Question

determine the amplitude and period of the following function without graphing.
y = -\frac{6}{7}\cos(\frac{7}{6}x)
for the function given, the amplitude is \frac{6}{7}.
(simplify your answer. type an exact answer, using pi as needed. use integers or fractions for any numbers in the expression.)
for the function given, \omega = \square, so that the period = t = \square.
(simplify your answers. type exact answers, using pi as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Identify the general form of cosine function

The general form of a cosine function is \(y = A\cos(\omega x)\), where \(A\) is the amplitude and the period \(T=\frac{2\pi}{\omega}\).
For the function \(y =-\frac{6}{7}\cos(\frac{7}{6}x)\), comparing with the general form \(y = A\cos(\omega x)\), we can see that \(\omega=\frac{7}{6}\).

Step2: Calculate the period

Since \(T = \frac{2\pi}{\omega}\) and \(\omega=\frac{7}{6}\), then \(T=\frac{2\pi}{\frac{7}{6}}\).
Using the rule for dividing by a fraction (\(a\div\frac{b}{c}=a\times\frac{c}{b}\)), we have \(T = 2\pi\times\frac{6}{7}=\frac{12\pi}{7}\).

Answer:

\(\omega=\frac{7}{6}\), \(T = \frac{12\pi}{7}\)