Problem 16 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\sin \frac{4}{5} x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=-\sin \frac{4}{5} x\) is 1 and the period is \(\frac{5\pi}{2}\) units.

Step 1: Find the amplitude

The coefficient of the sine function is -1. The amplitude is the absolute value of this coefficient, so the amplitude of the function is 1.

Step 2: Find the period

The period of the standard sine function is \(2\pi\). The coefficient of x in the function is \(\frac{4}{5}\). The period of the function \(y=-\sin \frac{4}{5} x\) should be the period of the standard sine function divided by the absolute value of the coefficient of x. Therefore, the period of the function \(y=-\sin \frac{4}{5} x\) is \(\frac{2\pi}{|\frac{4}{5}|} = \frac{5\pi}{2}\) units.

Step 3: Sketch the graph of the function

Now, sketch a sine wave starting from the origin. The crucial points on a sine wave are at its peak, its lowest point, and when it crosses the x-axis. Consider that the function is negative, so the wave starts from the origin and goes down first. Label the period on the x-axis which should be at \(\frac{5\pi}{2}\). Also, the amplitude is 1, so mark these points on the y-axis. Show a complete wave within one period, starting from the origin point, reaching the peak at \(\frac{1}{4}\)of the period, crossing back the x axis at half of the period, reaching the minimum at \(\frac{3}{4}\) of the period, and finally crossing the x axis at the end of the period.

Key Concepts

Amplitude

The amplitude of a trigonometric function describes the height of its peaks and the depth of its troughs from the central value, which is usually the horizontal axis. For any sine function expressed as \( y = a \sin(bx + c) + d \), the amplitude is determined by the absolute value of the coefficient \( a \). In the given function \( y = -\sin \frac{4}{5} x \), the coefficient of the sine term is \(-1\). Thus, the amplitude is the absolute value of \(-1\), which is 1.

In practical terms, this means that the sine wave will oscillate between 1 and -1, effectively creating a total oscillation height of 2 units. When dealing with trigonometric functions, knowing the amplitude helps in predicting the highest and lowest values the function can reach.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle of its wave. For a standard sine function, the period is \(2\pi\), meaning the wave completes a full cycle every \(2\pi\) units along the x-axis.

For a function like \( y = -\sin \frac{4}{5} x \), the period is determined by the factor multiplying \( x \) within the sine function—in this case, \( \frac{4}{5} \). To find the function's period, divide the standard period \(2\pi\) by the absolute value of this coefficient. So, calculate \( \frac{2\pi}{\left| \frac{4}{5} \right|} = \frac{5\pi}{2} \). This means the wave takes \(\frac{5\pi}{2}\) units to complete one full oscillation. Understanding the period is crucial when sketching and analyzing waves as it defines the repetition intervals along the x-axis.

Sine Function

The sine function, denoted as \( y=\sin(x) \), is one of the fundamental trigonometric functions used to model periodic phenomena. Its characteristic wave-like shape oscillates above and below the x-axis, capturing natural cycles we see in fields like physics and engineering. The base sine function has specific features:

• Amplitude of 1
• Period of \(2\pi\)
• It starts at zero, rises to a maximum, descends through zero to a minimum, and returns to zero to complete one cycle

The specific function \( y = -\sin \frac{4}{5} x \) flips the wave upside down due to the negative sign and compresses the cycle horizontally because of the coefficient \( \frac{4}{5} \). Despite these alterations, the function still retains the periodic nature of the sine wave, making it a versatile tool for modeling.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting a curve that represents the function over one or more periods. To graph \( y = -\sin \frac{4}{5} x \):

• Start at the origin to observe the function's behavior.
• Since the function is negative, begin by plotting the curve downward to the first trough.
• Next, the graph crosses the x-axis halfway through the period, \(\frac{5\pi}{4}\) in this case, then reaches a peak at \(\frac{5\pi}{2}\).

Because of the calculated amplitude of 1, the maximum height above and below the x-axis is 1 and -1, respectively.

Label the x-axis with significant points: the start, the x-axis crossing, and the end at \(\frac{5\pi}{2}\), ensuring the curve represents one full cycle. Graphing helps visualize the function's periodic nature, making it easier to comprehend wave characteristics.