209 2.1 Rectangular Coordinates and Graphs Step 2 We use the other ordered pairs found in Example 7(a): 1-2, -92 and 11, 32. Step 3 Plot the four ordered pairs from Steps 1 and 2 as shown in Figure 9. Step 4 Join the points plotted in Step 3 with a straight line. This line, shown in Figure 9, is the graph of the equation y = 4x - 1. –2 2 x y –6 –3 3 0 –9 y = 4x – 1 Figure 9 2 5 1 0 10 x y x = !y – 1 Figure 10 (b) For x = 2y - 1, the y-intercept 10, 12 was found in Example 7(b). Solve x = 20 - 1 Let y = 0. to find the x-intercept. When y = 0, the quantity under the radical symbol is negative, so there is no x-intercept. In fact, y - 1 must be greater than or equal to 0, so y must be greater than or equal to 1. We start by plotting the ordered pairs from Example 7(b) and then join the points with a smooth curve as in Figure 10. To confirm the direction the curve will take as x increases, we find another solution, 13, 102. (Point plotting for graphs other than lines is often inefficient. We will examine other graphing methods later.) (c) In Example 7(c), we made a table of five ordered pairs that satisfy the equation y = x2 - 4. 1-2, 02, 1-1, -32, 10, -42, 11, -32, 12, 02 x-intercept y-intercept x-intercept Plotting the points and joining them with a smooth curve gives the graph in Figure 11. This curve is called a parabola. S Now Try Exercises 47(b), 51(b), and 53(b). y 0 x y = x2 – 4 –2 2 –4 Figure 11 To graph an equation on a calculator, such as y = 4x - 1, Equation from Example 8(a) we must first solve it for y (if necessary). Here the equation is already in the correct form, y = 4x - 1, so we enter 4x - 1 for y1.* The intercepts can help determine an appropriate window, since we want them to appear in the graph. A good choice is often the standard viewing window for the TI-84 Plus, which has x minimum= -10, x maximum= 10, y minimum= -10, y maximum= 10, with x scale = 1 and y scale = 1. (The x and y scales determine the spacing of the tick marks.) Because the intercepts here are very close to the origin, we have chosen the x and y minimum and maximum to be -3 and 3 instead. See Figure 12. 7 * In this text, we use lowercase letters for variables when referring to graphing calculators. (Some models use uppercase letters.) y1 = 4x−1 −3 −3 3 3 Figure 12
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