1. Graphing Equations and Inequalities
Chapter 10
Graphing Equations and Inequalities

2. Chapter Sections
10.1 – Reading Graphs and the Rectangular Coordinate System
10.2 – Graphing Linear Equations
10.3 – Intercepts
10.4 – Slope and Rate of Change
10.5 – Equations of Lines
10.6 – Introduction to Functions
10.7 – Graphing Linear Inequalities in Two Variables
10.8 – Direct and Inverse Variation
Chapter 10 Outline

3. Reading Graphs and the Rectangular Coordinate System
§ 10.1
Reading Graphs and the Rectangular Coordinate System

4. Vocabulary
Ordered pair – a sequence of 2 numbers where the order of the numbers is important
Axis – horizontal or vertical number line
Origin – point of intersection of two axes
Quadrants – regions created by intersection of 2 axes
Location of a point residing in the rectangular coordinate system created by a horizontal (x-) axis and vertical (y-) axis can be described by an ordered pair. Each number in the ordered pair is referred to as a coordinate

5. Graphing an Ordered Pair
To graph the point corresponding to a particular ordered pair (a,b), you must start at the origin and move a units to the left of right (right if a is positive, left if a is negative), then move b units up or down (up if b is positive, down if b is negative).

6. Graphing an Ordered Pair
x-axis
y-axis
(5, 3)
5 units right
3 units up
(0, 5)
(-6, 0)
(2, -4)
(-4, 2)
(0, 0)
Quadrant I
Quadrant IV
Quadrant III
Quadrant II
origin
Note that the order of the coordinates is very important, since (-4, 2) and (2, -4) are located in different positions.

7. Vocabulary
Paired data are data that can be represented as an ordered pair
A scatter diagram is the graph of paired data as points in the rectangular coordinate system
An order pair is a solution of an equation in two variables if replacing the variables by the appropriate values of the ordered pair results in a true statement.

8. Solutions of an Equation
Example
Determine whether (3, – 2) is a solution of 2x + 5y = – 4.
Let x = 3 and y = – 2 in the equation.
2x + 5y = – 4
2(3) + 5(–2) = – (replace x with 3 and y with –2)
6 + (–10) = – (compute the products)
– 4 = – (True)
So (3, –2) is a solution of 2x + 5y = –4

9. Solutions of an Equation
Example
Determine whether (– 1, 6) is a solution of 3x – y = 5.
Let x = – 1 and y = 6 in the equation.
3x – y = 5
3(– 1) – 6 = (replace x with – 1 and y with 6)
– 3 – 6 = (compute the product)
– 9 = (False)
So (– 1, 6) is not a solution of 3x – y = 5

10. Solving an Equation
If you know one coordinate in an ordered pair that is a solution for an equation, you can find the other coordinate through substitution and solving the resulting equation.

11. Solving an Equation Example
Complete the ordered pair (4, ) so that it is a solution of –2x + 4y = 4.
Let x = 4 in the equation and solve for y.
–2x + 4y = 4
–2(4) + 4y = (replace x with 4)
–8 + 4y = (compute the product)
– y = (add 8 to both sides)
So the completed ordered pair is (4, 3).
4y = (simplify both sides)
y = (divide both sides by 4)

12. Solving an Equation Example 4x = 2 (simplify both sides)
Complete the ordered pair (__, – 2) so that it is a solution of 4x – y = 4.
Let y = – 2 in the equation and solve for x.
4x – y = 4
4x – (– 2) = (replace y with – 2)
4x + 2 = (simplify left side)
4x + 2 – 2 = 4 – (subtract 2 from both sides)
So the completed ordered pair is (½, – 2).
4x = (simplify both sides)
x = ½ (divide both sides by 4)

13. Graphing Linear Equations
§ 10.2
Graphing Linear Equations

14. Linear Equations Linear Equation in Two Variables
Ax + By = C
A, B, C are real numbers, A and B not both 0
This is called “standard form”
Graphing Linear Equations
Find at least 2 points on the line
y = mx + b crosses the y-axis at b (called “slope-intercept form”)

15. Graphing Linear Equations
Example
Graph the linear equation 2x – y = –4.
We find two ordered pair solutions (and a third solution as a check on our computations) by choosing a value for one of the variables, x or y, then solving for the other variable. We plot the solution points, then draw the line containing the 3 points.
Continued.

