Functions - Sinusoidal functions

Determine the amplitude, period, and principal axis for the function $f(x) = 4 \sin(2x - 60^{\circ}) + 3$.

1. Compare with the general form $y = a \sin(b(x - c)) + d$.
2. Identify $a = 4$. The amplitude is $|4| = 4$.
3. Identify $b = 2$. The period is $\frac{{360^{\circ}}}{{b}} = \frac{{360^{\circ}}}{{2}} = 180^{\circ}$.
4. Identify $d = 3$. The equation of the principal axis is $y = 3$.

The amplitude is the coefficient of the sine term. The period is calculated by dividing $360^{\circ}$ by the frequency coefficient $b$. The constant term $d$ determines the vertical shift and the principal axis.

A Ferris wheel has a maximum height of $30\text{ m}$ and a minimum height of $2\text{ m}$. It takes $40\text{ seconds}$ to complete one full revolution. Find a cosine model for the height $h$ in terms of time $t$, assuming the passenger starts at the bottom at $t = 0$.

1. Find amplitude $a$: $a = \frac{{30 - 2}}{{2}} = 14$
2. Find vertical shift $d$: $d = \frac{{30 + 2}}{{2}} = 16$
3. Find $b$ using the period $P = 40$: $b = \frac{{360^{\circ}}}{{40}} = 9$
4. Since the passenger starts at the bottom (the minimum), we use a negative cosine graph: $h(t) = -14 \cos(9t) + 16$

The amplitude is half the range. The vertical shift is the average of the max and min. Because the motion starts at the minimum point rather than the maximum or the midline, a negative cosine function is the simplest way to model the starting position.

Given the function $g(x) = 5 \cos(3x) - 2$, find the maximum and minimum values of the function.

1. The principal axis is $y = -2$ and the amplitude is $5$.
2. Maximum value: $d + a = -2 + 5 = 3$
3. Minimum value: $d - a = -2 - 5 = -7$

The maximum value is the principal axis plus the amplitude, and the minimum value is the principal axis minus the amplitude.

1. Amplitude: The coefficient $a = 3$, so the amplitude is $|3| = 3$.
2. Period: The coefficient $b = 0.5$. In radians, $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{0.5} = 4\pi \approx 12.57$.
3. Principal Axis: Since $d = 0$, the principal axis is $y = 0$ (the x-axis).
4. Range: The maximum value is $0 + 3 = 3$ and the minimum value is $0 - 3 = -3$.

To sketch the graph, we start at the origin $(0,0)$ because it is a sine function with no horizontal shift. One full cycle ends at $x = 4\pi$. The peak occurs at one-quarter of the period ($x = \pi$) and the trough at three-quarters ($x = 3\pi$).

1. Depth at 2:00 AM: Substitute $t = 2$ into the function: $D(2) = 2 \cos(30^{\circ} \times 2) + 10 = 2 \cos(60^{\circ}) + 10 = 2(0.5) + 10 = 11\text{ m}$.
2. Time when depth is 9m: Set $D(t) = 9$: $9 = 2 \cos(30^{\circ} t) + 10$ $-1 = 2 \cos(30^{\circ} t)$ $\cos(30^{\circ} t) = -0.5$ $30^{\circ} t = \arccos(-0.5) = 120^{\circ}$ $t = \frac{120}{30} = 4\text{ hours}$. So, the depth is $9\text{ m}$ at 4:00 AM.

The cosine function starts at its maximum value at $t=0$. The vertical shift $d=10$ means the average depth is $10\text{ m}$, and it fluctuates $2\text{ m}$ above and below this value.