Sinusoidal Functions

The height graph your group sketched during The Ferris Wheel rose, fell, and rose again — the same wave, over and over. Phenomena that repeat on a fixed schedule are periodic: one complete repeat is a cycle, the time it takes is the period, the horizontal midline is the axis, and the distance from axis to peak is the amplitude. The smoothest periodic functions of all come straight out of trigonometry.

From a spinning point to a graph

Take the rotating arm from Special Angles and record the height of its tip as the angle grows: at 0° the height is 0, at 90° it peaks at 1, at 180° it returns to 0, at 270° it bottoms out at $-1$. Plot height against angle and the unit circle unrolls into the wave $f(x)=\sin x$ — a genuine function, since one angle can only put the arm in one place. Its period is 360°, its amplitude 1, its range $-1\le y\le 1$. Tracking the horizontal position instead gives $f(x)=\cos x$: the identical wave, starting at its peak.

Reading the equation

Sinusoids are transformed sines and cosines, and the recipe is the same $a$, $k$, $d$, $c$ as Transformations of Functions — each letter now owning a wave-word:

Letter	Wave feature	How to read it
$a$	amplitude $=|a|$	axis-to-peak distance
$k$	period $=\frac{360°}{k}$	bigger $k$, faster repeats
$d$	phase shift	whole wave slides right by $d$
$c$	axis $y=c$	max is $c+|a|$, min is $c-|a|$

So $f(x)=3\sin (2(x-15°))+4$ has amplitude 3, period 180°, a 15° shift right, axis $y=4$ — hence a maximum of 7 and a minimum of 1, and range $1\le y\le 7$. To sketch it, transform a wave you know, exactly as you transformed parabolas; to check it, type it into Desmos and see whether the peaks land where you promised.

Modelling without angles

The input does not have to be an angle. Tides, daylight hours, breathing, a Ferris wheel seat — anything cyclic can ride a sinusoid whose input is time. A tide with amplitude 5 m, high tide at midnight, and a 12-hour cycle fits $h(t)=5\sin (30(t+3))$, with $t$ in hours: the $k=30$ delivers the 12-hour period ($360°÷30$), and the shift puts the peak at $t=0$. Build the equation from the story’s features — amplitude from half the high-low gap, $c$ from their average, $k$ from the period, $d$ last — and then interrogate it: heights at any time, times of any height.

That is precisely the job in The Tide Problem, and Sinusoidal Functions Practice rehearses every layer, from reading equations to predicting what changes when the wheel spins faster.

Curriculum connection

D2.4

sketch the graphs of $f(x)=\sin x$ and $f(x)=\cos x$ for angle measures expressed in degrees, and determine and describe their key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals)

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D2.6

determine the amplitude, period, phase shift, domain, and range of sinusoidal functions whose equations are given in the form $f(x)=a\sin (k(x-d))+c$ or $f(x)=a\cos (k(x-d))+c$

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D2.7

sketch graphs of $y=af(k(x-d))+c$ by applying one or more transformations to the graphs of $f(x)=\sin x$ and $f(x)=\cos x$, and state the domain and range of the transformed functions Sample problem: Transform the graph of $f(x)=\cos x$ to sketch $g(x)=3\cos 2x-1$, and state the domain and range of each function.

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D3.3

determine, through investigation, how sinusoidal functions can be used to model periodic phenomena that do not involve angles Sample problem: Investigate, using graphing technology in degree mode, and explain how the function $h(t)=5\sin (30(t+3))$ approximately models the relationship between the height and the time of day for a tide with an amplitude of 5 m, if high tide is at midnight.

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