16. Graphing Linear Equations
Example continued
Graph the linear equation 2x – y = – 4.
Let x = 1.
Then 2x – y = – 4 becomes
2(1) – y = – (replace x with 1)
2 – y = – (simplify the left side)
– y = – 4 – 2 = – (subtract 2 from both sides)
y = (multiply both sides by – 1)
So one solution is (1, 6)
Continued.

17. Graphing Linear Equations
Example continued
Graph the linear equation 2x – y = – 4.
For the second solution, let y = 4.
Then 2x – y = – 4 becomes
2x – 4 = – (replace y with 4)
2x = – (add 4 to both sides)
2x = (simplify the right side)
x = (divide both sides by 2)
So the second solution is (0, 4)
Continued.

18. Graphing Linear Equations
Example continued
Graph the linear equation 2x – y = – 4.
For the third solution, let x = – 3.
Then 2x – y = – 4 becomes
2(– 3) – y = – (replace x with – 3)
– 6 – y = – (simplify the left side)
– y = – = (add 6 to both sides)
y = – (multiply both sides by – 1)
So the third solution is (– 3, – 2)
Continued.

19. Graphing Linear Equations
Example continued x y
(1, 6)
(0, 4)
(– 3, – 2)
Now we plot all three of the solutions (1, 6), (0, 4) and (– 3, – 2).
And then we draw the line that contains the three points.

20. Graphing Linear Equations
Example
Graph the linear equation y = x + 3.
Since the equation is solved for y, we should choose values for x.
To avoid fractions, we should select values of x that are multiples of 4 (the denominator of the fraction).
Continued.

21. Graphing Linear Equations
Example continued
Graph the linear equation y = x + 3.
Let x = 4.
Then y = x + 3 becomes
y = (4) (replace x with 4)
y = = (simplify the right side)
So one solution is (4, 6)
Continued.

22. Graphing Linear Equations
Example continued
Graph the linear equation y = x + 3.
For the second solution, let x = 0.
Then y = x + 3 becomes
y = (0) (replace x with 0)
y = = (simplify the right side)
So a second solution is (0, 3)
Continued.

23. Graphing Linear Equations
Example continued
Graph the linear equation y = x + 3.
For the third solution, let x = – 4.
Then y = x + 3 becomes
y = (– 4) (replace x with – 4)
y = – = (simplify the right side)
So the third solution is (– 4, 0)
Continued.

24. Graphing Linear Equationsx y
Example continued
(4, 6)
(0, 3)
(– 4, 0)
Now we plot all three of the solutions (4, 6), (0, 3) and (– 4, 0).
And then we draw the line that contains the three points.

25. § 10.3
Intercepts

26. Intercepts Intercepts of axes (where graph crosses the axes)
Since all points on the x-axis have a y-coordinate of 0, to find x-intercept, let y = 0 and solve for x
Since all points on the y-axis have an x-coordinate of 0, to find y-intercept, let x = 0 and solve for y

27. Intercepts Example Find the y-intercept of 4 = x – 3y Let x = 0.
Then 4 = x – 3y becomes
4 = 0 – 3y (replace x with 0)
4 = – 3y (simplify the right side)
= y (divide both sides by -3)
So the y-intercept is (0, )

28. Intercepts Example Find the x-intercept of 4 = x – 3y Let y = 0.
Then 4 = x – 3y becomes
4 = x – 3(0) (replace y with 0)
4 = x (simplify the right side)
So the x-intercept is (4,0)

29. Graph by Plotting Intercepts
Example
Graph the linear equation 4 = x – 3y by plotting intercepts.
We previously found that the y-intercept is (0, ) and the x-intercept is (4, 0).
Plot both of these points and then draw the line through the 2 points.
Note: You should still find a 3rd solution to check your computations.
Continued.

30. Graph by Plotting Intercepts
Example continued
Graph the linear equation 4 = x – 3y.
Along with the intercepts, for the third solution, let y = 1.
Then 4 = x – 3y becomes
4 = x – 3(1) (replace y with 1)
4 = x – (simplify the right side)
4 + 3 = x (add 3 to both sides)
7 = x (simplify the left side)
So the third solution is (7, 1)
Continued.

