6.6 Graphing Linear Equations 345 Graphing Linear Equations by Using the Slope and y-Intercepts A linear equation given in the form y mx b = + is said to be in slope–intercept form. 4 5 3 3 2 2 5 4 1 22 23 24 25 23 24 25 y x (b) 4 5 3 3 2 2 1 5 4 1 21 22 23 24 25 23 24 25 y x (a) (x2, y2) (21, 23) (x1, y1) (1, 5) Horizontal change: 1 unit Vertical change: 4 units 21 1 21 21 22 22 Figure 6.13 7 Now try Exercise 65 Exploring Slope and y-intercept y 4 5 6 7 8 9 2 1 21 22 2423 25 1 2 3 4 5 x y-intercept y 5 3x 1 4 3 21 Figure 6.14 Slope–Intercept Form of the Equation of a Line y mx b = + where m is the slope of the line and b (0, ) is the y-intercept of the line. In the equation y mx b, = + b represents the value of y where the graph of the equation y mx b = + crosses the y-axis. Consider the graph of the equation y x3 4, = + which appears in Fig. 6.14. By examining the graph, we can see that the y-intercept is (0, 4). We can also see that the graph has a positive slope because it rises from left to right. Because the vertical change is 3 units for every 1 unit of horizontal change, the slope must be , 3 1 or 3. We could graph this equation by marking the y-intercept at (0, 4) and then moving up 3 units and to the right 1 unit to get another point. If the slope were 3, − which means , 3 1 − we could start at the y-intercept and move down 3 units and to the right 1 unit. Thus, if we know the slope and y-intercept of a line, we can graph the line. 3 1 4 21 22 23 24 25 26 27 y x y 5 23x 1 1 22 23 24 25 21 1 2 3 4 5 2 5 Figure 6.15 GRAPHING LINEAR EQUATIONS BY USING THE SLOPE AND Y- INTERCEPT 1. Solve the equation for y to place the equation in slope–intercept form. 2. Determine the slope and y-intercept from the equation. 3. Plot the y-intercept. 4. Obtain a second point using the slope. 5. Draw a straight line through the points. PROCEDURE Example 6 Graphing an Equation by Using the Slope and y-Intercept Graph y x3 1 = − + by using the slope and y-intercept. Solution The slope is 3, − or , 3 1 − and the y-intercept is (0, 1) (see Fig. 6.15). Plot (0, 1) on the y-axis. Then plot the next point by moving down 3 units and to the right 1 unit. A third point has been plotted in the same way. The graph of y x3 1 = − + is the line drawn through these three points. 7 Now try Exercise 73
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