843 8.7 Polar Equations and Graphs To graph polar equations, evaluate r for various values of u until a pattern appears, and then join the points with a smooth curve. The next four examples illustrate curves that are not usually discussed when rectangular coordinates are covered. (Using graphing calculators makes the task of graphing them quite a bit easier than using traditional point-plotting methods.) S Now Try Exercises 37 and 39. EXAMPLE 4 Graphing a Polar Equation (Cardioid) Graph r = 1 + cos u. ALGEBRAIC SOLUTION To graph this equation, find some ordered pairs as in the table. Once the pattern of values of r becomes clear, it is not necessary to find more ordered pairs. The table includes approximate values for cos u and r. U cos U r =1 +cos U U cos U r =1 +cos U 0° 1 2 135° -0.7 0.3 30° 0.9 1.9 150° -0.9 0.1 45° 0.7 1.7 180° -1 0 60° 0.5 1.5 270° 0 1 90° 0 1 315° 0.7 1.7 120° -0.5 0.5 330° 0.9 1.9 Connect the points in order — from 12, 0°2 to 11.9, 30°2 to 11.7, 45°2 and so on. See Figure 66. This curve is called a cardioid because of its heart shape. The curve has been graphed on a polar grid. 180° 0° 90° 270° 1 2 r = 1 + cos U Figure 66 GRAPHING CALCULATOR SOLUTION We choose degree mode and graph values of u in the interval 30°, 360°4. The screen in Figure 67(a) shows the choices needed to generate the graph in Figure 67(b). S Now Try Exercise 47. −2.05 −3.3 2.05 3.3 Polar graphing mode (b) Figure 67 (a) Figure 65 −4.1 −6.6 4.1 6.6 Polar graphing mode (b) x y 0 2 –2 –2 2 x2 + y2 = 4 (rectangular) r = 2 (polar) (a)
RkJQdWJsaXNoZXIy NjM5ODQ=