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Chapter 5: Problem 16

Find the period and amplitude. $$y=\frac{3}{2} \cos \frac{\pi x}{2}$$

Short Answer

Expert verified
The amplitude of the function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\) is \(\frac{3}{2}\) and the period is \(4\).

Step by step solution

01

Identify the Amplitude

The amplitude \(A\) is the multiplier of the cosine function, which determines its maximum magnitude or height. In the function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\), the amplitude is the coefficient in front of the cosine, which is \(\frac{3}{2}\).
02

Identify the Period

The period \(T\) is found by the logic \(B=\frac{2\pi}{T}\). In this function, the coefficient in front of \(x\) inside the cosine function is \(\frac{\pi}{2}\). So, equating it to \(\frac{2\pi}{T}\), we solve for \(T\): \(\frac{2\pi}{T} = \frac{\pi}{2} \implies T = \frac{2\pi}{(\pi/2)} = 4\). So, the period of the function is \(4\).

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Most popular questions from this chapter

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=180^{\circ}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-150^{\circ}$$

Use the procedure in Exercise 143 and a graphing utility to create a table of values and make a conjecture about the relationship between \(\cos \theta\) and \(\cos \left(180^{\circ}-\theta\right)\) for an acute angle \(\theta\).

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=-4+e^{3 x}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-300^{\circ}$$
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