Quadratics and inequalities

B if x 0 then y 16 2 48 84 84 the ordered pair is 0

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b) If x 0, then y 16 0 2 48 0 84 84. The ordered pair is (0, 84). To find x when y 20, replace y by 20 and solve the equation for x : 16 x 2 48 x 84 20 16 x 2 48 x 64 0 Subtract 20 from each side. x 2 3 x 4 0 Divide each side by 16. ( x 4)( x 1) 0 Factor. x 4 0 or x 1 0 Zero factor property x 4 or x 1 The ordered pairs are ( 1, 20) and (4, 20). Now do Exercises 7–10 Graphing Parabolas All equations of the form y ax 2 bx c with a 0 have graphs that are similar in shape. The graph of any equation of this form is called a parabola. Note that any real number can be used in place of x . dug22241_ch10a.qxd 11/10/2004 18:30 Page 639
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640 Chapter 10 Quadratic Equations and Inequalities 10-24 E X A M P L E 2 The simplest parabola Make a table of ordered pairs that satisfy y x 2 and then sketch the graph of y x 2 . Solution Make a table of values for x and y : x 2 1 0 1 2 y x 2 4 1 0 1 4 Plot the ordered pairs from the table and draw a parabola through the points as shown in Fig. 10.2. Now do Exercises 17–20 The parabola in Fig. 10.2 is said to open upward. In the next example we see a parabola that opens downward. If a 0 in the equation y ax 2 bx c , then the parabola opens upward. If a 0, then the parabola opens downward. Figure 10.2 y x 6 4 2 4 1 2 2 3 8 4 y x 2 1 2 3 Calculator Close-Up This close-up view of y x 2 shows how rounded the curve is at the bottom. When drawing a parabola by hand, be sure to draw it smoothly. 4 2 1 2 E X A M P L E 3 A parabola opening downward Graph y 4 x 2 . Solution We plot enough points to get the correct shape of the graph: y x 1 3 5 3 1 2 1 1 2 3 y 4 x 2 Figure 10.3 x 2 1 0 1 2 y 4 x 2 0 3 4 3 0 See Fig. 10.3 for the graph. Now do Exercises 21–26 dug22241_ch10a.qxd 11/10/2004 18:30 Page 640
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10-25 10.3 Graphing Parabolas 641 y x b b 2 4 ac 2 a b b 2 4 ac 2 a y ax 2 bx c b 2 a Vertex Figure 10.4 Study Tip Although you should avoid cram- ming, there are times when you have no other choice. In this case concen- trate on what is in your class notes and the homework assignments. Try to work one or two problems of each type. Instructors often ask some rela- tively easy questions on a test to see if you have understood the major ideas. Vertex of a Parabola The x -coordinate of the vertex of y ax 2 bx c is 2 a b , provided that a 0. The Vertex and Intercepts The lowest point on a parabola that opens upward or the highest point on a parabola that opens downward is called the vertex. The y -coordinate of the vertex is the minimum value of y if the parabola opens upward, and it is the maximum value of y if the parabola opens downward. For y x 2 the vertex is (0, 0), and 0 is the minimum value of y . For y 4 x 2 the vertex is (0, 4), and 4 is the maximum value of y . If y ax 2 bx c has x -intercepts, they can be found by solving ax 2 bx c 0 by the quadratic formula. The vertex is midway between the x -intercepts as shown in Fig. 10.4. Note that in the quadratic formula x , b 2 4 ac is added and subtracted from the numerator of 2 a b . So 2 a b , 0 is the point midway between the x -intercepts and the vertex has the same x -coordinate. Even if the parabola has no x -intercepts, the x -coordinate of the vertex is still 2 a b .
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