Performance Task Changing the Course Designers of motocross races use mathematics to create ramps and jumps for their courses. How could you modify their models so that riders will catch more air on your track? Motocross race course designers consider the experience level of the racers when designing the heights of the ramps for a course. The physical components of the ramp are the run u' of the ramp up to the launch point and the height h of the ramp. For jumps in amateur races, a standard ratio of the run to the height of the jump is 3:]. It h is 2:1 for professional races. The racer controls the speed s (in feet per second) at the beginning of the jump. These constants affect the height y (in feet) of the dirt bike with respect to the horizontal distance x (in feet) the dirt bike travels in the air after the launch, and can be modeled by the equation W2 + 173 11 7? X2 + X + h. S-Wh M\" y = 16 0 1. Which function family does the equation above belong to? What shape does this represent? 2. Compare the equation above to the standard form of a quadratic equation, y = ax + bx + c. How is the coefficient a defined? the coefficient b? the constant c? 3. The coefficient of the x- term includes a constant to account for gravity. What is the sign of this coefficient? What does this tell you about the shape of the graph of the function? What does this tell you about how gravity affects a jump? What happens to the x term as the height h of the ramp increases?Performance Task (continued) 4. How can you calculate the maximum height of the jump for the model? How is the height related to the shape of a parabola? 5. The term catching air is used to describe the portion of a jump that is completely in the air. How is the maximum height of a dirt bike during a jump related to how long the dirt bike is in the air? How can you calculate the total horizontal distance a dirt bike travels in the air? 6. Explore how different dimensions of the launch ramp on an amateur course affect the height of the dirt bike and the amount of air it catches. Using a spreadsheet, enter the formulas for each coefcient and compare the resulting quadratic equations. a. Assume the racer has a speed of 45 miles per hour. Convert the speed to the correct units. b. How are w and h related for an amateur course? c. Complete the table below. Round decimal answers to the nearest tenth. Amateur Track Results Total horizontal distance (feet) of dirt bike in air Vertex of Maximum parabolic height (feet)l graph of dirt bike Height (feet) Run length Quadratic of ramp, h (feet) of ramp, w equation Performance Task (continued) 7. Now analyze the jump height of riders on a course designed for professionals. Assume the drivers are driving the same speed of 50 miles per hour. a. How are 11' and it related for a professional course? b. Complete the table below. Round decimal answers to the nearest tenth. Total horizontal distance (feet) of dirt bike in air Height (feet) Run length Quadratic Vertex Of MaXImum . parabolic height (feet) of ramp, h (feet) of ramp. w equation graph of dirt bike 8. Compare the results of the amateur track to the professional track. How are the coefcients of the quadratic equations different? How does this affect the jumps