CHAPTER. Graphs of Linear Equations. 3.1 Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications

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2 Graphs of Linear Equations CHAPTER Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications Slide 2

3 3.1 Introduction to Graphing OBJECTIVES a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. b Find the coordinates of a point on a graph. c Determine whether an ordered pair is a solution of an equation with two variables. Slide 3

4 3.1 Introduction to Graphing Plot points associated with ordered pairs of numbers; a determine the quadrant in which a point lies. On the number line, each point is the graph of a number. On a plane, each point is the graph of a number pair. To form the plane, we use two perpendicular number lines called axes. They cross at a point called the origin. The arrows show the positive directions. Slide 4

5 3.1 Introduction to Graphing Plot points associated with ordered pairs of numbers; a determine the quadrant in which a point lies. The numbers in an ordered pair are called coordinates. In (3, 4) the first coordinate (the abscissa) is 3 and the second coordinate (the ordinate) is 4. To plot we start at the origin and move horizontally to the 3. Then we move up vertically 4 units and make a dot. Slide 5

6 3.1 Introduction to Graphing Plot points associated with ordered pairs of numbers; a determine the quadrant in which a point lies. Note that (3, 4) and (4, 3) represent different points. The order of the numbers in the pair is important. We use the term ordered pairs because it makes a difference which number comes first. The coordinates of the origin are (0, 0). Slide 6

7 3.1 Introduction to Graphing Plot points associated with ordered pairs of numbers; a determine the quadrant in which a point lies. EXAMPLE 1 Plot the point ( 5, 2). The first number, 5, is negative. Starting at the origin, we move units in the horizontal direction (5 units to the left). The second number, 2, is positive. We move 2 units in the vertical direction (up). Slide 7

8 3.1 Introduction to Graphing Caution! The first coordinate of an ordered pair is always graphed in a horizontal direction and the second coordinate is always graphed in a vertical direction. Slide 8

9 3.1 Introduction to Graphing Plot points associated with ordered pairs of numbers; a determine the quadrant in which a point lies. EXAMPLE 2 In which quadrant, if any, are the points ( 4, 5), (5, 5), (2, 4), ( 2, 5), and ( 5, 0) located? The point ( 5, 0) is on an axis and is not in any quadrant. Slide 9

10 b 3.1 Introduction to Graphing Find the coordinates of a point on a graph. EXAMPLE 3 Find the coordinates of points A, B, C, D, E, F, and G. Slide 10

11 b 3.1 Introduction to Graphing Find the coordinates of a point on a graph. EXAMPLE 3 Solution Point A is 3 units to the left (horizontal direction) and 5 units up (vertical direction). Its coordinates are ( 3, 5). Slide 11

12 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. Now we begin to learn how graphs can be used to represent solutions of equations. When an equation contains two variables, the solutions of the equation are ordered pairs in which each number in the pair corresponds to a letter in the equation. Unless stated otherwise, to determine whether a pair is a solution, we use the first number in each pair to replace the variable that occurs first alphabetically. Slide 12

13 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. EXAMPLE 4 Determine whether each of the following pairs is a solution of 4q 3p = 22: (2, 7), ( 1, 6). Slide 13

14 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. EXAMPLE 4 Solution Slide 14

15 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. EXAMPLE 5 Show that the pairs (3, 7), (0, 1), and ( 3, 5) are solutions of y = 2x + 1. Then graph the three points and use the graph to determine another pair that is a solution. Slide 15

16 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. EXAMPLE 5 Solution In each of the three cases, the substitution results in a true equation. Thus the pairs are all solutions. Slide 16

17 3.1 Introduction to Graphing Determine whether an ordered pair is a solution of an c equation with two variables. EXAMPLE 5 Solution The line appears to pass through (2, 5) as well. Let s see if this pair is a solution of y = 2x + 1. Slide 17

18 3.1 Introduction to Graphing Graph of an Equation The graph of an equation is a drawing that represents all of its solutions. Slide 18

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20 Graphs of Linear Equations CHAPTER Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications Slide 2

21 3.2 Graphing Linear Equations OBJECTIVES a Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. b Solve applied problems involving graphs of linear equations. Slide 3

22 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. Equations like y = 2x + 1 and 4q 3p = 22 are said to be linear because the graph of each equation is a straight line. In general, any equation equivalent to one of the form y = mx + b or Ax + By = C, where m, b, A, B, and C are constants (not variables) and A and B are not both 0, is linear. Slide 4

23 3.2 Graphing Linear Equations To graph a linear equation: 1. Select a value for one variable and calculate the corresponding value of the other variable. Form an ordered pair using alphabetical order as indicated by the variables. 2. Repeat step (1) to obtain at least two other ordered pairs. Two points are essential to determine a straight line. A third point serves as a check. 3. Plot the ordered pairs and draw a straight line passing through the points. Slide 5

24 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. EXAMPLE 1 Graph: y = 2x. First, we find some ordered pairs that are solutions. Slide 6

