MATH-A 100: Rectangular Coordinate System

Math A-100 IUSB Sec. 3.1 detailed notes – Week 5 Rectangular Coordinate System (a.k.a. graphing points) Chapter 3 is all about graphing. Section 3.1 starts out with the rectangular coordinate system, which we usually call a “graph”. That is actually not correct: the “graph” is the dots and lines you draw on it, the “grid” is the thing you draw on. There are 7 parts to the grid: 1) The x-axis, also called a. The horizontal variable b. The Input c. The Independent variable d. The domain e. The explanatory variable 2) The y-axis, also called a. The vertical variable b. The Output c. The Dependent variable d. The Range e. The Response variable 3) Quadrant I – in the upper right-hand corner, both the x and y values are positive here 4) Quadrant II – in the upper left-hand corner, the x values are negative and the y values are positive here 5) Quadrant III – in the lower left-hand corner, both x and y values are negative here 6) Quadrant IV – in the lower right-hand corner, the x values are positive while the y values are negative 7) The Origin - the center spot where the x-axis and y-axis cross, its coordinates are (0,0) NOTE: when listing the quadrant, you MUST use the Roman Numerals. They are each quadrant’s official name, i.e. Quadrant III, instead of Quadrant 3. It would be similar to using a nickname instead of the real name of a person. Writing Ordered pairs In real-life, we use the grid to chart data in which each x value has a corresponding y value, and write these relationships as ordered pairs (a.k.a. sets of coordinates, or just coordinates) in the form of (x, y), i.e. the x value listed first , then the y-value. Example 1: What list the order pair for a dot in Quadrant II with a x-value of -4 and a y-value of 6. X = -6 & y = 6  (x, y)  (-4, 6) NOTE: you must include the ( ). If you list a point without them, such as 5, 3 it would be interpreted as 2 answers: x = 5 and x = 3. Plotting points If you are given an ordered pair and asked to plot the point, go on the x-axis to the value of the x-coordinate, then go either up or down until you get to the value of the y-coordinate. Example 1: Plot the point (6, -9) x-value = 6, so go to 6 on the x-axis y-value = 9, so go straight down until you get to 9 on the y-axis Example 1: Plot the point (-7, 4) x-value = -7, so go to -7 on the x-axis y-value = 4, so go straight up until you get to 4 on the y-axis NOTE: make sure you start on the x-axis. If you get them switched, i.e. starting on the y-axis, your point may be very far off, i.e. (-7,4) is in Quadrant III, which is a MUCH different place than (4, -7) in Quad IV. Interpreting what an ordered pair means You will be asked to interpret what an ordered pair means. For example, (5, 7) which means that the y value is 7 when the x value is 5. HOWEVER, you must also use the specific details from that grid/question. Example 1: You have a graph in which the x variable is the number of hours spent studying for a test and the y variable is score on the test What does the ordered pair (3, 80) mean? x = number of hours studying, so x = 3 means 3 hours spent studying y = score on the test, so y = 80 means a test score of 80 the ordered pair (3, 80) means if you studied for 3 hours, you can expect a test score of 80 Example 2: You have a graph in which the x variable is how old your car is in years and the y variable is the value of the car in thousands of $$. What does the ordered pair (4, 12) mean? x = how old your car is, so x = 4 means your car is 4 years old y = value in thousands of $$, so y = 12 means your car is worth $12,000 the ordered pair (4, 12) means that your 4 year old car is worth $12,000 Example 3: You have a graph in which the x variable is the number of years since 1960 and the y variable is the average household income in thousands of $$. What does the ordered pair (5, 7) mean? x = number of years since 1960, so x = 5 means 5 years since 1960, so it is 1965 y = household income in thousands of $$, so y = 7 means a household income of $7000 the ordered pair (5, 7) means that in 1965, the average household income was $7000 Determining if you have a solution Recall from earlier that a “solution” in math is simply a value that makes an equation true, i.e. a number that works Here in Chapter 3, we have equations with 2 variables, “x” & “y”, not just one. So, for an ordered pair to be a “solution”, both values must work. Example 1: Determine which, if any, of these ordered pairs is a solution for 2x + 3y = 12 (0, 6) , (-3, -2), (3, 2) Plug in the values and see if you can a result that makes sense: (0, 6)  2x + 3y = 12 becomes 2(0) + 3(6) = 12  18 = 12??? Not true  NOT a solution (-3, -2)  2x + 3y = 12 becomes 2(-3) + 3(-2) = 12  -12 = 12??? Not true  NOT a solution (3, 2)  2x + 3y = 12 becomes 2(3) + 3(2) = 12  12 = 12!! TRUE  IS a solution Finding a solution when given one coordinate of tan ordered pair You can determine a solution if given only one coordinate of an ordered pair by plugging the value into your equation and solving for the missing coordinate. MML will give you several different scenarios to work with. NOTE: MML will dress them up differently, but in all cases they are giving you the value of either “x” or “y”. Example 1: when given the value for a variable Determine the solution for 2x + 3y = 12 when x = -3 Plug in -3 for “x” and solve for “y”: 2x + 3y = 12 becomes 2(-3) + 3y = 12  -6 + 3y = 12 Solve for y: -6 + 3y = 12  3y = 18  y = 6  solution = (-3, 6) Example 2: when given the value for a variable Determine the solution for 2x + 3y = 12 for (___, - 4) Here they are stating that the y-coordinate = - 4, so plug in -4 for “y” and solve for “x”: 2x + 3y = 12 becomes 2x + 3(- 4) = 12  2x + -12 = 12 Solve for x: 2x + -12 = 12  2x = 24  x = 12  solution = (12, - 4) Example 3: when given the value for a variable Determine the solution for 2x + 3y = 12 when x 9 y Here they are stating that the x-coordinate = 9, so plug in 9 for “x” and solve for “y”: 2x + 3y = 12 becomes 2(9) + 3y = 12  18 + 3y = 12 Solve for x: 18 + 3y = 12  3y = -6  y = -2  solution = (9, -2) NOTE: Often MML will ask for the answer as an ordered pair, so if you answer “y = -2”, MML will mark it incorrect Videos that may help: Plotting Points on the Coordinate Plane http://www.youtube.com/watch?v=s7NKLWXkEEE Graphing by Plotting Points http://www.youtube.com/watch?v=OmZ9O1xM0KQ http://www.youtube.com/watch?v=EqcpY98k-AM