Question

Determine the amplitude and period of each function without graphing. $$ y=-\frac{1}{7} \cos \left(\frac{7}{2} x\right) $$

Short answer

Amplitude: \( \frac{1}{7} \), Period: \( \frac{4\pi}{7} \)

Step-by-step solution

Identify the standard form of the cosine function

The standard form of the cosine function is given by \[ y = a \, \text{cos}(b x) \]Here, the given function is \[ y = -\frac{1}{7} \, \text{cos}\left(\frac{7}{2} x\right) \]

Determine the amplitude

The amplitude of the cosine function \[ y = a \, \text{cos}(b x) \]is given by \[ |a| \]In this case, \[ a = -\frac{1}{7} \]Therefore, the amplitude is \[ \left| -\frac{1}{7} \right| = \frac{1}{7} \]

Determine the period

The period of the cosine function \[ y = a \, \text{cos}(b x) \]is given by \[ \frac{2\pi}{b} \]Here, \[ b = \frac{7}{2} \]Therefore, the period is \[ \frac{2\pi}{\frac{7}{2}} = \frac{2\pi \cdot 2}{7} = \frac{4\pi}{7} \]

Key concepts

cosine function properties

The cosine function is one of the fundamental trigonometric functions, often used to model periodic phenomena. Here are some important properties:

The standard form of the cosine function is written as \[ y = a \, \text{cos}(b x) \]
The variable \ a \ controls the amplitude of the function. It determines how high and low the curve goes.
The variable \ b \ affects the period of the function. This determines how long it takes for the function to repeat its pattern.

The function \ \text{cos}(x) \ oscillates between -1 and 1. When multiplied by a constant \ a \, this range changes to \ [-a, a] \. The period of the basic cosine function is \ 2\pi \, meaning it repeats its values every \ 2\pi \ units along the x-axis.

The given function \ y = -\frac{1}{7} \, \text{cos}\left(\frac{7}{2} x\right) \ follows the same basic properties, adjusted by its specific values of \ a \ and \ b \.

amplitude calculation

The amplitude of a trigonometric function like the cosine function represents the maximum distance from the midline to the peak or trough of the curve.

In the standard form \ y = a \, \text{cos}(b x) \, the amplitude is given by the absolute value of \ a \: \[ \text{Amplitude} = |a| \]
For the function \ y = -\frac{1}{7} \, \text{cos}\left(\frac{7}{2} x\right) \, the coefficient \ a \ is \ -\frac{1}{7} \.
The absolute value of \ -\frac{1}{7} \ is \ \frac{1}{7} \, so the amplitude is \ \frac{1}{7} \.

Since amplitude is always positive, it doesn't matter if the coefficient is negative; only its absolute value is considered.

Thus, the amplitude for our given cosine function is \ \frac{1}{7} \.

period calculation

The period of a cosine function is the distance along the x-axis for one complete cycle of the wave.

In the standard form \ y = a \, \text{cos}(b x) \, the period is calculated as follows:
\[ \text{Period} = \frac{2\pi}{b} \]
For our function \ y = -\frac{1}{7} \, \text{cos}\left(\frac{7}{2} x\right) \, the coefficient \ b \ is \ \frac{7}{2} \.

Substituting this value in, we get:

\[ \text{Period} = \frac{2\pi}{\frac{7}{2}} = \frac{2\pi \cdot 2}{7} = \frac{4\pi}{7} \]

Thus, the period of our given function is \ \frac{4\pi}{7} \, meaning the function repeats its pattern every \ \frac{4\pi}{7} \ units along the x-axis.