Periodic Behaviour Can Be Modelled With Sinusoids

1. Trigonometric modelling uses sine and cosine functions to represent situations that repeat in a regular cycle (for example, swings, waves, daylight hours, tides, rotating wheels, and seasonal temperature).
2. In real-life modelling, the axes usually represent meaningful units (such as time on the $x$-axis and height on the $y$-axis), not degrees.

The Key Features of a Sinusoidal Model

A general sinusoidal model can be written as $$y = A\sin(B(x-C))+D \quad \text{or} \quad y = A\cos(B(x-C))+D$$

The parameters have clear geometric meanings:

1. Amplitude: $|A|$.
   1. Larger $|A|$ means taller waves.
   2. A negative $A$ reflects the graph in the principal axis (equivalently a reflection in the $x$-axis before shifting up/down).
2. Vertical shift (principal axis): $D$.
   1. The principal axis is the line $y=D$.
3. Period: how long one full cycle takes.
   1. In radians, period $T=\frac{2\pi}{|B|}$.
   2. In degrees, period $T=\frac{360^\circ}{|B|}$.
4. Phase shift: $C$.
   1. Shifts the graph right by $C$ if $C>0$ (because of $(x-C)$).

How Transformations Build a Model

It helps to start from the parent graph $y=\sin x$ or $y=\cos x$ and apply transformations.

Vertical Transformations Are Usually Read Off First

For $y=A\sin(\cdots)+D$:

1. Multiply by $A$ (a vertical dilation by factor $|A|$, and possibly a reflection if $A<0$).
2. Add $D$ (a vertical translation).

These determine the range immediately: $$D-|A| \le y \le D+|A|.$$

Horizontal Transformations Control Speed and Timing

For $y=\sin(B(x-C))$:

1. $B$ produces a horizontal dilation (often described as "frequency change").
   1. Larger $|B|$ means the graph cycles faster (smaller period).
2. $C$ translates the graph horizontally.

Choosing Sine or Cosine in Context

Both can model the same situation, but one may match the "starting position" more naturally:

1. Use cosine when the cycle starts at a maximum or minimum.
2. Use sine when the cycle starts on the principal axis going up or down.

Modelling a Real Situations

Ferris Wheel Height

Sinusoidal functions are powerful because many repeating phenomena come from circular motion.

1. Suppose a Ferris wheel has radius 10 m and its lowest point is 2 m above the ground.
   1. The rider's height varies between 2 m (bottom) and $2 + 2\cdot 10 = 22$ m (top).
   2. The midline is halfway between: $\frac{2+22}{2} = 12$ m.
   3. The amplitude is 10 m.
2. To create a model in terms of time $t$, you also need the period (how long for one rotation).
3. If one rotation takes $P$ seconds, then the height can be modeled by $$h(t) = 10 \cos\left(\frac{360^\circ}{P}t\right) + 12$$
4. This choice starts at maximum height when $t = 0$.
5. If the rider starts at the bottom, you could use $$h(t) = -10 \cos\left(\frac{360^\circ}{P}t\right) + 12$$ or a sine function with a suitable phase shift.

A Swinging Wrecking Ball

1. A swinging object moves back and forth in a repeating pattern, so a sinusoid is a natural model.
2. From a displacement-time graph:
   1. The amplitude is the maximum distance from the central position.
   2. The period is the time for one complete back-and-forth cycle.
3. Consider a wrecking ball graph where:
   1. Amplitude is $5\text{ m}$, so the object reaches $5\text{ m}$ left and $5\text{ m}$ right of its centre.
   2. Period is $6\text{ s}$.
4. If the graph is a sine curve and one cycle is 6 seconds, then using degree measure: $$B=\frac{360}{6}=60$$ so a suitable model could be $$y=-5\sin(60t) \quad \text{or} \quad y=5\sin(-60t)$$ where $t$ represents time in seconds and the angle is in degrees.

Interpreting Parameters in Real Contexts

A model is useful only if each parameter makes sense in the situation.

1. $|A|$ tells you the size of variation (how far a quantity moves from its average).
2. $D$ tells you the average level.
3. $T$ (or $B$) tells you how quickly it repeats.
4. $C$ tells you when the cycle starts relative to your chosen time origin.

Does the Order of Transformations Matter?

1. When building $y=A\sin(B(x-C))+D$, you are combining:
   1. horizontal changes (from $B$ and $C$), and
   2. vertical changes (from $A$ and $D$).
2. Within the same direction, order can matter:
   1. Horizontal: applying a horizontal stretch and a horizontal shift can give different results if you swap the order (because stretching also stretches the shift).
   2. Vertical: applying a vertical stretch and then a vertical shift is not the same as shifting first and then stretching (because stretching would also stretch the shifted amount).
3. A safe approach is to treat the function as written:
   1. Start with the parent $\sin x$ or $\cos x$.
   2. Do what is inside first (work with $B(x-C)$ to understand period and shift).
   3. Apply the outside changes (multiply by $A$, then add $D$).