SECTION 3.3 Quadratic Functions and Their Properties 165 Solution For f x x x a b 4 5, 1, 4, 2 ( ) = − − = = − and c 5. = − Because a 0, > the graph of f is concave up, which means the vertex is a minimum point. The minimum occurs at x b a 2 4 2 1 4 2 2 = − = − − ⋅ = = a b 1, 4 ↑ = =− The minimum value of f is f b a f 2 2 2 4 2 5 4 8 5 9 2 ( ) ( ) − = = −⋅−=−−=− Now Work PROBLEM 69 Now Work PROBLEM 87 Seeing the Concept Use a graphing utility to graph h x x x 1 5000 500 0 x 5500 2 ( ) = − + + ≤ ≤ Use the appropriate commands to find the maximum height of the projectile and the distance from the base of the cliff to where it strikes the water. Compare your results with those obtained in Example 8. Analyzing the Motion of a Projectile A projectile is fired from a cliff 500 feet above the water at an inclination of 45° to the horizontal, with a muzzle velocity of 400 feet per second. From physics, the height h of the projectile above the water can be modeled by h x x x 32 400 500 2 2 ( ) = − + + where x is the horizontal distance of the projectile from the base of the cliff. See Figure 24. EXAMPLE 8 Figure 24 1000 2000 3000 4000 500 1000 1500 2500 2000 h (x) x 458 5000 (a) Find the maximum height of the projectile. (b) How far from the base of the cliff will the projectile strike the water? Solution (a) The height of the projectile is given by the quadratic function h x x x x x 32 400 500 1 5000 500 2 2 2 ( ) = − + + = − + + We are looking for the maximum value of h. Because a 0, < the vertex is the maximum point and occurs at x b a2 1 2 1 5000 5000 2 2500 = − = − ⋅ − = = The maximum height of the projectile is h 2500 1 5000 2500 2500 500 1250 2500 500 1750 ft 2 ( ) = − ⋅ + + = − + + = (b) The projectile strikes the water when the height h is zero. To find the distance x traveled, solve the equation h x x x 1 5000 500 0 2 ( ) = − + + = The discriminant of this quadratic equation is − = − ⋅ − ⋅ = b ac 4 1 4 1 5000 500 1.4 2 2 Then = − ± − = − ± ⋅ − x b b ac a 4 2 1 1.4 2 1 5000 2 x x 458 or 5458 ≈ − ≈ Discard the negative solution. The projectile strikes the water about 5458 feet from the base of the cliff.
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