IB Mathematics Applications & Interpretation 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

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

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 $$f(t)=4\cos (2t-\frac{\pi }{4})+1$$ 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

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