91Ó°ÊÓ

The amplitude of a trigonometric function is a measure of its vertical stretch or compression.
For a cosine function of the form \(y = A \cos(Bx + C) + D \), the amplitude is represented by \(A\).
It indicates the maximum distance the graph of the function rises or falls from its central axis, which is determined by the value of \(D\).

In the function you provided, \(y = \frac{2}{3} \cos \frac{\pi x}{10}\), the amplitude is explicitly given as \(\frac{2}{3}\).
This tells us that the graph of this cosine function will reach a peak value of \(\frac{2}{3}\) and a minimum value of \(-\frac{2}{3}\).

This is important because:

• It shows the extent of oscillation from the mean position.
• The amplitude remains the same regardless of the function's period or horizontal shifts.

Remember, amplitude is always a positive value, even when the coefficient is negative, as it refers to a distance.

The period of a trigonometric function indicates the interval after which the function's pattern repeats itself.
For cosine functions, such as \(y = A \cos(Bx + C) + D \), the period is calculated by the formula \(\frac{2\pi}{|B|}\).
Here, the period is determined by the coefficient \(B\) in the argument of the cosine.

In the function \(y = \frac{2}{3} \cos \frac{\pi x}{10}\), \(B\) is \(\frac{\pi}{10}\).
Using the formula, the period is \(\frac{2\pi}{\frac{\pi}{10}}\).
Simplify this to get \(20\), meaning the function's cycle repeats every 20 units on the x-axis.

Understanding the period:

• Aids in graphing; helps to know how long one cycle is.
• Informs about the frequency; longer periods mean a lower frequency and vice versa.

This regular repetition is a fundamental aspect of trigonometric functions and helps in predicting function behavior over extended intervals.

The cosine function is one of the basic trigonometric functions, often written as \(\cos(x)\).
It describes the relationship between the angles and sides of a right-angle triangle, but its application extends beyond geometry into various fields like physics and engineering.
In its simplest form, the function oscillates between -1 and 1.

The standard form of a cosine function is expressed as \(y = A \cos(Bx + C) + D\), where:

• \(A\) affects the amplitude (height of peaks and troughs).
• \(B\) affects the period (length of one cycle).
• \(C\) provides the phase shift (horizontal shift).
• \(D\) determines the vertical shift (moves the graph up or down).

In the given exercise \(y = \frac{2}{3} \cos \frac{\pi x}{10}\), we see a transformation of the standard cosine:

• With amplitude \(\frac{2}{3}\), it is vertically compressed.
• A period of \(20\) signifies slower oscillation due to the \(\frac{\pi}{10}\) factor affecting \(x\).

These modifications help in tailoring the function to fit specific modeling needs and conditions.