1. 5 minute check 5 Click the mouse button or press the Space Bar to display the answers.

2. 5 minute check 5a

3. 4-2 Graphing Linear Functions

4. Geogebra Parallel Lines Perpendicular Lines Slope from a Graph

5. 4-2 Videos Describe positive and negative slopes In this lesson you will learn to describe positive and negative slopes by using directed measurements. Standards: 8.EE.B.6 Derive y=mx using similar triangles In this lesson you will learn to derive the equation y=mx by using similar triangles. Standards: 8.EE.B.6 Derive y=mx+b using similar triangles In this lesson you will learn to derive the equation y=mx+b by using similar triangles. Standards: 8.EE.B.6

6. Define a linear function In this lesson you will learn to define linear functions by comparing graphs. Standards: 8.F.A.3 Analyze graphs of y = x and y = mx In this lesson you will learn to define linear functions by analyzing graphs of the forms y = x and y = mx. Standards: 8.F.A.3 Analyze Tables of y = x and y = mx In this lesson you will learn to define linear functions by analyzing tables of the forms y = x and y = mx. Standards: 8.F.A.3 Analyze graphs and tables of y = mx + b In this lesson you will learn to define linear functions by analyzing graphs and tables of the form y = mx + b. Standards: 8.F.A.3

7. Recognize the charactertistics of a linear function In this lesson you will learn how to recognize a linear function by examining the four representations of a function. Standards: 8.F.A.3

8. Video Tutor Help Graph using slope intercept form Brain Pop Slope and intercept Linear functions Graphing Functions of the Form y=mx Khan Academy 4-2 Graphing Linear Functions Course 3 Slope-Intercept Form Linear equations of the form y = mx+b can describe any non-vertical line in the Cartesian plane. The constant m determines the line's slope, and the constant b determines the y intercept and thus the line's vertical position. 4-2 Graphing Linear Functions Course 3 Linear Equations in the Real World Linear equations can be used to solve many types of real-world problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its behavior. Unfortunately, ace Algebra student A.V. Geekman ends up in hot water anyway. 4-2 Graphing Linear Functions Course 3 Solving Problems with Linear Equations How do we create linear equations to solve real-world problems? This video explains the process.

9. Make lines from right triangles In this lesson you will learn how to make a straight line by using right triangles. Standards: 8.EE.B.6 Find coordinates of points on a straight line In this lesson you will learn to determine coordinates of points on a straight line by using similar triangles. Standards: 8.EE.B.6 Describe a line with a unique slope In this lesson you will learn to describe a straight line by naming its unique slope. Standards: 8.EE.B.6 Find the slope of a line on the coordinate plane In this lesson you will learn to find the slope of a line on a coordinate plane by drawing a right triangle. Standards: 8.EE.B.6

10. Describe positive and negative slopes In this lesson you will learn to describe positive and negative slopes by using directed measurements. Standards: 8.EE.B.6 Derive y=mx using similar triangles In this lesson you will learn to derive the equation y=mx by using similar triangles. Standards: 8.EE.B.6 Derive y=mx+b using similar triangles In this lesson you will learn to derive the equation y=mx+b by using similar triangles. Standards: 8.EE.B.6

11. Understand a function as a type of relation In this lesson you will learn what a function is by evaluating examples of relations. Standards: 8.F.A.1 Define a function by looking at its parts In this lesson you will learn the definition of a function by breaking down its parts. Standards: 8.F.A.1 Determine whether a set of points plotted on a graph is a function In this lesson you will learn how to tell if a set of points represents a function by looking at points plotted on a graph. Standards: 8.F.A.1 Determine whether a graph is a function In this lesson you will learn how to tell if a line on a graph represents a function by mapping inputs to outputs. Standards: 8.F.A.1 Determine whether a set of ordered pairs represents a function In this lesson you will learn to tell if a set of ordered pairs represents a function by matching the x-values to the y-values. Standards: 8.F.A.1

12. Identify a function In this lesson you will learn how to identify a function by investigating a real-world example. Standards: 8.F.A.1 Identify function properties In this lesson you will learn how to identify function properties by examining the input and output of real world examples. Standards: 8.F.A.1 Identify a function from a graph In this lesson you will learn identify a function by analyzing its graph. Standards: 8.F.A.1 Compare two functions by analyzing an equation and a graph In this lesson you will learn how to compare two functions by analyzing an equation and a graph. Standards: 8.F.A.2

13. Compare two functions by analyzing an equation and a table In this lesson you will learn how to compare two functions by analyzing an equation and a table. Standards: 8.F.A.2 Compare two functions by analyzing an equation and a verbal description In this lesson you will learn how to compare two functions by analyzing an equation and a verbal description. Standards: 8.F.A.2

14. Define a linear function In this lesson you will learn to define linear functions by comparing graphs. Standards: 8.F.A.3 Analyze graphs of y = x and y = mx In this lesson you will learn to define linear functions by analyzing graphs of the forms y = x and y = mx. Standards: 8.F.A.3 Analyze Tables of y = x and y = mx In this lesson you will learn to define linear functions by analyzing tables of the forms y = x and y = mx. Standards: 8.F.A.3 Analyze graphs and tables of y = mx + b In this lesson you will learn to define linear functions by analyzing graphs and tables of the form y = mx + b. Standards: 8.F.A.3

