SECOND-DEGREE INEQUALITIES

Transcription

1 60 (-40) Chapter Nonlinear Sstems and the Conic Sections FIGURE FOR EXERCISE GETTING MORE INVOLVED. Cooperative learning. Let (, ) be an arbitrar point on an ellipse with foci (c, 0) and ( c, 0) for c 0. The following equation epresses the fact that the distance from (, ) to (c, 0) plus the distance from (, ) to ( c, 0) is the constant value a (for a 0): ( c ) ( 0 ) ( ( c) ) ( 0 ) a Working in groups, simplif this equation. First get the radicals on opposite sides of the equation, then square both sides twice to eliminate the square roots. Finall, let b a c to get the equation 4. Sonic boom. An aircraft traveling at supersonic speed creates a cone-shaped wave that intersects the ground along a hperbola, as shown in the accompaning figure. A thunderlike sound is heard at an point on the hperbola. This sonic boom travels along the ground, following the aircraft. The area where the sonic boom is most noticeable is called the boom carpet. The width of the boom carpet is roughl five times the altitude of the aircraft. Suppose the equation of the hperbola in the figure is, where the units are miles and the width of the boom carpet is measured 40 miles behind the aircraft. Find the altitude of the aircraft. 4 or 6.9 miles. a b 6. Cooperative learning. Let (, ) be an arbitrar point on a hperbola with foci (c, 0) and ( c, 0) for c 0. The following equation epresses the fact that the distance from (, ) to (c, 0) minus the distance from (, ) to ( c, 0) is the constant value a (for a 0): ( c ) ( 0 ) ( ( c) ) ( 0 ) a Working in groups, simplif the equation. You will need to square both sides twice to eliminate the square roots. Finall, let b c a to get the equation. a b GRAPHING CALCULATOR EXERCISES Width of boom carpet 0 40 Most intense sonic boom is between these lines FIGURE FOR EXERCISE 4 7. Graph,,, and 4 to get the graph of the hperbola along with its asmptotes. Use the viewing window and. Notice how the branches of the hperbola approach the asmptotes. 8. Graph the same four functions in Eercise 7, but use 0 0 and 0 0 as the viewing window. What happened to the hperbola?. SECOND-DEGREE INEQUALITIES In this section Graphing a Second-Degree Inequalit Sstems of Inequalities In this section we graph second-degree inequalities and sstems of inequalities involving second-degree inequalities. Graphing a Second-Degree Inequalit A second-degree inequalit is an inequalit involving squares of at least one of the variables. Changing the equal sign to an inequalit smbol for an of the equations of the conic sections gives us a second-degree inequalit. Second-degree inequalities are graphed in the same manner as linear inequalities.

2 . Second-degree Inequalities (-4) 6 E X A M P L E E X A M P L E + < FIGURE. A second-degree inequalit Graph the inequalit. We first graph. This parabola has -intercepts at (, 0) and (, 0), -intercept at (0, ), and verte at (, 4). The graph of the parabola is 4 4 drawn with a dashed line, as shown in < Fig... The graph of the parabola divides the plane into two regions. Ever + point on one side of the parabola satisfies the inequalit, and ever point on the other side satisfies the inequalit FIGURE.. To determine which side is which, we test a point that is not on the parabola, sa (0, 0). Because is false, the region not containing the origin is shaded, as in Fig... A second-degree inequalit Graph the inequalit 9. The graph of 9 is a circle of radius centered at the origin. The circle divides the plane into two regions. Ever point in one region satisfies 9, and ever point in the other region satisfies 9. To identif the regions, we pick a point and test it. Select (0, 0). The inequalit is true. Because (0, 0) is inside the circle, all points inside the circle satisf the inequalit 9, as shown in Fig... The points outside the circle satisf the inequalit 9. 4 E X A M P L E > FIGURE.4 A second-degree inequalit Graph the inequalit. First graph the hperbola. Because the hperbola shown in Fig..4 divides the plane into three regions, we select a test point in each region and check to see whether it satisfies the inequalit. Testing the points (, 0), (0, 0), and (, 0) gives us the inequalities ( ) 0, 0 0, and 0. Because onl the first and third inequalities are correct, we shade onl the regions containing (, 0) and (, 0), as shown in Fig..4.

3 6 (-4) Chapter Nonlinear Sstems and the Conic Sections Sstems of Inequalities A point is in the solution set to a sstem of inequalities if it satisfies all inequalities of the sstem. We graph a sstem of inequalities b first determining the graph of each inequalit and then finding the intersection of the graphs. E X A M P L E 4 Sstems of second-degree inequalities Graph the sstem of inequalities: 9 6 Figure.(a) shows the graph of the first inequalit. In Fig..(b) we have the graph of the second inequalit. In Fig..(c) we have shaded onl the points that satisf both inequalities. Figure.(c) is the graph of the sstem (a) (b) FIGURE. (c) WARM-UPS True or false? Eplain our answer.. The graph of 4 is a circle of radius. False. The graph of 9 9 is an ellipse. True. The graph of is a hperbola. True 4. The point (0, 0) satisfies the inequalit. True. The graph of the inequalit contains the origin. False 6. The origin should be used as a test point for graphing. False 7. The solution set to 8 0 includes the origin. False 8. The graph of 4 is the region inside a circle of radius. True 9. The point (0, 4) satisfies and. True 0. The point (0, 0) satisfies and. True

4 . Second-degree Inequalities (-4) 6. EXERCISES Graph each inequalit. See Eamples ( ) ( ) 4 4. ( ) ( )

5 64 (-44) Chapter Nonlinear Sstems and the Conic Sections Graph the solution set to each sstem of inequalities. See Eample

6 . Second-degree Inequalities (-4) Solve the problem. 4. Buried treasure. An old pirate on his deathbed gave the following description of where he had buried some treasure on a deserted island: Starting at the large palm tree, I walked to the north and then to the east, and there I buried the treasure. I walked at least 0 paces to get to that spot, but I was not more than 0 paces, as the crow flies, from the large palm tree. I am sure that I walked farther in the northerl direction than in the easterl direction. With the large palm tree at the origin and the positive -ais pointing to the north, graph the possible locations of the treasure FIGURE FOR EXERCISE GRAPHING CALCULATOR EXERCISES 44. Use graphs to find an ordered pair that is in the solution set to the sstem of inequalities:.( 4) Verif that our answer satisfies both inequalities. 4. Use graphs to find the solution set to the sstem of inequalities: 8 No solution