Greenwood press
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book-20600
QUADRATIC FUNCTIONS
89 Graph of the quadratic function f (x) = −3x 2 + 2x + 5. Rewrite the expanded trinomial as a perfect square and simplify: f (x) = a(x + b 2a ) 2 − b 2 4a + c. Compare this expression to the vertex form of a quadratic function and notice that the vertex can be represented as (− b 2a , − b 2 4a + c). This coordinate will serve as a shortcut to find the highest point when a < 0, and lowest point when a > 0, in an application that uses quadratic functions. Take, for instance, a business setting that sells sport memorabilia. The demand, and hence the price, for a baseball player’s autographed ball may decline as more of them become available. Suppose that the price of an auto- graphed ball, a, from a new hall-of-famer begins at $200 and declines by five cents, or $0.05, for every ball, x, sold. This relationship would be represented by the equation a = 200 − 0.05x. The revenue, r, obtained from selling the balls would be the product of the number of balls sold and the price for each ball, or r = x(200 − 0.05x) = 200x − 0.05x 2 . The business owner will have to pay for general start-up costs such as hiring the baseball player to sign autographs and renting a place to sell the merchandise, as well as paying for the materials, such as the cost of each ball. Suppose the start-up costs are $1,300 and the business owner pays $1.25 for each new ball. Then the cost, c, that the business assumes in terms of the number of balls sold will be c = 1.25b + 1300. The profit, p, is the difference between the revenue and cost, or r − c, which equals (200x − 0.05x 2 ) − (1.25x + 1300), and simplifies to p = -0.05x 2 + 199.75x − 1300. In a quadratic function in the form of f(x) = ax 2 + bx + c, a maximum value will occur when x = – b 2a , since a < 0. In this case, a maximum profit will occur when approximately 1,997 balls are sold (x = − 199.75 2( −0.05) = 1997.5). In that case, a reasonable sale price of the “limited edition” ball should be around 200 − 0.05(1997) = $100.15. Although, to appease the human psy- che, a more reasonable price might be twenty cents cheaper at $99.95 so that con- sumers feel like they are getting a deal by paying less than $100. A graph and table of values can also support this sale price as a means of producing almost a maximum possible profit. The vertical height, h, of an object is determined by the quadratic equation h = –0.5gt 2 + v o t + h o , where g is the acceleration due to earth’s gravity (9.8 m/sec 2 ), v o is the initial vertical velocity, and h o is the initial height of the object. Therefore an object with an initial vertical velocity of 45 meters per second, thrown at a height of 0.4 meters, can be modeled with the equation h = –4.9t 2 + 45t + 0.4. Engineers of fireworks can use this type of function so that the rockets explode at a time where optimal height offers safety as well as viewing pleasure. This quadratic equation can also be used to measure the initial vertical veloc- ity of an object thrown in the air, such as a ball, assuming that it reaches the ground with minimal air resistance. For example, if a ball thrown at a height of 1.45 meters is airborne for 3.84 seconds, then the values can be substituted into the equation h = –0.5gt 2 + v o t + h o to solve for v o . In this case, the height after 3.84 seconds is equal to 0 meters, because that is the amount of time it takes for Download 1.81 Mb. Do'stlaringiz bilan baham: |
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