15. Recognize the charactertistics of a linear function In this lesson you will learn how to recognize a linear function by examining the four representations of a function. Standards: 8.F.A.3 Describe the rate of change of a linear function In this lesson you will learn how to describe the rate of change in a linear function by using the four representations. Standards: 8.F.A.3 Identify a linear function by analyzing characteristics of a linear f... In this lesson you will learn how to identify a linear function relation by analyzing characteristics of a linear function. Standards: 8.F.A.3 Locating and describing the y-intercept of a linear function In this lesson you will learn how to describe the y-intercept of a linear function by examining the four representations of a function. Standards: 8.F.A.3

16. Interpret linear relationships in word problems In this lesson you will learn to interpret linear relationships by analyzing word problems. Standards: 8.F.B.4 Interpret direction by analyzing graphs Interpret direction in linear relationships by analyzing graphs Standards: 8.F.B.4 Determine rate of change in linear relationships In this lesson you will learn to determine rate of change in linear relationships by comparing slopes in a graph. Standards: 8.F.B.4 Determine initial value in a linear relationship In this lesson you will learn to interpret the initial value in a linear relationship by analyzing graphs. Standards: 8.F.B.4

17. Define and interpret linear relationships in tables In this lesson you will learn to define and interpret linear relationships by analyzing tables. Standards: 8.F.B.4 Define a linear function In this lesson you will learn to define a linear function by constructing an equation. Standards: 8.F.B.4 Construct a function by solving for unknown information In this lesson you will learn to construct a linear function by solving for unknown information. Standards: 8.F.B.4

18. Determining the constant rate of change In this lesson you will learn calculate the rate of change of a linear function by examining the four representations of a function. Standards: 8.F.B.4 Determining the y-intercept In this lesson you will learn how to determine the initial value of a linear function by interpreting all four representations of a function. Standards: 8.F.B.4 Create equation, table, and graph from a situation In this lesson you will learn how to create an equation, table, and graph by using a contextual situation. Standards: 8.F.B.4 Interpreting slope and y-intercept in context In this lesson you will learn how to interpret the rate of change and the y-intercept in context by examining each of the four representations. Standards: 8.F.B.4 Construct a linear function In this lesson you will learn how to construct a linear function by calculating the slope and y-intercept given any representation. Standards: 8.F.B.4

19. Compare distance-time graphs with distance-time equations In this lesson you will learn how to compare distance-time graphs with distance- time equations. Standards: 8.F.A.2 Construct linear functions from tables In this lesson you will learn how to construct linear functions from tables. Standards: 8.F.B.4 Construct linear functions from a graph In this lesson you will learn how to construct linear functions from a graph. Standards: 8.F.B.4

20. Worksheets 4-2 Note-Taking Guide 4-2 Practice 4-2 Guided Problem Solving

21. Vocabulary Practice Chapter 4 Vocabulary (Electronic) Flash Cards

22. Additional Lesson Examples 4-2 Step-by-Step Examples

23. Lesson Readiness 4-2 Problem of the Day 4-2 Lesson Quiz

24. Understanding Slope 1.Find the slope of line s. 2.Find the slope of line t. 3.The data in the table is linear. Find the slope. 4. Graph Exercise 3 data and the line. LESSON 4-1 undefined 3 2323 Lesson Quiz

29. Finding Slope and y-intercept

30. continued

40. Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

44. If you know the slope of a line and the y-intercept, you can write an equation that describes the line. Step 1 If a line has a slope of m and the y-intercept is b, then (0, b) is on the line. Substitute these values into the slope formula.

45. Step 2 Solve for y: Simplify the denominator. Multiply both sides by x. Add b to both sides. +b mx = y– b mx + b = y, or y = mx + b

46. Slope-Intercept Form Slope-Intercept Form: y = mx + b slope Y-intercept When the equation is written in this form, m is the slope, and b is the y-intercept.

47. You have seen that you can graph a line if you know two points on the line. Another way is to use the point that contains the y-intercept and the slope of the line.

48. Additional Example 1A: Graphing by Using Slope and y-intercept Graph the line given the slope and y-intercept. y intercept = 4 Step 1 The y-intercept is 4, so the line contains (0, 4). Plot (0, 4). Step 3 Draw the line through the two points. Run = 5 Rise = –2 Step 2 Slope = Count 2 units down and 5 units right from (0, 4) and plot another point. y

49. Not all linear equations describe functions. The graphs of some linear equations are vertical lines, which do not pass the vertical line test. Helpful Hint

53. Example 5-1a State the slope and the y-intercept of the graph of the equation of Answer: The slope of the graph is and the y-intercept is –5. Write the equation in the form Find Slopes and y-intercepts of Graphs

54. Example 5-2a Answer: The slope of the graph is –2 and the y-intercept is 8. Write the original equation. State the slope and the y-intercept of the graph of the equation of Subtract 2x from each side. Simplify. Write the equation in the form Find Slopes and y-intercepts of Graphs

