Question 1206910
1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.

{{{y = - (1/2) cos ((6pi/5)*x + 3)-7}}}


The general form of the cosine function is:


{{{y = a *cos(b*x+c) + d}}}, where:

amplitude = {{{abs(a)}}} => amount of travel above and below midpoint

period = {{{2pi/abs(b)}}} => time to complete one cycle

phase shift = {{{-c/b}}} => how far from zero the cycle starts

{{{d}}} = vertical shift of the function


in your case

the period: {{{2pi/(6pi/5)= 5/3}}}

amplitude: {{{abs(-1/2)=1/2}}}

vertical displacement: {{{d=-7}}}

phase shift :  {{{-(c/b) = -3/(6pi/5)=-1/(2pi/5)=-5/2pi}}}



2. A Ferris wheel has a radius of {{{16m}}} and its center is {{{18m}}} above the ground. It rotates once every {{{60s}}}. Ethan gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. 

Determine both a sine and cosine function that gives the height, {{{h}}}, of point {{{P}}} above the ground at any time, {{{t}}}, where {{{h}}} is in meters and {{{t }}}is in seconds. Both functions must have the smallest possible phase shift.

Since the Ferris wheel starts with the rider at the bottom (minimum point in the cycle), we can represent the rider’s position with a {{{negative}}} cosine function.


{{{h(t) = -a *cos(bt+c) + d}}}


given:

 amplitude is {{{a=abs(16)}}}

Since we have a negative cosine function, {{{a = -16}}}

It rotates once every {{{60s}}}=> {{{2pi/60=pi/30}}}

The rider starts at the bottom, so there is no phase shift, so {{{c=0}}}

We are given that the center of the wheel is {{{18m}}} from the ground, so {{{d=18}}}


{{{h(t) = -16cos((pi/30)t+0) + 18}}}

{{{h(t) = -16cos((pi/30)t) + 18}}}


or sine function:

{{{h(t)=16sin((pi/30)t)+18}}}