Section 3.1 Reading graphs and the Rectangular Coordinate System
Learning objectives Read bar & line graphs Introduce the rectangular coordinate system Graph paired data points on the rectangular coordinate system Graph points to create a scatter diagram Find the missing coordinate of an ordered pair solution, given one coordinate of the pair Vocabulary: bar graph, broken line graph, ordered pair, origin, quadrant, x-axis, y-axis, rectangular coordinate system, coordinate plane, x-coordinate, y-coordinate, paired data, scatter diagram
Example 1 Reading bar & line graphs A bar graph is a series of Worldwide Internet Users bars 4th Qtr They are used to compare different quantities in 3rd Qtr different categories The quantity being 2nd Qtr compared here is number of internet users 1st Qtr The category is region (ie. US/Canada) Points Which region has the 0 5 10 15 20 25 30 35 most internet users? Series1
The graph shows points scored per quarter for a basketball game Use the bar graph to find the final score. Take the readings from each color for each quarter and add them up Celtics: 35 30 25 20 15 Points Per Quarter Lakers: 10 5 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Celtics Lakers
Line graph A line graph consists of a series of points connected by a line. This graph shows the relationship between the time spent smoking a cigarette, and pulse rate What point is the pulse rate the highest? The lowest? 100 90 80 70 60 50 40 30 20 10 Pulse rate Pulse rate 0 Time (minutes)
Smoking pulse rate line graph What is the pulse rate 15 minutes after the cigarette is lit? Find the 15 along the bottom Then trace straight up until you hit the line graph The pulse rate at 15 minutes is 82 bpm There is a point on the graph with the ordered pair of (15,82) 100 90 80 70 60 50 40 30 20 10 0 Pulse rate 82-5 0 5 10 15 20 25 30 35 40
Monthly rent in Peru How much did monthly rent increase in Peru between 1980 and 2000? What was monthly rent in 1990? Was there a bigger change between 1980 and 1990, or between 1990 and 2000? 1600 1400 1200 1000 800 600 400 200 0 Monthly rent in Peru 1980 1990 2000
Ordered pairs of numbers We use the idea of ordered pairs of numbers to describe the location of a point on a chart. In math terms, the ordered pair represents a point on the coordinate plane, just a fancy term for on the chart. In math, the most commonly used type of coordinate plane is the rectangular coordinate system We have used number lines in the past. This shows points going from left to right: -- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- - -5-4 -3-2 -1 0 1 2 3 4 5 6 7
The rectangular coordinate system The rectangular coordinate system is like a number line going in two directions. The horizontal line is called the x-axis The vertical line is called the y-axis The two lines intersect at the origin, also known as the point (0,0)
Quadrants I, II, III, and IV The two axes, the x-axis and the y-axis, divide the diagram into 4 quadrants. Anything in quadrant I has an x-value greater than 0, as well as a y-value greater than 0. See the point P(3,5)? This represents the point where x = 3 and y = 5 The first number is always x, and the 2 nd always y Since both are gt 0, P(3,5) lies in Quadrant I
Rectangular coordinate system The point (0,0) is called what? It is the only point that lies on both axes. What quadrant would the point (2,-5) fall in? First, x=2. Start at the origin, and go 2 spaces to the right. Then y=-5. Go 5 spaces down. You are in quadrant IV
Quadrants explained Quadrant I x is positive, y is positive. (3,5) Quadrant II x is negative, y is positive. (-2,6) Quadrant III x is negative, y is negative (-1,- 2 Quadrant IV x is positive, y is negative
Quadrants exercise Each ordered pair represents 1 point on the graph. (3,5) is one point in quadrant I Which quadrant is (-2,5)? (0,-4) (2.6, 4.5) (-1, 4)
Scatter diagrams Each ordered pair represents exactly one point on the plane Data that can be represented as ordered pairs are called paired data A plot of paired data points on a rectangular diagram is called a scatter plot Example of paired data: a child s height by age Y-Values 4 3 2 1 0-5 -1 0 5-2 -3-4 -5
Example 4 Target net sales In 1999, net sales was 34 billion, and 2000 was 37. We can pair these values with x = year and y = billions (1999,34) (2000,37) (2001,40) (2002,44) and (2003,48) The x-axis shows the years along the bottom The points represent the net sales amounts in billions Net Sales (billions) 50 45 40 35 30 25 20 15 10 5 0 1998 2000 2002 2004
Scatter diagram exercise Bookstore sales by year: 2000 19 thousand 2001 22 thousand 2002 21 thousand 2003 23 thousand 2004 25 thousand 2005 26 thousand Write the ordered pairs Create a scatter diagram with these ordered pairs
Completing ordered pair solutions We can use ordered pairs to record solutions of equations containing TWO variables The equation x + 3 = 6 has only one solution: x = 3 How about the equation 2x + y = 8? These two-variable equations have solutions consisting of two values, one for x and one for y We can replace x with 3 and y with 2. This is a solution to our equation, and it is written as an ordered pair: (3, 2) In general, an ordered pair is a solution of an equation in two variables if replacing the variables by the values of the ordered pair results in a true statement
Completing ordered pair solutions, cont However, there can be more than one ordered pair as a solution. Generally, if we give you one number in the pair, substitute it into the equation to find the other number Complete each ordered pair as a solution for 3x + y = 12 (0, )(,6) (-1, ) First, (0, ) : 3(0) + y = 12; 0 + y = 12; y=12; (0,12) (,6): 3x + 6 = 12; 3x = 6; x = 2; (2,6) (-1, ): 3(-1) + y = 12; y = 15; (-1, 15) The ordered pairs (0,12), (2,6), and (-1,15) are all solutions
Complete each ordered pair so that it is a solution of the given linear equation x + 2y = 6; (2, ) (, -3) y = 1/3 x 2; (6, ) (,-1/3)