55. Example 6-1a State the slope and the y -intercept of the graph of. Write the original equation. Answer: The slope of the graph is, and the y -intercept is 3. Find the Slope and y-Intercept

56. Example 6-3a Graph using the slope and y -intercept. Step 1 Find the slope and y -intercept. Write the original equation. Subtract 3x from each side to write in slope- intercept form. Graph an Equation Graphing Functions of the Form y = mx + b

57. Example 6-3a Step 2 Graph the y -intercept point at ( 0, 9 ) Step 3 Write the slope –3 as. Use it to locate a second point on the line. (0, 9)

58. Example 6-3a (0, 9) Another point on the line is at (1, 6). Step 4Draw a line through the two points. (1, 6)

60. Graph the linear function y = 3x + 1. Input RuleOutput Ordered Pair x3x + 1y(x, y) 0 1 –1 3(0) + 1 3(1) + 1 3(–1) + 1 1 4 –2 (0, 1) (1, 4) (–1, –2) Check It Out! Example 1 Make a table. Substitute positive, negative, and zero values for x.

61. Check It Out! Example 1 Continued Graph the linear function y = 3x + 1. Plot each ordered pair on the coordinate grid. Then connect the points with a line. x y 0 –2 –4 24 2 4 –2 –4 (0, 1) (1, 4) (–1, –2)

62. The fastest-moving tectonic plates on Earth move apart at a rate of 15 centimeters per year. Scientists began studying two parts of these plates when they were 30 centimeters apart. Write a linear function that describes the movement of the plates over time. Then make a graph to show the movement over 4 years. Additional Example 2: Earth Science Application Let x represent the input and y represent the output. The function is y = 15x + 30, where x is the number of years and y is the total distance apart the two plates are.

63. Additional Example 2 Continued InputRuleOutput 15(x) + 30 x 0 2 4 15(0) + 30 15(2) + 30 15(4) + 30 y 30 60 90 Multiply the input by 15 and then add 30.

64. Additional Example 2 Continued Graph the ordered pairs (0, 30), (2, 60), and (4, 90) from your table. Connect the points with a line. x Distance (cm) Years

65. Number ofTotal Cost Gallons 1$2.19 2$4.38 3$6.57 4$8.76 Juice costs $2.19 per gallon. The total cost of g gallons is a function of the price of a single gallon. Make a table and a graph. Graphing Linear Functions LESSON 4-2 You cannot buy part of a container, so the data are discrete. Use points for each input value. Connect the points with a dashed line. Additional Examples

69. Amber earns $7 per hour. Make a table to describe Amber’s earnings (output) as a function of hours she works (input). Graph the function. Graphing Linear Functions LESSON 4-2 Input (h)012345 Output ($)0714212835 Additional Examples

70. Example 5-3a Graph using the slope and y-intercept. Step 1 Find the slope and y-intercept. Step 2 Graph the y-intercept (0, 2). (0, 2) y-intercept Graph an Equation

71. Example 5-3b Step 4 Draw a line through the two points. Answer: right 3 up 2 change in y: up 2 units change in x: right 3 units Step 3 Use the slope to locate a second point on the line. (0, 2)

72. State the slope and the y -intercept of the graph of. Example 6-2a Add 4x to each side. Write the original equation. Simplify. Divide each side by 5. Write and Equation in Slope-Intercept Form

75. Lesson Review: Part I Graph the linear functions. 1. y = 3x – 4 2. y = –x + 4 3. y = 2 y = 3x – 4 y = –x +4 y = 2

76. Partner Share! Example 4 Continued b. Identify the slope and y-intercept and describe their meanings. The y-intercept is 200. This is the cost for 0 people, or the initial fee of $200. The slope is 18. This is the rate of change of the cost: $18 per person. c. Find the cost of catering an event for 200 guests. y = 18m + 200 = 18(200) + 200 = 3800 Substitute 200 for m in the equation. The cost of catering for 200 people is $3800.

77. To sell at a particular farmers’ market for a year, there is a $100 membership fee. Then you pay $3 for each hour that you sell at the market. However, if you were a member the previous year, the membership fee is reduced to $50. The red line shows the total cost if you are a new member. The blue line shows the total cost if you are a returning member. Parallel Lines

78. These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect.

80. Perpendicular lines are lines that intersect to form right angles (90 °). Perpendicular Lines

81. Identify which lines are perpendicular: y = 3; x = –2; y = 3x;. The graph given by y = 3 is a horizontal line, and the graph given by x = –2 is a vertical line. These lines are perpendicular. y = 3 x = –2 y =3x The slope of the line given by y = 3x is 3. The slope of the line described by is. Additional Example 3: Identifying Perpendicular Lines

82. Identify which lines are perpendicular: y = 3; x = –2; y = 3x;. y = 3 x = –2 y =3x Additional Example 3 Continued These lines are perpendicular because the product of their slopes is –1.

83. Lesson Review: Part I Write the equation that describes each line in the slope-intercept form. 1. slope = 3, y-intercept = –2 y = 3x – 2 2. slope = 0, y-intercept = y = 3. slope =, (2, 7) is on the line y = x + 4