The Cartesian Coordinate System
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1 The Cartesian Coordinate System The xy-plane Although a familiarity with the xy-plane, or Cartesian coordinate system, is expected, this worksheet will provide a brief review. The Cartesian coordinate system is form by two perpendicular real number lines: the x- axis, which is horizontal, and the y-axis, which is vertical. On the x-axis, the values increase from left to right, and on the y-axis, the values increase from bottom to top. Small lines, or tickmarks, are used to indicate the location of the numbers on the axis. The point at which the axes intersect is called the origin, and it corresponds to the number 0 on each axis. Each point in the Cartesian coordinate system is assigned a set of coordinates corresponding to its location relative to the x and y axes. The coordinates of the origin are x = 0 and y = 0. These are written as the ordered pair (x, y) = (0, 0). The axes divide the xy-plane into four regions, called quadrants. In the first quadrant, or quadrant I, both x and y coordinates are positive. In the second quadrant, or quadrant II, the x coordinate is negative and the y-coordinate is positive. In the third quadrant, quadrant III, both coordinates are negative. In the fourth quadrant, quadrant IV, the x-coordinate is positive while the y-coordinate is negative. Plotting Points To plot a point in the xy-plane: Start at the origin and move to the right or left to the indicated x-value. From the x-axis move up or down until the point is at the level indicated by the y-coordinate. Example: To plot the point (2, 4) in the first quadrant, from the origin move to the right two units and then up four units. To plot the point (-4, 0), move to the left 4 units from the origin. Because the y- coordinate is 0, this point is on the x-axis. To plot the point (-3, -5) in the third quadrant, move to the left 3 units from the origin and then down 5 units. Scale The xy-plane extends in all directions forever. Whenever a portion of the plane is drawn, one must decide what values of x and y to include. If a large range of values is selected on an axis, the tickmarks could represent more than one unit. The scale is the number of units between two adjacent tickmarks on an axis. This can be different for the x-axis and the y- axis, but the scale must be consistent on the entire axis. On the graphing calculator, the window settings define what part of the xy-plane will be shown. The window settings are: xmin = the smallest value shown on the x-axis xmax = the largest value shown on the x-axis xscl = the number of units between adjacent tickmarks on the x-axis ymin = the smallest value shown on the y-axis ymax = the largest value shown on the y-axis yscl = the number of units between adjacent tickmarks on the y-axis Example: Give the coordinates of each point. Pay close attention to the scale on each axis. Because there are 5 tickmarks between the origin and the value 15 on the x-axis, the scale on the x-axis is 3. Because there are 5 tickmarks between the origin and the value 50 on the y- axis, the scale on the y-axis is 10. Therefore, the points are: A = (-3, 10), B = (-10, 0), and C = (9, -20)
2 Graph Interpretation Relationships between x and y values are often represented by graphs in the xy-plane. For example, consider the following: The values on the x-axis correspond to the age of a washing machine in years; the values on the y-axis correspond to the value of the washing machine in dollars. Because the point (x, y) = (2, 300) is on the graph, after x = 2 years, the value of the washing machine is y = $300. What is the value of the washing machine after 4 years? Because the point (x, y) = (4, 100) is on the graph, the washing machine is worth y = $100 after x = 4 years. When will the value of the washing machine be $250? Because the point (x, y) = (2.5, 250) is on the graph, the value of the washing machine will be $250 after 2.5 years. Example: The graph below shows the number of staff employed by a local business for the first six years. How many staff members were employed by the company after the first year? Because the point (x, y) = (1, 10) is on the graph, after x = 1 year the company has y = 10 staff members. After how many years does the company employ 60 staff members? Because the point (x, y) = (5, 60) is on the graph, the company has y = 60 staff members after x = 5 years. Example: The graph below shows the height of cherry trees. This type of graph is called a histogram. The height of each bar represents the number (frequency) of trees with heights between the values defining the width of the bar. For example, there are 10 trees between 75 and 80 feet tall. How many trees are between 65 and 70 feet tall? Because the bar between the x values 65 and 70 is 8 units high, there are 8 black cherry trees between 65 and 70 feet tall. How many trees are more than 80 feet tall? Because there are 5 trees between 80 and 85 feet tall and there are 2 tree between 85 and 90 feet tall, there are 7 trees that are at least 80 feet tall. Example: The graph below shows the types of pets owned by the 34 students in a first grade class. The height of each bar represents the number of students with the indicated type of pet. This type of graph is called a bar graph. How many students have a goldfish? Because the green bar above the word goldfish is 6 units high, 6 of the students have a goldfish. How many of the students have either a cat or a dog? Because 8 students have a dog and 11 students have a cat, a total of 19 students have either a cat or a dog.
3 The Cartesian Coordinate System Examples (Solutions provided at bottom of page) On the set of axes provided, plot the points below and state in which quadrant or on what axis each lies. 1. A = (-1, 3) 4. D = (15, 40) 2. B = (2, -1) 5. E = (10, 0) 3. C = (0, 1) 6. F = (-10, -20) The graph below shows the number (frequency) of students in a senior class with the indicated grade point averages (GPA). 7. How many students have a GPA of 2.50? 8. What is the most common GPA in the class? 9. How many students have a GPA of 3.50 or higher? The graph to the right shows the average amount (in dollars) spent on Valentine s Day for the years 2003 to What was the average amount spent on Valentine s Day in 2004? 11. During what year(s) was the average amount spent more than $120? The graph to the right shows the fuel usage of 17 automobiles. 12. How many of the vehicles get between 10 and 15 miles per gallon? 13. How many of the vehicles get between 25 and 40 miles per gallon? The graph to the right shows the birthday months of a group of students. 14. How many of the students have a birthday in October? 15. In what month is there only one birthday? Solutions: 1. A Q2 2. B Q4 3. C y-axis 4. D Q1 5. E x-axis 6. F Q , August
4 Worksheet 7 The Cartesian Coordinate System 1. Determine in which quadrant or on which coordinate axis each point lies. a. (-2, -5) b. (-3, 1) c. (0, 5) d. (2, 7) e. (1, -5) 2. Give the coordinates of the points in the graph to the right. a. b. c. d. e. 3. Give the coordinates of the points in the graph to the right. a. b. c. d. e. 4. The graph to the right relates a car s value with its mileage. a. What is the value of a car that has been driven miles? b. What is the mileage of a car worth $4000? 5. The graph to the right shows the math scores for a certain student in grades 7 through 12. a. What was the student s math score in the 9 th grade? b. In what grade was the student s math score 83%? 6. The graph to the right shows the high temperature (in degrees Celsius) at a given location for one week. a. What was the high temperature on Monday? b. On what day(s) was the high temperature less than 23 C?
5 7. The graph to the right shows the U.S. unemployment rate (as a percent) from 1998 to a. What was the unemployment rate in 2003? b. During what year(s) was the unemployment rate higher than 7%? 8. The graph to the right shows the daily lecture attendance at a major conference for one week. a. How many people attended on Friday? b. On what day(s) did exactly 300 people attend? 9. The graph to the right gives a breakdown of Monday s cafeteria orders in a certain high school. a. How many students ordered a salad? b. What was the most popular item? 10. The graph to the right gives the distribution of final exam scores in a certain class. a. How many students scored between 60% and 70%? b. How many students scored above 80%?
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