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Problem 518
518
The diagram shows part of the graph of \(y = \sin(a(x + b))\), where \(a\) and \(b\) are positive constants. The graph is plotted with the x-axis ranging from \(-\frac{2}{3}\pi\) to \(2\pi\) and the y-axis ranging from -1 to 1. State the value of \(a\) and one possible value of \(b\).
Solution
The function \(y = \sin(a(x + b))\) is a sine wave with an amplitude of 1. The period of the sine function is given by \(\frac{2\pi}{a}\). From the graph, the period is \(4\pi\), so:
\(\frac{2\pi}{a} = 4\pi\)
Solving for \(a\), we get:
\(a = \frac{1}{2}\)
The phase shift is determined by \(b\). The graph starts at \(x = -\frac{\pi}{3}\), which corresponds to the phase shift. Therefore, \(b = \frac{\pi}{3}\).