Lesson 8: Graphs and Graphing Linear Equations
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1 A critical skill required for the study of algebra is the ability to construct and interpret graphs. In this lesson we will learn how the Cartesian plane is used for constructing graphs and plotting data. We will interpret the points and behavior of a graph with respect to the input and output variables. We will also learn the rules/guidelines for constructing good graphs so that the graphs we create can be easily read and understood. Lesson Topics Section 8.1: The Cartesian plane Input and Output Variables Ordered pairs Plotting points Quadrants Section 8.2: Constructing good graphs from Data Section 8.3: Linear Equations Two Variables Section 8.4: Graphing linear equations by plotting points Section 8.5: Intercepts Section 8.6: Horizontal and Vertical Lines
2 Lesson 8 Notes
3 Name: Date: Mini-Lesson 8 Section 8.1: The Cartesian Plane In this chapter, we will begin looking at the relationships between two variables. Typically one variable is considered to be the INPUT, and the other is called the OUTPUT. The input is the value that is considered first, and the output is the value that corresponds to or is matched with the input. The input/output designation may represent a cause/effect relationship, but that is not always the case. Ordered Pairs Example 1: Ordered Pairs (input value, corresponding output value) Input Output Ordered Pairs (input, output) (0, 4) ( 2, 6) Example 2: The Rectangular Coordinate System (Cartesian Coordinate System)
4 Mini-Lesson Plot and label the points. A. ( 4, 2) B. (3, 8) C. (0, 5) D. ( 6, 4) E. (5, 0) F. (2, 8) G. (0, 0) Quadrants Quadrant Coordinates I (+, +) II (, +) III (, ) IV (+, ) You Try Plot and label the points. A. (6, 3) B. (1, 9) C. ( 4, 0) D. ( 2, 8) E. (0, 5)
5 Mini-Lesson Section 8.2: Constructing a Graph from Data Criteria for a Good Graph 1. The horizontal axis should be properly labeled with the name and units of the input variable. 2. The vertical axis should be properly labeled with the name and units of the output variable. 3. Use an appropriate scale. Start at or just below the lowest value. End at or just above the highest value. Scale the graph so the adjacent tick marks are equal distance apart. Use numbers that make sense for the given data set. The axes meet at (0,0) Use a // between the origin and the first tick mark if the scale does not begin at All points should be plotted correctly, and the graph should make use of the available space. Example 1: The table below shows the total distance (including reaction time and deceleration time) it takes a car traveling at various speeds to come to a complete stop. Speed (miles per hour) Stopping Distance (ft)
6 Mini-Lesson You Try 2. Consider the following data set. Elapsed time (seconds) Height of Golf Ball (feet) a. What is the input variable? b. What was the height of the ball after 3 seconds? c. After how many seconds was the ball 77 feet in the air? d. In a complete sentence, interpret the meaning of the ordered pair (1, 59). e. Construct a good graph of this data.
7 Mini-Lesson Section 8.3: Linear Equations - Two Variables Linear Equations Example 1: Verify that the ordered pairs below satisfy the equation y = 2x + 3. ( 2, 1) (0, 3) (1, 5)
8 Mini-Lesson Example 2: Verify that the ordered pairs below satisfy the equation 3x + 2y = 6. ( 2, 6) (0, 3) (2, 0) You Try 3. Verify that the ordered pairs below satisfy the equation y = 3x 2. ( 2, 8) (3, 7) (0, 2)
9 Mini-Lesson Section 8.4: Graphing Linear Equations - Two Variables Graphing Linear Equations by Plotting Points Step 1: Choose two or more values for the input variable, and list them in a table of values. Step 2: Substitute each input value into the equation and compute the corresponding output values. List these values in the table. Step 3: Write each input-output pair in the table as an ordered pair. Step 4: Plot the ordered pairs, connect them, and extend the line beyond the points to show that the pattern continues. Example 1: Graph the equation y = 3x 2 x y Ordered Pair Example 2: Graph the equation x y Ordered Pair
10 Mini-Lesson Example 3: Graph the equation 4x + 2y = 10 x y Ordered Pair 4. Graph the equation You Try x y Ordered Pair
11 Mini-Lesson Section 8.5: Intercepts Vertical and Horizontal Intercepts The vertical intercept (y-intercept) is the point at which the graph crosses the vertical axis. The input value of the vertical intercept is always The coordinates of the vertical intercept will be To determine the vertical intercept: The horizontal intercept (x-intercept) is the point at which the graph crosses the horizontal axis. The output value of the horizontal intercept is always The coordinates of the horizontal intercept will be To determine the horizontal intercept: Example 1: Determine the vertical and horizontal intercepts for y = 3x 2. Example 2: Determine the vertical and horizontal intercepts for 4x 2y = 10.
