Algebra Applications of quadratic functions Name: 1. The path of a diver off a springboard is modeled by the function h( x) ( x 3) 3, where h is the height of the diver and x is the distance of the diver from the board. a. Sketch the path of the diver by plotting the vertex, y-intercept, and additional points as needed. b. What is the height of the board? c. How far is the diver from the board when they reach maximum height? d. What is the maximum height of the diver? e. Estimate how far the diver travels horizontally before entering the water.. The path of a ski jumper after takeoff can be modeled by the function y 0.036 x( x 35), where x is the horizontal distance (in feet) and y is the height (in feet). a. Sketch the path of the ski jumper by plotting the vertex, x-intercepts, and additional points as needed. Be sure to label the axes.
b. How far does the ski jumper travel before landing? c. How far does the ski jumper travel before reaching maximum height? d. What is the ski jumper s maximum height? 3. A rocket is launched from the ground. The path of the rocket is given by the equation h( t) 16t 18t, where h is the height of the rocket in feet and t is the time in seconds. a. Sketch the path of the rocket by plotting the vertex, x-intercepts, and additional points as needed. Be sure to label the axes. b. When does the rocket reach its maximum height? c. What is the maximum height of the rocket? d. How long does it take the rocket to hit the ground?
4. The height of a flare is a function of the elapsed time since it was fired. An expression for its height is h( x) 5( t 10) 501, where h is the height in meters and t is the time in seconds. a. When did the flare reach its maximum height? b. What was the maximum height of the flare? c. From what height was the flare launched? 5. The path of a table-tennis ball after being hit and hitting the surface of the table can be modeled by the function h( t) 4.9 t( t 0.4), where h is the height in meters above the table and t is the time in seconds. a. After how long does the ball hit the table? b. How long does it take the ball to reach maximum height? d. What is the ball s maximum height? 6. A ball is thrown from the top of a hill to the ground below. The height of the ball is given by the function h( t) 16t 96t 11, where h is the height of the ball in feet and t is the time in seconds. a. When does the ball reach its maximum height? b. What is the maximum height of the ball? c. What is the height of the hill? d. How long does it take the ball to hit the ground?
7. The profit earned by a small theatre company for one of its productions is given by the function p( x) 0.5( x 0) 150, where p is the profit in thousands of dollars and x is the ticket price in dollars. a. At what price should tickets be sold to maximize profit? b. What is the maximum possible profit? c. If no tickets are sold, will the theatre company make a profit? Explain your answer. 8. The path of the opening of a tunnel can be modeled by the function y 0.06 x( x 46), where x is the horizontal distance (in feet) from the left edge of the opening and y is the height (in feet) above the road. a. What is the width of the tunnel opening? b. What is the maximum height of the tunnel opening? 9. The path that a motocross dirt bike rider follows during a jump is given by y 0.4x 4x 9.6, where x is the horizontal distance (in feet) from the edge of the ramp and y is the height (in feet). a. How far does the rider travel horizontally before reaching maximum height? b. What is the maximum height of the rider during the jump? c. What is the height of the edge of the ramp? d. How far from the ramp does the rider travel before landing on the ground?
Answers: 1. a. The vertex is (3, 3). The y-intercept is (0, 14) and its matching point is (6, 14). b. The height of the board is 14 feet. c. The diver is 3 feet from the board at maximum height. d. The maximum height of the diver is 3 feet. e. The diver enters the water just under 8 feet from the board.. a. The vertex is (17.5, 11.05). The x-intercepts are (0, 0) and (35, 0). b. The ski jumper travels 35 feet before landing. c. The ski jumper travels 17.5 feet to reach maximum height. d. The maximum height is 11.05 feet. 3. a. The vertex is (4, 56). The x-intercepts are (0, 0) and (8, 0). b. The rocket reaches its maximum height at seconds. c. The maximum height of the rocket is 56 feet. d. It takes 8 seconds for the rocket to hit the ground. 4. a. The flare reaches its maximum height at 10 seconds. b. The maximum height is 501 meters. c. The flare was launched from a height of 1 meter. 5. a. The ball hits the table at 0.4 seconds. b. The ball takes 0.1 seconds to reach maximum height. c. The maximum height is about 0. meters. 6. a. The ball reaches its maximum height at 3 seconds. b. The maximum height of the ball is 56 feet. c. The height of the hill is 11 feet. d. It takes the ball 7 seconds to hit the ground. 7. a. The tickets should be sold at a price of $0 each. b. The maximum profit is $150,000. c. If no tickets are sold the theatre will suffer a loss of $50,000. 8. a. The width of the tunnel opening is 46 feet. b. The maximum height of the tunnel opening is 13.754 feet. 9. a. The rider travels 5 feet horizontally before reaching maximum height. b. The maximum height of the rider is 19.6 feet. c. The height of the edge of the ramp is 9.6 feet. d. The rider travels 1 feet from the ramp before landing on the ground.