6.10 Functions and Their Graphs 391 The graph of every quadratic function is a parabola. Two parabolas are illustrated in Fig. 6.45. Note that both graphs represent functions, since they pass the vertical line test. A parabola opens upward when the coefficient of the squared term, a, is greater than 0, as shown in Fig. 6.45(a). A parabola opens downward when the coefficient of the squared term, a, is less than 0, as shown in Fig. 6.45(b). y 5 ax2 1 bx 1 c a . 0 y 5 ax2 1 bx 1 c a , 0 Vertex Vertex Axis of symmetry Axis of symmetry (a) (b) x y x y Figure 6.45 The vertex of a parabola is the lowest point on a parabola that opens upward and the highest point on a parabola that opens downward. Every parabola is symmetric with respect to a vertical line through its vertex. This line is called the axis of symmetry of the parabola. Solution a) = = + = + = + = v f t t f ( ) 3.2 0.45 (3) 3.2(3) 0.45 9.6 0.45 10.05 = = + = + = + = + = h g t t t g ( ) 1.6 0.45 (3) 1.6(3) 0.45(3) 1.6(9) 1.35 14.4 1.35 15.75 2 2 The velocity 3 seconds before touchdown was 10.05 meters per second, and the height above the moon’s surface 3 seconds before touchdown was 15.75 meters. b) = = + = + = + = v f t t f ( ) 3.2 0.45, (0) 3.2(0) 0.45 0 0.45 0.45 = = + = + = + = h g t t t g ( ) 1.6 0.45 (0) 1.6(0) 0.45(0) 0 0 0 2 2 The touchdown velocity was 0.45 meter per second. At touchdown, the Eagle is on the moon, and the height above the moon’s surface is therefore 0 meters. 7 Now try Exercise 73 Learning Catalytics Keyword Angel-SOM-6.10 (See Preface for additional details.) Axis of Symmetry of a Parabola The axis of symmetry of the graph of an equation of the form = + + y ax bx c 2 can be determined by the following formula. = − x b a2 This formula also gives the x-coordinate of the vertex of a parabola. Once the x-coordinate of the vertex has been determined, the y-coordinate can be determined by substituting the value determined for the x-coordinate into the quadratic equation and evaluating the equation. This procedure is illustrated in Example 6.
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