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

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

$y = 3\sin\bigl(2x + \tfrac{\pi}{4}\bigr) - 1$

Use a graphing utility to sketch the function and list the coordinates of one full cycle for

$y = -2\cos\bigl(\tfrac{\pi}{3}x - \tfrac{\pi}{6}\bigr) + 4.$

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

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

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

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.

Use a graphing utility to 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 $6$.

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

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.

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.

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

Given the data 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

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