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
$\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x.$