31. Graph by Plotting Interceptsx y
Example continued
Now we plot the two intercepts (0, ) and (4, 0) along with the third solution (7, 1).
(4, 0)
(7, 1)
(0, )
And then we draw the line that contains the three points.

32. Graph by Plotting Intercepts
Example
Graph 2x = y by plotting intercepts
To find the y-intercept, let x = 0
2(0) = y
0 = y, so the y-intercept is (0,0)
To find the x-intercept, let y = 0
2x = 0
x = 0, so the x-intercept is (0,0)
Oops! It’s the same point. What do we do?
Continued.

33. Graph by Plotting Intercepts
Example continued
Since we need at least 2 points to graph a line, we will have to find at least one more point
Let x = 3
2(3) = y
6 = y, so another point is (3, 6)
Let y = 4
2x = 4
x = 2, so another point is (2, 4)
Continued.

34. Graph by Plotting Interceptsx y
Example continued
(3, 6)
(0, 0)
(2, 4)
Now we plot all three of the solutions (0, 0), (3, 6) and (2, 4).
And then we draw the line that contains the three points.

35. Graph by Plotting Intercepts
Example
Graph y = 3
Note that this line can be written as y = 0·x + 3
The y-intercept is (0, 3), but there is no x-intercept!
(Since an x-intercept would be found by letting y = 0, and 0  0·x + 3, there is no x-intercept)
Every value we substitute for x gives a y-coordinate of 3
The graph will be a horizontal line through the point (0,3) on the y-axis
Continued.

36. Graph by Plotting Intercepts
Example continued x y
(0, 3)

37. Graph by Plotting Intercepts
Example
Graph x = – 3
This equation can be written x = 0·y – 3
When y = 0, x = – 3, so the x-intercept is (– 3,0), but there is no y-intercept
Any value we substitute for y gives an x-coordinate of –3
So the graph will be a vertical line through the point (– 3,0) on the x-axis
Continued.

38. Graph by Plotting Intercepts
Example continued x y
(-3, 0)

39. Vertical and Horizontal Lines
Vertical lines
Appear in the form of x = c, where c is a real number
x-intercept is at (c, 0), no y-intercept unless c = 0 (y-axis)
Horizontal lines
Appear in the form of y = c, where c is a real number
y-intercept is at (0, c), no x-intercept unless c = 0 (x-axis)

40. Slope and Rate of Change
§ 10.4
Slope and Rate of Change

41. Slope
Slope of a Line

42. Slope Example Find the slope of the line through (4, -3) and (2, 2)
If we let (x1, y1) be (4, -3) and (x2, y2) be (2, 2), then
Note: If we let (x1, y1) be (2, 2) and (x2, y2) be (4, -3), then we get the same result.

43. Slope of a Horizontal Line
For any 2 points, the y values will be equal to the same real number.
The numerator in the slope formula = 0 (the difference of the y-coordinates), but the denominator  0 (two different points would have two different x-coordinates).
So the slope = 0.

44. Slope of a Vertical Line
For any 2 points, the x values will be equal to the same real number.
The denominator in the slope formula = 0 (the difference of the x-coordinates), but the numerator  0 (two different points would have two different y-coordinates),
So the slope is undefined (since you can’t divide by 0).

45. Summary of Slope of Lines
If a line moves up as it moves from left to right, the slope is positive.
If a line moves down as it moves from left to right, the slope is negative.
Horizontal lines have a slope of 0.
Vertical lines have undefined slope (or no slope).

46. Parallel Lines
Two lines that never intersect are called parallel lines.
Parallel lines have the same slope
unless they are vertical lines, which have no slope.
Vertical lines are also parallel.

47. Parallel Lines Example
Find the slope of a line parallel to the line passing through (0,3) and (6,0)
So the slope of any parallel line is also –½

48. Perpendicular Lines
Two lines that intersect at right angles are called perpendicular lines
Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other
The product of their slopes will be –1
Horizontal and vertical lines are perpendicular to each other

49. Perpendicular Lines Example
Find the slope of a line perpendicular to the line passing through (1,3) and (2,-8)
So the slope of any perpendicular line is

50. Parallel and Perpendicular Lines
Example
Determine whether the following lines are parallel, perpendicular, or neither.
5x + y = 6 and x + 5y = 5
First, we need to solve both equations for y.
In the first equation,
y = 5x – (add 5x to both sides)
In the second equation,
5y = x (subtract x from both sides)
y = x (divide both sides by 5)
The first equation has a slope of 5 and the second equation has a slope of , so the lines are perpendicular.

