Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name __________________________________ Period __________ Date: Essential Question: How does a Cartesian coordinate system incorporate the concept of number lines? Topic: 3-2 Graphs of Linear Equations in Two Variables Standard: A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Objective: To graph a linear equation in two variables. Graphing: Solutions of open sentences in x and y can be graphed in an xy-coordinate plane. To set up a plane rectangular coordinate system, draw two number lines, or axes, meeting at right angles at a point O, the origin. The horizontal axis is called the x-axis, and the vertical axis, the y-axis. The axes divide the plane into four quadrants. With each ordered pair of numbers, you can associate a unique point in the plane. To associate a point with the pair (4, 3), imagine drawing a vertical line through 4 on the x-axis and a horizontal line through 3 on the y-axis, as shown below. These lines intersect at a point P, the graph of (4, 3). Locating a point in this way is called graphing the ordered pair, or plotting the point. Summary Coordinates: By reversing the process just described, you can associate with each point P in the plane a unique ordered pair (a, b) of real numbers, called the coordinates of P (see the diagram below). The first coordinate, a. is called the x-coordinate, or abscissa, of P; the second coordinate, b, is called the y-coordinate, or ordinate of P. This one-to-one correspondence between ordered pairs of real numbers and points of the plane can be summarized as follows: 1. There is exactly one point in the coordinate plane associated with each ordered pair of real numbers. 2. There is exactly one ordered pair of real numbers associated with each point in the coordinate plane. A rectangular coordinate system is often called a Cartesian coordinate system in honor of René Descartes (1596-1650), the French philosopher and mathematician who introduced coordinates. 2 Example 1: Graph the ordered pairs (8, 1), (0, 2), (2. 8), (2, 8), and (4, 0) in the same coordinate plane. (2, 8) (0, 2) (8, 1) (4, 0) (2, 8) Exercise 1: Graph the ordered pairs (5, 10), (0, 6), (2. 8), (8, 8), and (4, 12) in the same coordinate plane. 3 Example 2: Solution: Find and graph five solutions of . Solve for one variable, say y, in terms of the other. Choose convenient values of x and find the corresponding values of y. Then graph the resulting ordered pairs. Exercise 2: Find and graph five solutions of . 4 Graph: The graph of an open sentence in two variables is the set of all points in the coordinate plane whose coordinates satisfy the open sentence. Example 2 suggests that if we were to graph all the ordered pairs satisfying , we would obtain the line shown in blue. The following theorem, which is proved in more advanced courses, tells us that the graph of the equation is in fact a line. Theorem The graph of every equation of the form (A and B not both 0) is a line. Conversely, every line in the coordinate plane is the graph of an equation of this form. Because of this property, any equation that can be expressed in the form (A and B not both 0) is called a linear equation in two variables. The equation is linear since it can be written as . The following equations are not linear: Although you need only two points to determine the graph of a linear equation, it is a good practice to plot a third point as a check. Points where the graph crosses the axes are often easy to find and plot. 5 Example 3: Solution: Graph . The graph crosses the y-axis at a point whose x-coordinate is 0. The graph crosses the x-axis at a point whose y-coordinate is 0. Let . Let ( ) Solution (0, 3) . ( ) Solution ( , 0) The graph is the line through the points with coordinates (0, 3) and ( , 0). As a check, note that (3, 5) is a solution of graph lies on the line. and its 6 Exercise 3: Graph Example 4: Graph in a coordinate plane. . a. Solution: The equation written as can be The graph consists of all points having y-coordinate 4 and is therefore a horizontal line. b. Solution: The equation be written as can The graph consists of all points having x-coordinate and is therefore a vertical line. 7 Exercise 4: Graph in a coordinate plane. a. b. Example 4 illustrates the fact that the graph of Ax + By = C is a horizontal line if A = 0, and is a vertical line if B = 0. From now on, we will follow the common practice of using the simpler phrase "the point (a, b)" instead of "the point whose coordinates are (a, b)." Also, the phrase "the line whose equation is Class work: p 110 Oral Exercises: 1-31 Homework: p 111 Written Exercises: 2-24 even " will mean "the line ." P 105 Written Exercises: 34-38 even 8