BYHS AFM INVESTIGATION NAME __________________________ 1. SET UP: Consider the Ferris wheel below. Label all the seats with the appropriate degree measure. The radius of the wheel is 1 decameter (10 meters) and the wheel is 2 meters off the ground. a) Suppose you are riding on the Ferris wheel. Find your height off the ground at each seat. Seat (angle) Height b) At what other angles would your seat be at the same height as the 30⁰ seat? What about the 240° seat? c) What will happen to the heights as you continue to rotate around the Ferris wheel? GRAPH: Use the information above to plot your height versus seat location on the graph provided (from 0° to 360°). (LABEL AXES) d) If the Ferris wheel started moving while you were at the 0° seat but this time it started going the other direction, how would the heights be affected? Graph these heights on the graph above. e) How could we use the graph to help us find the height of a seat if the Ferris wheel were to stop at 40°? Estimate this height. 2. What would the graph look like if the radius of the Ferris wheel was 1 decameter and the center of the Ferris wheel was at the origin? In other words, the Ferris wheel is half above ground and half below ground! Find the new heights. Seat (angle) Height Ground GRAPH: Use the information above graph height vs angle on the graph provided (from -­‐360° to 360°). (LABEL AXES) CONCLUSION: You have just graphed y = sin x (where x represents the ____________ and y represents the ________________). a) Use the graph to estimate the height at 315°. b) How does problem 1 relate to problem 2? c) If problem 2 is the graph 𝑦 = 𝑠𝑖𝑛𝑥, what is the equation for the graph in problem 1? d) How do you think the graph of problem 2 would change if the radius of the Ferris wheel was 2 decameters? e) How do you think this new radius would change the equation? 3. SET-­‐UP: Consider the Ferris wheel below. Label the x-­‐ and y-­‐axes by using the center of the Ferris wheel as the origin. Label the seats with the appropriate degree measures. Consider the center of the wheel to be the origin and the radius of the wheel to be 1 decameter. a) Find out how far each seat is from the y-­‐axis: Seat (angle) Displacement from y-­‐axis Ground b) At which other seat will your displacement from the y-­‐axis be the same as that of the 60° seat? c) What will happen to the displacements from the y-­‐axis as you continue to rotate around the Ferris wheel? GRAPH: Use the information above to graph horizontal displacement vs angle on the graph provided. (LABEL AXES) CONCLUSION: You have just graphed y = cos x (x represents the _____________ and y represents the ___________________). d) What would happen to the displacements from the y-­‐axis if the Ferris wheel started rotating in the opposite direction? Graph these displacements on the graph above if you didn’t already do so. e) How would the graph change if the radius was 3 decameters and the bottom of the Ferris wheel was 1 decameter off the ground? f) What is the equation of the Ferris wheel described in part e? Putting it all together We can refer to the LOCATION of the seat, not just by the angle measurement, but also by considering coordinates on a coordinate plane. 1. If the Ferris wheel shown below has a radius of 1 decameter and is centered at the origin, find the (x,y) coordinates for each seat using the information from previous problems we worked in class. 2. How would the coordinates be effected by changing the radius of the Ferris wheel to 2 decameters? What about 3 decameters? 3. Describe how you could find the coordinates of any point, P, on the Ferris wheel. 4. If a point starts at (1, 0) and has rotated through the following angle, find the new coordinates: a) 20° b) 140° c) 345° d) 115° 5. When the Ferris wheel has rotated through an angle of 40°, the seat that started at (1,0) is now at about (0.77, 0.64). How can the symmetry of the circle help you quickly find the location of that seat after rotations of 140°, 220°, 320°, etc.? Find the coordinates. EXTENSIONS: Use your knowledge of basic transformations to graph the following (LABEL THE AXES appropriately): 1) y = sin x + 3 2) y = 2 sin x 4) y = 4 cos x 6) y = 3 sin x − 1 3)
y = cos x − 2 5) y = − cos x