The height of a rider on a ferris wheel, in metres can be modeled using the function H(0)=10sin(0-90•)+12 A) state the amplitude, phase shift and vertical translation B) sketch the function C)...
Given `h(theta)=10sin(theta-90^circ)+12` where h gives the height in meters:
(a) The amplitude is 10. The phase shift is a horizontal translation of 90 to the right. The vertical translation is 12 up.
(if f(x)=AsinB(x-h)+k ; |A| gives the amplitude, if A<0 then reflect over the x-axis; B gives the horizontal stretch/compression [change in period]; h is the horizontal translation; k is the vertical translation)
(b) The sketch:
(The peak in the first quadrant is at 180; the first trough after 0 is at 360)
(c) The radius is 1/2 the diameter of the ferris wheel. The height of the rider ranges from 2 meters to 22 meters, so the diameter of the wheel is 20 meters. The radius is 10m.
(d) When the ride begins at `theta=0` the rider is 2m above the ground.
Look at the graph or plug in `theta=0` . The rider's maximum height is 22m.
(e) The graph will not change if the wheel spins the other direction. Consider the situation: the rider begins at the lowest point, the height increases until the maximum is reached, and the decreases until the minimum is reached, and repeat; regardless of which direction the wheel turns.
Mathematically you could also write the function given as `h(theta)=-10cos(theta)+12` . Since `cos(-theta)=cos(theta)` the function is the same. (`cos(-theta)` describes what happens if the wheel reverses direction)