12 Mini-Lesson You Try 5. Complete the table below. Write the intercepts as ordered pairs. Equation Vertical Intercept Horizontal Intercept y = 24 6x 5x 3y = 30
13 Mini-Lesson Section 8.6: Horizontal and Vertical Lines Horizontal Lines y = b, where b is a real constant Example 1: Graph the equation y = 2 x y Ordered Pair Vertical Lines x = k, where k is a real constant Example 2: Graph the equation x = 3 x y Ordered Pair You Try 6. a. Graph the equation y = 2 b. Graph the equation x = 4
14 Mini-Lesson
15 Name: Date: Lesson 8 Practice Problems Skills Practice 1. Plot and label the points. A. (8, 2) B. (0, 0) C. (0, 5) D. (10, 10) E. ( 4, 4) F. ( 9, 1) G. ( 5, 0) H. (2, 8) 2. Give the coordinates of each of the points shown below. A. B. C. D. E. F.
16 Practice Problems 3. Identify the graph that best represents the speed of a car coming to a stop at a red light. a. b. c. 4. Identify the graph that best represents the height of an arrow that has been shot straight up in the air, and lands on the ground. a. b. c. 5. Identify the graph that best represents the distance traveled by a car driving at a constant speed. a. b. c.
17 Practice Problems 6. Which of the following ordered pairs satisfy the equation y = 2x 4. Circle all that apply, and show all supporting work (9, 22) (6, 5) ( 9, 14) (2, 0) ( 4, 0) 7. Which of the following ordered pairs satisfy the equation 3x 2y = 8. Circle all that apply, and show all supporting work (2, 1) ( 4, 0) (1, 8) ( 2, 7) ( 16, 8) 8. Which of the following ordered pairs satisfy the equation y = 1 x. Circle all that apply, and show all supporting work ( 7, 8) (0, 1) (3, 2) ( 1, 0) ( 20, 21) 9. Which of the following ordered pairs satisfy the equation y = 2x. Circle all that apply, and show all supporting work (6, 12) ( 1, 2) (4, 8) (0, 2) (0, 0)
18 Practice Problems 10. Graph the equation y 4x 2. x y Ordered Pair 11. Graph the equation 2 y x 3. 5 x y Ordered Pair 12. Graph the equation y = 3 x. x y Ordered Pair
19 Practice Problems 13. Graph the equation 4x 2y = 12. x y Ordered Pair 14. Graph the equation x y = 4. x y Ordered Pair 15. Graph the equation y = x. x y Ordered Pair
20 Practice Problems Graph the equation y x. 3 x y Ordered Pair 17. Graph the equation y = 4. x y Ordered Pair 18. Graph the equation x = 3 x y Ordered Pair
21 Practice Problems 19. Complete the table below. Write the intercepts as ordered pairs. Equation Vertical Intercept Horizontal Intercept y = 5x 3 y = 4 x y = 4x y = 3 5x + 6y = 12 3x 4y = 24 x 2y = 8 x = 5
22 Practice Problems 20. Consider the linear equation y = 5 2x a. Enter this linear equation into your graphing calculator. Use your graphing calculator to complete the table below x y b. Use your graphing calculator to sketch the graph of y = 5 2x. Use the standard viewing window (ZOOM 6) Xmin= 10, Xmax=10, Ymin= 10, Ymax=10. Draw what you see on your calculator screen. c. Use your graphing calculator to sketch the graph of y = 5 2x. Use viewing window Xmin= 0, Xmax= 3, Ymin= 0, Ymax= 5. Draw what you see on your calculator screen.
23 Practice Problems 21. Consider the equation y = 2x 2 + 9x 51. a. Enter this equation into your graphing calculator. Use your graphing calculator to complete the table below x y b. Use your graphing calculator to sketch the graph of y = 2x 2 + 9x 51. Use the viewing window Xmin= 10, Xmax=10, Ymin= 70, Ymax=200. Draw what you see on your calculator screen Consider the equation y = 6 x. 3 a. Enter this equation into your graphing calculator. Use your graphing calculator to complete the table below. Round to the nearest hundredth as needed. x y b. Use your graphing calculator to sketch the graph of this linear equation. Use the viewing window Xmin= 0, Xmax=9, Ymin=0, Ymax=6. Draw what you see on your calculator screen.
24 Practice Problems Applications 23. The graph below shows Sally s distance from home over a 25 minute time period. a. What is the input variable? b. What are the units of the input variable? c. What is the output variable? d. What are the units of the output variable? e. Sally is 4 miles from home after minutes. f. After 15 minutes, Sally is miles from home. g. Interpret the meaning of the ordered pair (10, 12). h. Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. i. Identify the horizontal intercept. Write it as an ordered pair and interpret its meaning in a complete sentence.
