The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.
With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.
Passengers board 2 m above ground level, so the center of the wheel must be located
$\text{\hspace{0.17em}}67.5+2=69.5\text{\hspace{0.17em}}$ m above ground level. The midline of the oscillation will be at 69.5 m.
The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.
Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.
Amplitude:
$\text{\hspace{0.17em}}\text{67}\text{.5,}\text{\hspace{0.17em}}$ so
$\text{\hspace{0.17em}}A=67.5$
Midline:
$\text{\hspace{0.17em}}\text{69}\text{.5,}\text{\hspace{0.17em}}$ so
$\text{\hspace{0.17em}}D=69.5$
Period:
$\text{\hspace{0.17em}}\text{30,}\text{\hspace{0.17em}}$ so
$\text{\hspace{0.17em}}B=\frac{2\pi}{30}=\frac{\pi}{15}$
Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of
$\text{\hspace{0.17em}}2\pi .$
The function
$\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is odd, so its graph is symmetric about the origin. The function
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, so its graph is symmetric about the
y -axis.
The graph of a sinusoidal function has the same general shape as a sine or cosine function.
In the general formula for a sinusoidal function, the period is
$\text{\hspace{0.17em}}P=\frac{2\pi}{\left|B\right|}.\text{\hspace{0.17em}}$ See
[link] .
In the general formula for a sinusoidal function,
$\text{\hspace{0.17em}}\left|A\right|\text{\hspace{0.17em}}$ represents amplitude. If
$\text{\hspace{0.17em}}\left|A\right|>1,\text{\hspace{0.17em}}$ the function is stretched, whereas if
$\text{\hspace{0.17em}}\left|A\right|<1,\text{\hspace{0.17em}}$ the function is compressed. See
[link] .
The value
$\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ in the general formula for a sinusoidal function indicates the phase shift. See
[link] .
The value
$\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ in the general formula for a sinusoidal function indicates the vertical shift from the midline. See
[link] .
Combinations of variations of sinusoidal functions can be detected from an equation. See
[link] .
The equation for a sinusoidal function can be determined from a graph. See
[link] and
[link] .
A function can be graphed by identifying its amplitude and period. See
[link] and
[link] .
A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See
[link] .
Sinusoidal functions can be used to solve real-world problems. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Why are the sine and cosine functions called periodic functions?
The sine and cosine functions have the property that
$\text{\hspace{0.17em}}f\left(x+P\right)=f\left(x\right)\text{\hspace{0.17em}}$ for a certain
$\text{\hspace{0.17em}}P.\text{\hspace{0.17em}}$ This means that the function values repeat for every
$\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ units on the
x -axis.
How does the graph of
$\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ compare with the graph of
$\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x?\text{\hspace{0.17em}}$ Explain how you could horizontally translate the graph of
$\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to obtain
$\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x.$
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.