25 3.2 Graphing Linear Equations Slide 7

26 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. EXAMPLE 2 Graph: y = 3x + 1 Remember: (1) Choose x. (2) Compute y. (3) Form the pair (x, y). (4) Plot the points. Slide 8

27 3.2 Graphing Linear Equations Slide 9

28 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. If x is replaced with 0 in the equation y = mx + b, then the corresponding y-value is m 0 + b, or b. Thus any equation of the form y = mx + b has a graph that passes through the point (0, b). Since (0, b) is the point at which the graph crosses the y-axis, it is called the y- intercept. Sometimes, for convenience, we simply refer to b as the y- intercept. Slide 10

29 3.2 Graphing Linear Equations y-intercept Slide 11

30 3.2 Graphing Linear Equations a Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. EXAMPLE 3 Graph and identify the y- intercept. Slide 12

31 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. EXAMPLE 3 Solution Slide 13

32 3.2 Graphing Linear Equations a Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. EXAMPLE 4 Graph and identify the y- intercept. To find an equivalent equation in the form y = mx+ b, solve for y. Slide 14

33 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. EXAMPLE 4 Solution Slide 15

34 3.2 Graphing Linear Equations Graph linear equations of the type y = mx + b and Ax + a By = C, identifying the y-intercept. EXAMPLE 4 Solution Slide 16

35 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. EXAMPLE 6 World Population The world population, in billions, is estimated and projected by where x is the number of years since That is, x = 0 corresponds to 1980, x = 12 corresponds to 1992, and so on. Slide 17

36 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. EXAMPLE 6 Solution a) Determine the world population in 1980, in 2005, and in b) Graph the equation and then use the graph to estimate the world population in c) In what year would we estimate the world population to be billion? Slide 18

37 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. EXAMPLE 6 Solution a) The years 1980, 2005, and 2030 correspond to x = 0, x = 25, and x = 50, respectively. The world population in 1980, in 2005, and in 2030 is estimated to be billion, billion, and billion, respectively. Slide 19

38 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. EXAMPLE 6 Solution b) Slide 20

39 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. EXAMPLE 6 Solution c) In 44 yr after 1980, or in 2024, the world population will be approximately billion. Slide 21

40 3.2 Graphing Linear Equations Solve applied problems involving graphs of linear b equations. Many equations in two variables have graphs that are not straight lines. Three such nonlinear graphs are shown below. We will cover some such graphs in the optional Calculator Corners throughout the text and in Chapter 11. Slide 22

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42 Graphs of Linear Equations CHAPTER Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications Slide 2

43 3.3 More with Graphing OBJECTIVES a Find the intercepts of a linear equation, and graph using intercepts. b Graph equations equivalent to those of the type x = a and y = b. Slide 3

44 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. In Section 3.2, we graphed linear equations of the form Ax + By = C by first solving for y to find an equivalent equation in the form y = mx + b. We did so because it is then easier to calculate the y-value that corresponds to a given x-value. Another convenient way to graph Ax + By = C is to use intercepts. Slide 4

45 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. Look at the graph of 2x + y = 4 shown below. The y-intercept is (0, 4). It occurs where the line crosses the y-axis and thus will always have 0 as the first coordinate. The x-intercept is ( 2, 0). It occurs where the line crosses the x-axis and thus will always have 0 as the second coordinate. Slide 5

46 3.3 More with Graphing Intercepts Slide 6

47 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. EXAMPLE 1 Consider 4x + 3y = 12. Find the intercepts. Then graph the equation using the intercepts. Slide 7

48 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. EXAMPLE 1 Solution Slide 8

49 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. EXAMPLE 1 Solution Slide 9

50 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. EXAMPLE 1 Solution Slide 10

51 3.3 More with Graphing Find the intercepts of a linear equation, and graph a using intercepts. EXAMPLE 2 Graph: y = 3x. Slide 11

52 3.3 More with Graphing Graph equations equivalent to those of the type x = a b and y = b. EXAMPLE 3 Graph: y = 3. Slide 12

53 3.3 More with Graphing Graph equations equivalent to those of the type x = a b and y = b. EXAMPLE 3 Solution Slide 13

54 3.3 More with Graphing Graph equations equivalent to those of the type x = a b and y = b. EXAMPLE 4 Graph: x = 4. Slide 14

55 3.3 More with Graphing Graph equations equivalent to those of the type x = a b and y = b. EXAMPLE 4 Solution Slide 15

56 3.3 More with Graphing Horizontal and Vertical Lines Slide 16

57 3.3 More with Graphing Graphing Linear Equations 1. If the equation is of the type x = a or y = b the graph will be a line parallel to an axis; x = a is vertical and y = b is horizontal. Slide 17

58 3.3 More with Graphing Graphing Linear Equations 2. If the equation is of the type y = mx, both intercepts are the origin, (0, 0). Plot (0, 0) and two other points. Slide 18

