Question Type 17: Using sinusoidal models to create graphs using technology Exercises

Given the points $(0,5), (2,9), (4,5), (6,1)$ which suggest a sinusoidal pattern, use a graphing calculator to find a model of the form

$y = A \sin(Bx + C) + D$

that fits these four points.

Sketch the graph of the function $y = -2\cos\left(\frac{\pi}{3}x - \frac{\pi}{6}\right) + 4$ for one full cycle. List the coordinates of the key points (maximum, minima, and points of intersection with the central axis).

Sketch by technology the function

$y = 1.5\sin\bigl(0.5x + \tfrac{\pi}{3}\bigr) - 2$

and state its amplitude, period, phase shift, and midline.

A water wave oscillates with amplitude $0.5\text{ m}$, period $8\text{ s}$, and has a crest $0.5\text{ m}$ above the still-water level at $t=2\text{ s}$. Write a model

$y = A\sin(Bt + C)$

for the displacement $y$ from still-water level.

Identify the amplitude, period, midline, and phase shift of the function

$y = 3\sin\left(2x + \frac{\pi}{4}\right) - 1$

Determine a sinusoidal model of the form

$y = A\cos(Bx + C) + D$

that has maximum $y=7$ at $x=1$, minimum $y=-1$ at $x=4$, and period $6$.

A Ferris wheel of radius 10 m has its center 12 m above ground and completes one revolution every 4 minutes. If a passenger boards at the lowest point at $t=0$, write a sinusoidal model $h(t)$ for the height above ground after $t$ minutes.

Model the daily temperature in a desert by a sinusoid with maximum $40^\circ\text{C}$ at 15:00, minimum $20^\circ\text{C}$ at 03:00, and period $24\text{ h}$. Write $T(h)=A\cos\left(B(h-h_0)\right)+D$ where $h$ is hours after midnight.

The function $f(t)=4\cos\bigl(2t - \frac{\pi}{4}\bigr)+1$ models a vibration, where $t$ is time in seconds. Use technology to find the smallest positive $t$ such that $f(t)=5$.

Given the data points $(t, h)$ for tidal height over time: $(0, 2), (3, 5), (6, 2), (9, -1)$, use technology to fit a sinusoidal curve and state the equation in the form

$h(t) = A\sin(Bt + C) + D.$