51. § 10.5
Equations of Lines

52. Slope-Intercept Form Slope-Intercept Form of a line
y = mx + b has a slope of m and has a y-intercept of (0, b).
This form is useful for graphing, since you have a point and the slope readily visible.

53. Slope-Intercept Form Example
Find the slope and y-intercept of the line –3x + y = –5.
First, we need to solve the linear equation for y.
By adding 3x to both sides, y = 3x – 5.
Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept.
slope is 3
y-intercept is (0, – 5)

54. Slope-Intercept Form Example
Find the slope and y-intercept of the line 2x – 6y = 12.
First, we need to solve the linear equation for y.
– 6y = – 2x (subtract 2x from both sides)
y = x – (divide both sides by –6)
Since the equation is now in the form of y = mx + b, slope is
y-intercept is (0, –2)

55. Point-Slope Form
The slope-intercept form uses, specifically, the y-intercept in the equation.
The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line.
m is the slope
(x1, y1) is a point on the line

56. Point-Slope Form Example
Find an equation of a line with slope –2, through the point (–11, –12). Write the equation in standard form.
First we substitute the slope and point into the point-slope form of an equation.
y – (– 12) = – 2(x – (– 11))
y + 12 = – 2x – (use distributive property)
2x + y + 12 = – (add 2x to both sides)
2x + y = – 34 (subtract 12 from both sides)

57. Point-Slope Form Example
Find the equation of the line through (–4,0) and (6, –1). Write the equation in standard form.
First find the slope.
Continued.

58. Point-Slope Form Example continued
Now substitute the slope and one of the points into the point-slope form of an equation.
(clear fractions by multiplying both sides by 10)
(use distributive property)
(add x to both sides)

59. Point-Slope Form Example
Find the equation of the line passing through points (2, 5) and (–4, 3). Write the equation using function notation.
Continued.

60. Point-Slope Form
Example continued

61. Introduction to Functions
§ 10.6
Introduction to Functions

62. Vocabulary
An equation in 2 variables defines a relation between the two variables.
A set of ordered pairs is also called a relation.
The domain is the set of x-coordinates of the ordered pairs
The range is the set of y-coordinates of the ordered pairs

63. Domain and Range Example
Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)}
Domain is the set of all x-values, {4, –4, 2, 10}
Range is the set of all y-values, {9, 3, –5}

64. Domain and Range Exampley
Example
Find the domain and range of the function graphed to the right. Use interval notation.
Domain is [–3, 4]
Domain
Range is [–4, 2]
Range

65. Domain and Range Exampley
Example
Range is [– 2, )
Range
Find the domain and range of the function graphed to the right. Use interval notation.
Domain is (– , )
Domain

66. Functions Some relations are also functions.
A function is a set of order pairs that assigns to each x-value exactly one y-value.

67. Functions
Example
Given the relation {(4,9), (–4,9), (2,3), (10, –5)}, is it a function?
Since each element of the domain is paired with only one element of the range, it is a function.
Note: It’s okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (has to be assigned to ONLY one y-value).

68. Vertical Line Test
Relations and functions can also be described by graphing their ordered pairs.
Graphs can be used to determine if a relation is a function.
If an x-coordinate is paired with more than one y-coordinate, a vertical line can be drawn that will intersect the graph at more than one point.
If no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function. This is the vertical line test.

69. Vertical Line Test Exampley
Example
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

70. Vertical Line Test Exampley
Example
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since no vertical line will intersect this graph more than once, it is the graph of a function.

71. Vertical Line Test Exampley
Example
Use the vertical line test to determine whether the graph to the right is the graph of a function.
Since vertical lines can be drawn that intersect the graph in two points, it is NOT the graph of a function.

72. Vertical Line Test
Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line
Note: An equation of the form y = c is a horizontal line and IS a function.
An equation of the form x = c is a vertical line and IS NOT a function.

73. Function Notation
Specialized notation is often used when we know a relation is a function and it has been solved for y.
For example, the graph of the linear equation y = –3x + 2 passes the vertical line test, so it represents a function.
We often use letters such as f, g, and h to name functions.
We can use the function notation f(x) (read “f of x”) and write the equation as f(x) = –3x + 2.
Note: The symbol f(x) is a specialized notation that does NOT mean f • x (f times x).