59 3.3 More with Graphing Graphing Linear Equations 3. If the equation is of the type y = mx + b, plot the y- intercept (0, b) and two other points. Slide 19

60 3.3 More with Graphing Graphing Linear Equations 4. If the equation is of the type Ax + By = C, but not of the type x = a or y = b, then either solve for y and proceed as with the equation y = mx + b, or graph using intercepts. If the intercepts are too close together, choose another point or points farther from the origin. Slide 20

61 3.3 More with Graphing Graphing Linear Equations Slide 21

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63 Graphs of Linear Equations CHAPTER Introduction to Graphing 3.2 Graphing Linear Equations 3.3 More with Graphing 3.4 Slope and Applications Slide 2

64 3.4 Slope and Applications OBJECTIVES a Given the coordinates of two points on a line, find the slope of the line, if it exists. b Find the slope of a line from an equation. c Find the slope, or rate of change, in an applied problem involving slope. Slide 3

65 3.4 Slope and Applications a Given the coordinates of two points on a line, find the slope of the line, if it exists. We have considered two forms of a linear equation, We found that from the form of the equation y = mx + b, we know that the y-intercept of the line is (0, b). What about the constant m? Does it give us information about the line? Slide 4

66 3.4 Slope and Applications a Given the coordinates of two points on a line, find the slope of the line, if it exists. Can you make any connection between the constant m and the slant of the line? Slide 5

67 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. Call the change in y the rise and the change in x the run. The ratio rise/run is the same for any two points on a line. We call this ratio the slope of the line. Slope describes the slant of a line. The slope of the line in the graph is given by Slide 6

68 3.4 Slope and Applications Slope Slide 7

69 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. EXAMPLE 1 Graph the line containing the points ( 4, 3) and (2, 6), and find the slope. Slide 8

70 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. EXAMPLE 1 Solution Slide 9

71 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. The slope of a line tells how it slants. A line with positive slope slants up from left to right. The larger the slope, the steeper the slant. Slide 10

72 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. A line with negative slope slants downward from left to right. Slide 11

73 3.4 Slope and Applications Given the coordinates of two points on a line, find the a slope of the line, if it exists. A horizontal line has a slope of 0. For a vertical line, the slope is undefined. Slide 12

74 3.4 Slope and Applications Determining Slope from the Equation y = mx + b The slope of the line y = mx + b is m. To find the slope of a nonvertical line, solve the linear equation in x and y for y and get the resulting equation in the form y = mx + b. The coefficient of the x-term, m, is the slope of the line. Slide 13

75 3.4 Slope and Applications b Find the slope of a line from an equation. EXAMPLE Find the slope of each line. Slide 14

76 3.4 Slope and Applications b Find the slope of a line from an equation. EXAMPLE 6 Find the slope of the line 2x + 3y = 7. Slide 15

77 3.4 Slope and Applications b Find the slope of a line from an equation. EXAMPLE 6 Solution Slide 16

78 3.4 Slope and Applications b Find the slope of a line from an equation. EXAMPLE 7 Find the slope of the line y = 5. Consider the points ( 3, 5) and (4, 5), which are on the line. The slope of any horizontal line is 0. Slide 17

79 b 3.4 Slope and Applications Find the slope of a line from an equation. EXAMPLE 8 Find the slope of x = 4. Consider the points ( 4, 3) and ( 4, 2), which are on the line. The slope of any vertical line is not defined. Slide 18

80 3.4 Slope and Applications Slope 0; Slope Not Defined Slide 19

81 3.4 Slope and Applications Find the slope, or rate of change, in an applied c problem involving slope. Numbers like 2%, 3%, and 6% are often used to represent the grade of a road, a measure of how steep a road on a hill or mountain is. For example, a 3% grade means that for every horizontal distance of 100 ft. Slide 20

82 3.4 Slope and Applications c Find the slope, or rate of change, in an applied problem involving slope. Architects and carpenters use slope when designing and building stairs, ramps, or roof pitches. Another application occurs in hydrology. When a river flows, the strength or force of the river depends on how far the river falls vertically compared to how far it flows horizontally. Slide 21

83 3.4 Slope and Applications Find the slope, or rate of change, in an applied c problem involving slope. EXAMPLE 9 Skiing Among the steepest skiable terrain in North America, the Headwall on Mt. Washington, in New Hampshire, drops 720 ft over a horizontal distance of 900 ft. Find the grade of the Headwall. Slide 22

84 3.4 Slope and Applications Find the slope, or rate of change, in an applied c problem involving slope. EXAMPLE 9 Solution Slide 23

85 3.4 Slope and Applications Find the slope, or rate of change, in an applied c problem involving slope. EXAMPLE 10 Masonry Jacob, an experienced mason, prepared a graph displaying data from a recent day s work. Use the graph to determine the slope, or the rate of change of the number of bricks he can lay with respect to time. Slide 24

86 3.4 Slope and Applications Find the slope, or rate of change, in an applied c problem involving slope. EXAMPLE 10 Solution Slide 25