Question

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

Short answer

Amplitude: 1, Period: 4\pi

Step-by-step solution

- Identify Amplitude

The amplitude of a sinusoidal function of the form \( y = a \, \text{sin}(bx) \) is given by the absolute value of the coefficient 'a'. In this problem, \( y = -\text{sin}\big(\frac{1}{2}x\big) \), so the coefficient 'a' is -1. Therefore, the amplitude is: \[ |a| = |-1| = 1 \]

- Identify Period

The period of a sinusoidal function of the form \( y = a \, \text{sin}(bx) \) is given by \( \frac{2\pi}{|b|} \). In this problem, \( y = -\text{sin}\big(\frac{1}{2}x\big) \), so the coefficient 'b' is \( \frac{1}{2} \). Therefore, the period is: \[ \text{Period} = \frac{2\pi}{|\frac{1}{2}|} = \frac{2\pi}{\frac{1}{2}} = 4\pi \]

Key concepts

sinusoidal function

A sinusoidal function is a mathematical function that describes a smooth, periodic oscillation. These types of functions are very common in various fields, including physics, engineering, and signal processing. Technically speaking, a sinusoidal function is a form of either the sine or cosine function.

The general form of a sinusoidal function can be written as:
y = a \, \text{sin}(bx + c) + d

Where:

• a is the amplitude
• b affects the period
• c affects the phase shift
• d affects the vertical shift

A deeper understanding of the parameters 'a' and 'b' is essential to grasp the concepts of amplitude and period fully.

In the problem we are solving, the sinusoidal function is
\( y = -\sin \left( \frac{1}{2} x \right) \). This is a typical sine function with 'a' as -1 and 'b' as \( \frac{1}{2} \), and no phase shift or vertical shift. We will use these values to calculate the amplitude and period.

amplitude

The amplitude of a sinusoidal function is a measure of how 'tall' the waves are. In other words, it is the maximum distance from the middle of the wave (the mean or equilibrium position) to the peak or trough.

Technically, the amplitude is determined by the 'a' coefficient in the standard formula \( y = a \, \text{sin}(bx) \). Specifically, it is the absolute value of 'a':
\[ |a| \]

In our exercise, we have:
\( y = -\sin \left( \frac{1}{2} x \right) \),
and the coefficient 'a' is -1.

Therefore, the amplitude is:

\[ \text {Amplitude} = |-1| = 1 \]

This means that the maximum height of the wave above the middle line (or its depth below the middle line) is 1 unit. Remember, the amplitude is always a positive number.

period calculation

The period of a sinusoidal function refers to the length of one complete cycle of the wave. In simpler terms, it's the distance (along the x-axis) it takes for the wave to start repeating itself.

The general formula for finding the period of a sine function \( y = a \, \text {sin}(bx) \) is:
\[ \text {Period} = \frac{2\pi}{|b|} \]

For the given problem, we have:
: \( y = -\sin \left( \frac{1}{2} x \right) \),
and the coefficient 'b' is \( \frac{1}{2} \).

Substituting \( b = \frac{1}{2} \) into the formula gives us:
\[ \text {Period} = \frac{2\pi}{|\frac{1}{2}|} = \frac{2\pi}{\frac{1}{2}} = 4\pi \]

So, the period of the function \( y = -\sin \left( \frac{1}{2} x \right) \) is \( 4\pi \). This means that it takes \( 4\pi \) units along the x-axis for the function to complete one full cycle and repeat itself. Knowing this helps in graphing these functions accurately and understanding their behavior over specific intervals.