74. Function Notation
When we want to evaluate a function at a particular value of x, we substitute the x-value into the notation.
For example, f(2) means to evaluate the function f when x = 2. So we replace x with 2 in the equation.
For our previous example when f(x) = –3x + 2, f(2) = –3(2) + 2 = –6 + 2 = –4.
When x = 2, then f(x) = –4, giving us the order pair (2, –4).

75. Function Notation Example
Given that g(x) = x2 – 2x, find g(– 3). Then write down the corresponding ordered pair.
g(– 3) = (– 3)2 – 2(– 3) = 9 – (– 6) = 15.
The ordered pair is (– 3, 15).

76. Graphing Linear Inequalities in Two Variables
§ 10.7
Graphing Linear Inequalities in Two Variables

77. Linear Equations in Two Variables
Linear inequality in two variables
Written in the form Ax + By < C
A, B, C are real numbers, A and B are not both 0
Could use (>, , ) in place of <
An ordered pair is a solution of the linear inequality if it makes the inequality a true statement.

78. Linear Equations in Two Variables
To Graph a Linear Inequality
Graph the related linear equality (forms the boundary line).
 and  are graphed as solid lines
< and > are graphed as dashed lines
Choose a point not on the boundary line & substitute into original inequality.
If a true statement results, shade the half-plane containing the point.
If a false statements results, shade the half-plane that does NOT contain the point.

79. Linear Equations in Two Variablesx y
Example
Graph 7x + y > –14
Graph 7x + y = –14 as a dashed line.
Pick a point not on the graph: (0,0)
(0, 0)
Test it in the original inequality.
7(0) + 0 > –14, 0 > –14
True, so shade the side containing (0,0).

80. Linear Equations in Two Variablesx y
Example
Graph 3x + 5y  –2
Graph 3x + 5y = –2 as a solid line.
Pick a point not on the graph: (0,0), but just barely
(0, 0)
Test it in the original inequality.
3(0) + 5(0) > –2, 0 > –2
False, so shade the side that does not contain (0,0).

81. Linear Equations in Two Variablesx y
Example
Graph 3x < 15
Graph 3x = 15 as a dashed line.
(0, 0)
Pick a point not on the graph: (0,0)
Test it in the original inequality.
3(0) < 15, 0 < 15
True, so shade the side containing (0,0).

82. Linear Equations in Two Variables
Warning!
Note that although all of our examples allowed us to select (0, 0) as our test point, that will not always be true.
If the boundary line contains (0,0), you must select another point that is not contained on the line as your test point.

83. Direct and Inverse Variation
§ 10.8
Direct and Inverse Variation

84. Direct Variation
y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx.
The family of equations of the form y = kx are referred to as direct variation equations.
The number k is called the constant of variation or the constant of proportionality.

85. Direct Variation
If y varies directly as x, find the constant of variation k and the direct variation equation, given that y = 5 when x = 30.
y = kx
5 = k·30
k = 1/6
y = 1 6 x
So the direct variation equation is

86. Direct Variation Example
If y varies directly as x, and y = 48 when x = 6, then find y when x = 15.
y = kx
48 = k·6
8 = k
So the equation is y = 8x.
y = 8·15
y = 120

87. Direct Variation Example
At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places.
Continued.

88. Direct Variation Example continued
We substitute our given value for the elevation into the equation.
So our equation is

89. Inverse Variation
y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that y = k/x.
The family of equations of the form y = k/x are referred to as inverse variation equations.
The number k is still called the constant of variation or the constant of proportionality.

90. Inverse Variation Example
If y varies inversely as x, find the constant of variation k and the inverse variation equation, given that y = 63 when x = 3.
y = k/x
63 = k/3
k = 63·3
k = 189
y =
189 x
So the inverse variation equation is

91. Powers of x y can vary directly or inversely as powers of x, as well.
y varies directly as a power of x if there is a nonzero constant k and a natural number n such that y = kxn
y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that

92. Powers of x
Example
The maximum weight that a circular column can hold is inversely proportional to the square of its height.
If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold.
Continued.

93. Powers of x Example continued
We substitute our given value for the height of the column into the equation.
So our equation is