Modeling and Analyzing a Shifted Trigonometric Function from Real Data
A set of real data indicates periodic behavior and suggests that it can be modeled by a function of the form \(y = A \sin(B(t – C)) + D\). The data shows a maximum value of 15 at \(t = 2\) seconds and a minimum value of 5 at \(t = 5\) seconds. (a) Construct […]
Tidal Modeling using Cosine Function Transformation
A coastal engineer is modeling the tidal height, \(H(t)\), in meters at a certain location as a function of time, \(t\) (in hours), using a cosine function of the form: \( H(t) = A \cos\left(\frac{\pi}{6}t + \phi\right) + D \). Given that the tidal range (the difference between high tide and low tide) is 4 […]
Part a
For the function transformation, deduce the new amplitude of \(g(x) = 3 \sin(2x – \frac{\pi}{3})\) when compared to \(f(x) = \sin(x)\).
Graphing Transformed Sine Functions
Sketch the graph of the function \(y = 3\sin(x – \tfrac{\pi}{4})\). In your explanation, clearly indicate the amplitude, phase shift, and period of the sine curve.
Phase Shift Analysis for Sine Functions
Consider the function \(y=\sin(x-\pi/3)\). Answer the following: (a) Determine the phase shift of the function. (b) Sketch the graph of \(y=\sin(x-\pi/3)\) over the interval \(0\leq x\leq 2\pi\), indicating key intercepts and extrema.