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§ 1.3 Graphing Equations The Rectangular Coordinate System In the rectangular coordinate system, the horizontal number line is the x-axis. The vertical number line is the y-axis. The point of intersection of these axes is their zero points, called the origin. The axes divide the plane into 4 quarters, called quadrants. y-axis 2nd quadrant 1st quadrant x-axis 3rd quadrant 4th quadrant Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.3 The Rectangular Coordinate System Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers (x,y). Note the word “ordered” because order matters. The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number, the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axis Itself. Point Movement from origin (3.5) 3 units right and 5 units up (4,-3) 4 units right and 3 units down (-2,-7) 2 units left and 7 units down (0,0) 0 right or left and 0 up or down In plotting points, we move across first (either left or right), and then move either up or down, always starting from the origin. Blitzer, Algebra for College Students, 6e – Slide #3 Section 1.3 Plotting Points EXAMPLE Plot the points (3,2) and (-2,-4). SOLUTION (3,2) (-2,-4) Blitzer, Algebra for College Students, 6e – Slide #4 Section 1.3 The Graph of an Equation The graph of an equation in two variables is the set of points whose coordinates satisfy the equation. An ordered pair of real numbers (x,y) is said to satisfy the equation when substitution of the x and y coordinates into the equation makes it a true statement. For example, in the equation y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point the sentence reads 8 = 8, which is true. The ordered pair (2,3) is not a solution. When we substitute this point, the sentence reads 3 = 10, which is not true. Blitzer, Algebra for College Students, 6e – Slide #5 Section 1.3 Graphing an Equation EXAMPLE Graph y = 2|x| + 1. SOLUTION x y = 2|x| + 1 Ordered Pair (x,y) -2 y = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5 (-2,5) -1 y = 2|-1| + 1 = 2(1) + 1 = 2 + 1 = 3 (-1,3) 0 y = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1 (0,1) 1 y = 2|1| + 1 = 2(1) + 1 = 2 + 1 = 3 (1,3) 2 y = 2|2| + 1 = 2(2) + 1 = 4 + 1 = 5 (2,5) Blitzer, Algebra for College Students, 6e – Slide #6 Section 1.3 Graphing an Equation CONTINUED (-2,5) (2,5) (-1,3) (1,3) In graphing an equation, we try to get enough ordered pairs to get a good idea of what the graph looks like. Next, we plot these points. Finally, we connect these points with a smooth curve or line, always moving from left to right. This often gives us a picture of all ordered pairs that satisfy the equation. (0,1) Blitzer, Algebra for College Students, 6e – Slide #7 Section 1.3 Graphing an Equation EXAMPLE Graphs in the rectangular coordinate system can also be used to tell a story. Try to select the graph that best illustrates the story of the population of the U.S.A. Years Population (c) Population (b) Population (a) Years SOLUTION Graph (c) Blitzer, Algebra for College Students, 6e – Slide #8 Section 1.3 Years