Beginning with the graphs of y=sinx or y=cosx, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. q(x)=3.6cos(24πx)+2
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Start with the basic graph of y = \cos(x), which is a cosine wave with an amplitude of 1, a period of 2\pi, and a midline at y = 0.
Apply a horizontal scaling transformation to the function. The term \frac{\pi x}{24} inside the cosine function indicates a horizontal stretch. The period of the cosine function is given by \frac{2\pi}{\frac{\pi}{24}} = 48. This means the graph completes one full cycle over an interval of 48 units on the x-axis.
Apply a vertical scaling transformation. The coefficient 3.6 in front of the cosine function indicates a vertical stretch. This changes the amplitude of the cosine wave from 1 to 3.6, meaning the wave will oscillate between -3.6 and 3.6.
Apply a vertical shift. The +2 at the end of the function indicates a vertical shift upwards by 2 units. This moves the midline of the cosine wave from y = 0 to y = 2, so the wave will now oscillate between -1.6 and 5.6.
Combine all transformations to sketch the graph of q(x) = 3.6\cos\left(\frac{\pi x}{24}\right) + 2. The graph is a cosine wave with a period of 48, an amplitude of 3.6, and a midline at y = 2. Use a graphing utility to verify the transformations and the final graph.
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