394 CHAPTER 6 Algebra, Graphs, and Functions Did You Know? 3. y-coordinate of the vertex: Substitute = − x 1, and solve for y. = − − + = − − − − + = − + + = y x x2 8 ( 1) 2( 1) 8 (1) 2 8 9 2 2 Thus, the vertex is −( 1,9). 4. y-intercept: Substitute = x 0 and solve for y. = − − + = − − + = y x x y 2 8 (0) 2(0) 8 8 2 2 Thus, the y-intercept is (0,8). 5. x-intercepts: Substitute = y 0 and solve for x. y x x x x x x x x 2 8 0 2 8 2 8 0 ( 4)( 2) 0 2 2 2 = − − + = − − + + − = + − = + = x 4 0 or − = x 2 0 x 4 = − or = x 2 Thus, the x-intercepts are ( 4, 0) − and (2, 0). 6. Plot the vertex, −( 1, 9), the axis of symmetry, x 1, = − the y-intercept, (0, 8), and the x-intercepts, ( 4, 0) − and (2, 0). Then graph the function (Fig. 6.47). Note, if you substitute −2 for x in the original equation, the corresponding value of y is 8. Thus, the point −( 2, 8) is also a point on the graph. (24, 0) (22, 8) (21, 9) (0, 8) (2, 0) f(x) 5 2x2 2 2x 1 8 y 7 8 9 10 6 5 4 1 22 21 22 1 3 4 21 23 25 26 x 2 3 24 2 Figure 6.47 b) The domain, the values that can be used for x, is the set of all real numbers, .R The range, the resulting values obtained for y, is ≤ y 9. 7 Now try Exercise 59 Added x , 2 added x2 , and subtracted 8 from both sides of the equation. If we use the quadratic formula to determine the x-intercepts of a graph, and if the radicand, − b ac 4 , 2 is a negative number, the graph has no x-intercepts. The graph will lie totally above or below the x -axis. Gravity and the Parabola Any golfer knows that a golf ball will travel in the same path going down as it did going up. Early gunners knew it, too. To hit a distant target, the cannon barrel was pointed skyward, not directly at the target. A cannonball fired at a ° 45 angle will travel the greatest horizontal distance. What the golfer and gunner alike were allowing for is the effect of gravity on a projectile. Projectile motion follows a parabolic path. Galileo was neither a gunner nor a golfer, but he gave us the formula that effectively describes that motion and the distance traveled by an object if it is projected at a specific angle with a specific initial velocity. Vanessa Nel/Shutterstock
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