25 Practice Problems 24. The graph below shows the number of calories burned while riding a stationary bike. a. What is the output variable? b. Interpret the meaning of the ordered pair (8, 32). c. calories are burned in 10 minutes. d. 60 calories are burned in minutes. e. calories are burned in 16 minutes. f. 100 calories are burned in minutes. g. Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence.
26 Practice Problems 25. Consider the following data set. Years Since 1980 Sales (in millions of dollars) a. What is the input variable? b. What is the output variable? c. What were the sales in 2001? d. In what year did sales total $1,280,000? e. In a complete sentence, interpret the meaning of the ordered pair (10, 1.91). f. Use the values in the table to construct a properly scaled and labeled graph of the data.
27 Practice Problems 26. The following data set gives the value of a car over time. Years since purchase Value in Dollars 0 20, , , , , ,883 a. What was the purchase price of the car? b. After one year the car will be worth what percent of its original value? c. After five years the car will be worth what percent of its original value? d. Use the values in the table to construct a properly scaled and labeled graph of the data.
28 Practice Problems 27. A pebble falls from a bridge into the river below. Time (seconds) Height above the water (feet) a. What is the input variable? b. What is the output variable? c. How far did the pebble fall during the first second? d. In a complete sentence, interpret the meaning of the ordered pair (2, 68). e. Use the values in the table to construct a properly scaled and labeled graph of the data.
29 Practice Problems 28. Jordan is saving money for emergencies (or a trip to Europe). She has $420 under her mattress, and is adding $60 to it each week. a. Let A represent the total amount of money under her mattress, and w represent the number of weeks. Write an algebraic equation to represent this situation. b. Use the equation in part a. to complete the table below. w A c. Interpret the meaning of the ordered pair (18, 1500). d. Use the values in the table to construct a properly scaled and labeled graph of the linear equation found in part a.
30 Practice Problems 29. Jill is planning to sell bottled water at the local carnival. She buys 10 packages of water (240 bottles) for $66 and plans on selling the bottles for $1.50 each. Jill s profit, P in dollars, from selling b bottles of water is given by the formula P = 1.50b 66. a. Use your graphing calculator to complete the table below. b P b. Interpret the meaning of the ordered pair (84, 60). c. Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. d. Determine the horizontal intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. e. Use your graphing calculator to generate a graph of this linear equation. Use the values in the table to determine your viewing window. In the space below, sketch what you see on your calculator screen and write down the viewing window you used. Xmin = Xmax = Ymin = Ymax=
31 Practice Problems Extension 30. Plot and label the points. A. ( 800, 1.8) B. (550, 0.2) C. (180, 0) D. (0, 1.5) E. (425, 0.4) F. ( 950, 1) 31. The graph below shows the distance traveled by a car. Draw a graph to represent the speed of the car during the same time period. 32. The graph below shows the speed of a car. Draw a graph to represent the distance traveled by the car during the same time period
32 Practice Problems 33. Draw a graph to represent each situation. a. The height above the ground of a child swinging on a swing. b. Bill is walking to school when he realizes that he forgot his math book. He runs home to get it, and then jogs to school. c. The speed of a car stuck morning traffic.
33 Name: Date: Lesson 8 Assessment 1. Complete the table below. Show all work. Write the intercepts as ordered pairs. Equation Vertical Intercept Horizontal Intercept y = 2 5x Ordered Pair: Ordered Pair: 3x y = 9 Ordered Pair: Ordered Pair: 2. John is a door to door vacuum salesman. His weekly salary is given by the linear equation S = v, where v is the number of vacuums sold. Determine the vertical intercept of this linear equation. Write it as an ordered pair and interpret its meaning in a complete sentence.
34 Assessment 3. The maximum heart rate is the highest heart rate achieved during maximal exercise. In general, you get the most benefits and reduce the risks when you exercise near your target heart rate. Usually this is when your exercise heart rate (pulse) is about 80% percent of your maximum heart rate. For adults 19 years of age and older, the formula T = a, gives the target heart rate, T, in beats per minute, for a person who is a years of age. a. Complete the table below. Age (years) Target Heart Rate (bpm) b. In a complete sentence, interpret the meaning of the ordered pair (25, 156). c. Use the values in the table to construct a properly scaled and labeled graph of this linear equation.
Lesson 8 Practice Problems
Name: Date: Lesson 8 Skills Practice 1. Plot and label the points. A. (8, 2) B. (0, 0) C. (0, 5) D. (10, 10) E. ( 4, 4) F. ( 9, 1) G. ( 5, 0) H. (2, 8) 2. Give the coordinates of each of the points shown
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