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Rectangular Coordinate System • In pre-algebra, we used number lines to plot numbers and equations and inequalities of 1 variable (x = -3, x < 4 => one-dimensional) -5 -4 -3 -2 -1 • 1 2 3 4 5 To examine equations involving 2 variables, we graph on a rectangular (Cartesian) coordinate system (y = x, y = x - 1 => two-dimensional) On a plane, each point is a pair of numbers => (1,2); (-2,-3); (3,-1) 4 4 3 3 2 2 1 x - axis -4 -3 -2 -1 1 -1 y - axis • 0 origin (0,0) 2 3 4 1 -4 -3 -2 -1 -2 -3 -4 (1,2) 1 -1 -2 (-2,-3) -3 -4 2 3 4 (3,-1) Quadrants and Finding Coordinates • • Coordinates like (2,3) are called ordered pairs and are of the form (x,y), where x is the x-coordinate, and y is the y-coordinate Graphs can be divided into 4 quadrants – – – – Quadrant Quadrant Quadrant Quadrant I => both coordinates are positive II => 1st-coordinate negative / 2nd-coordinate positive III => both coordinates are negative II => 1st-coordinate positive / 2nd-coordinate negative Bonus: Plot the points (-1,-3) , (3/2,5/2) , and (3,4) 4 II 4 3 3 I 2 2 1 -4 -3 -2 -1 1 -1 III -2 R: ( B: ( G: ( P: ( 1 2 3 IV 4 -4 -3 -2 -1 1 -1 -2 -3 -3 -4 -4 2 3 4 , , , , ) ) ) ) Solutions of Equations • • • To determine if an ordered pair is a solution of an equation, we use the 1st number in the pair to replace the variable that occurs 1st alphabetically The solution of an equation in 2 variables (typically x and y) is an ordered pair which when substituted into the equation give a true statement Because of this, we can generate ordered-pair solutions to equations to graph – – – Substituting x = 2 into y = 3 – x2 gives an ordered pair solution of (2, ) Show that (-2,-1) is a solution to y = 3 – x2 Graph y = 3 – x2 by using numbers from -2 to 2 for x and finding the coordinates x y -2 -1 4 3 2 -1 1 0 0.5 -4 -3 -2 -1 1 -1 1 -2 2 -3 -4 2 3 4 – Graph y = |x + 1| and y = 2x – 1 on the board with time Identifying Intercepts • • • • A y-intercept of a graph is a point where the graph intersects the y-axis (this is also the point where x = 0) An x-intercept of a graph is a point where the graph intersects the x-axis (this is also the point where y = 0) Two graphs intersect each other at any point where their x-coordinates and ycoordinates are the same Find the x and y intercepts for the following… 6 5 Red: 4 Blue: 3 Green: 2 Purple: 1 2x + 5y = 10: -6 -5 -4 -3 -2 -1 1 -1 -2 y = 4 – x2: -3 -4 -5 • Bonus: At what points do the blue and green graph intersect each other? -6 2 3 4 5 6 Interpreting Graphs Book problems: 1, 3, 5, 9, 11, 13, 17, 23, 39, 45, 57 Line graphs are often used to show trends over time or a range By identifying points on line graphs and their coordinates, you can interpret specific information given in the graph – How many rushing yards did the Hokies have in 2006? In 2009? Total Yards (VA Tech Hokies) 3500 – How many passing yards did the Hokies have in 2003? – For the period given, what is the biggest total of rushing yards in a season for the Hokies and in what season? – Around how many more passing yards did VT have than rushing yards in 2008? – How many more yards passing did the Hokies have in 2010 than the previous season (y)? 3000 Yards • • 2500 2000 1500 1000 2002 2003 2004 2005 2006 2007 2008 Year Rushing Yards Passing Yards 2009 2010