the smaller slope?

19. 10/5/2015V. J. Motto 19 Example 6 Find the slope of the line that passes through the points (0, 1) and ( 3, 4). Often it is best to think of slope as a fraction

20. 10/5/2015V. J. Motto 20 Comments on Slope Vertical lines Points on these types of lines have the same x- coordinate. Hence, their slope is undefined because we would be dividing by 0 when we calculate it. Horizontal lines Points on these types of lines have the same y- coordinate. Hence, their slope is 0 because the numerator is 0.

21. 10/5/2015V. J. Motto 21 Parallel and Perpendicular Lines Parallel lines have the same slope Perpendicular lines have slope that are the negative reciprocal of each other: that is, the product of their slopes is -1 or m 1 *m 2 = -1.

22. 10/5/2015V. J. Motto 22 Slope-Intercept Form Linear equations of the form y = mx + b have slope-intercept form. The coefficient of the x term is the slope and the b value is the y-coordinate of the y-intercept which is (0, b). We can easily read the slope and discover the coordinate for the y-intercept by inspection of the equation.

23. 10/5/2015V. J. Motto 23 Example 7 Find the slope and y-intercept for the following linear equations:

24. 10/5/2015V. J. Motto 24 Transposing Often we are given the equation of a line in the standard form Ax + By = C Then we are asked to change it to the slope-intercept form y = mx + b This is easily accomplished by solving the standard form for the variable y.

25. 10/5/2015V. J. Motto 25 Example 3 Write 3x – 4y = 4 in slope-intercept form. What is the slope for this line? What are the coordinates of the y-intercept?

26. 10/5/2015V. J. Motto 26 Parallel or Perpendicular Other types of problems will ask us to compare the slopes of two lines in order to decide if the lines are parallel (same slope) or perpendicular (the product of slopes is -1).

27. 10/5/2015V. J. Motto 27 Example 8 T Thus, the line represented by these equations are parallel.

28. 10/5/2015V. J. Motto 28 Finding the equation We explore in the next example the first type of “finding the linear equation” type of problems. Here we are given the slope and the y-intercept. Using the slope-intercept form, y = mx +b, renders these type of problems quickly. We will explore other types in our next lecture.

29. 10/5/2015V. J. Motto 29 Example 9

30. 10/5/2015V. J. Motto 30 Finding equations In the previous study we have developed strategies for finding the equation of a line. Given the slope, m, and the coordinates of the y- intercept, (0, b) we can use the slope-intercept form, y = mx + b, to generate the equation of a line. Here we explore some other situations. You are given: The slope and a point of the line Two points. For these situations we use the point-slope form.

31. 10/5/2015V. J. Motto 31 Point-Slope Form The point-slope form of the equation of a line is y – y 1 = m(x – x 1 ) where m is the slope of the line and (x 1, y 1 ) is a point on the line.

32. 10/5/2015V. J. Motto 32 Example 10: Find the equation of the line passing through (-1, 5) with slope -2. Since the slope of the line is given, we know that m = -2. Thus, we have y – y 1 = -2(x – x 1 ) Now using the coordinates of the point (-1, 5), we have the following y – 5 = -2(x – (-1)) which becomes y – 5 = -2(x + 1)

33. 10/5/2015V. J. Motto 33 Example 10 (continued) If we want to express the equation in slope-intercept form, we have y – 5 = - 2x - 2 y = - 2x + 3 But if we want to express the equation in standard form, we have y = - 2x + 3 2x + y = 3

34. 10/5/2015V. J. Motto 34 Two Points Most often when we are looking for the equation of a line we are only given two points. Thus, the technique becomes Finding the slope Using the slope-point form Our next example illustrates this situation.

35. 10/5/2015V. J. Motto 35 Example 11: Find an equation of the line through (2, 5) and (-3, 4). First we must find the slope.

36. 10/5/2015V. J. Motto 36 Example 11 (continued) Now we use the point-slope form to generate the equation. We will use the point (2, 5). Thus

37. 10/5/2015V. J. Motto 37 Example 11 (continued) In the previous slide the equation is in slope-intercept form. What does the equation look like in standard form?

38. 10/5/2015V. J. Motto 38 Special Lines The equation of a vertical line can be written in the form x = c. You should recall that this line is parallel to y-axis The equation of a horizontal line can be written y = c. You should recall that this line is parallel to the x-axis.

39. 10/5/2015V. J. Motto 39 Example 12 Find an equation of the vertical line through ( -3, 5). The equation of a vertical line can be written in the form x = c, so an equation for a vertical line passing through (-3, 5) is x = -3.

40. 10/5/2015V. J. Motto 40 Example 12 (continued)

41. 10/5/2015V. J. Motto 41 Example 13: Find an equation of the line parallel to the line y = 5 and passing through the point (-2, -3). Since the graph of y = 5 is a horizontal line, any line parallel to it is also horizontal. The equation of a horizontal line can be written in the form y = c. An equation for the horizontal line passing through (-2, -3) is y = -3.

42. 10/5/2015V. J. Motto 42 Example 3 (continued)

43. Table of Values 1. Enter the linear function y = 3x – 5 2. Now press the graph key. 3. To see a table of values, press the 2 nd key followed by the graph key  table view. 10/5/2015V. J. Motto 43

44. Table Functions You can alter the setup values in the table by pressing the 2 nd key followed by the Window key  table set function 10/5/2015V. J. Motto 44

45. Intersecting Lines Consider the two equations: y = 3x – 5 y = -2x + 6 The graph is shown at the right. You should observe that they intersect. 10/5/2015V. J. Motto 45

46. Find the Intersection Press 2 nd Trace. Choose option 5:intersection You want to use the first equation. You want to use the second equation. Yes, you want the calculator to “guess” or calculate the intersection 10/5/2015V. J. Motto 46

47. 10/5/2015V. J. Motto 47 Forms of Linear Equations FormDescription Ax + By = CStandard Form. A and B are not both 0 y = mx + bSlope-intercept Form. Slope is m; y-intercept is (0, b) y – y1 = m(x - x1)Point-slope Form Slope is m; (x1, y1) is a point of the line y = cHorizontal Line Slope is 0; y-intercept is (0, c) x = cVertical Line Slope is undefined; x-intercepet is (c, 0)