152 Chapter 3 Graphs, Relations, and Functions Student Check 2 Determine if each ordered pair is a solution of the given equation. a. 3x - y = 0; (-2, 6), (2, 6), a1 3 , 1b b. y = 2x - 1; a1 2 , 0b , (0, -1), a11 2 , 10b c. y = |x + 3|; (-3, 0), (-4, 1), (-7, -4) d. y = 3x2 - 4x + 2; (0, -2), (-1, 9), (1, 3) Graphing Equations in Two Variables Example 2 shows that an equation in two variables has more than one solution. In fact, an equation in two variables can have infinitely many ordered pairs that satisfy it. It is impossible to state every solution of this type of equation. Our goal then is to visualize the solution set of an equation in two variables by graphing the equation. To graph an equation in two variables means to draw a picture of all the solutions of the equation. We plot several ordered pairs that satisfy the equation and use these points to determine the shape of the graph. Some of the graphs we will obtain are shown as follows. Line Parabola V-Shaped Graph 6 2 –4 –2 2 x 4 4 –2 y 2 2 4 4 –2 –4 –2 –4 x y 2 2 4 4 –2 –4 –2 –4 x y Each type of graph comes from a special type of equation. A line comes from a linear equation in two variables. A parabola comes from a quadratic equation in two variables. A V-shaped graph comes from an absolute value equation in two variables. These special equations will be studied in more detail in later chapters. At this point, our goal is to construct a graph of an equation in two variables by plotting points. Procedure: Graphing an Equation in Two Variables Step 1: Determine at least three solutions of the equation. a. Substitute a numerical value for x. Choose values that are negative, positive, and zero so that we get a complete picture of the graph. b. Simplify the resulting expression or solve the resulting equation to find y. c. The ordered pair (x, y) is a solution of the equation. d. Repeat this process as many times as needed. e. Organize the information in a chart. Step 2: Plot the solutions found in step 1 on a coordinate system. Step 3: Connect the points and use their pattern to sketch the graph. Objective 3 Examples Graph each equation by plotting points. 3a. y = x - 2 3b. y = x2 - 4 3c. y = |x + 1| Objective 3 ▶ Graph an equation in two variables.
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