Problem 1 - Algebra 2

Question

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{1}{2} \sin \theta $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{2}\), Period: \(2\pi\).

Step 1: Identify the Amplitude

The amplitude of a sine function in the form \( y = a \sin b\theta \) is the absolute value of \( a \). Here, \( a = \frac{1}{2} \), so the amplitude is \( |a| = \frac{1}{2} \).

Step 2: Determine the Period of the Function

The period of a sine function in the form \( y = a \sin b\theta \) is \( \frac{2\pi}{b} \). In this case, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Graph the Function

Using the amplitude \( \frac{1}{2} \) and period \( 2\pi \), graph the function. Since this is a sine function, start at \( (0,0) \), reach the maximum \( \frac{1}{2} \) at \( \frac{\pi}{2} \), return to 0 at \( \pi \), reach the minimum \( -\frac{1}{2} \) at \( \frac{3\pi}{2} \), and complete one cycle by returning to 0 at \( 2\pi \). Plot similar cycles on both sides if drawing over a wider interval.

Key Concepts

Amplitude

The concept of amplitude is crucial when studying sine functions. In a sinusoidal function like \( y = a \sin b\theta \), the amplitude is the peak value that the function reaches. It shows how "tall" the wave is, and is a measure of how far the function's values deviate from their central position, which is typically zero.

The amplitude is determined from the coefficient \( a \) in the function. Specifically, it is the absolute value of \( a \), noted as \(|a|\). This means the amplitude is always a positive number, ensuring it reflects the size of the wave no matter the direction or phase.

• If \( a > 0 \), the wave is above the midline.
• If \( a < 0 \), the wave inverts, putting its peak below the midline.

In our example, \( y = \frac{1}{2} \sin \theta \), the amplitude is \( \frac{1}{2} \). This indicates that the wave oscillates between \( \frac{1}{2} \) and \( -\frac{1}{2} \) as it repeats.

Understanding amplitude helps not only in predicting the waves' motion but also in applications such as sound waves, where it influences volume, or in alternating current electricity, where it affects voltage power.

Period of a Function

The period of a function is essentially the duration it takes for a function's values to repeat. For trigonometric functions like sine, the concept of period indicates how long the wave takes to complete one full cycle and start over. This is vital for understanding oscillating systems.

Mathematically, for the sine function \( y = a \sin b\theta \), the period is determined by the formula \( \frac{2\pi}{b} \). This equation tells us how the scale of the \( \theta \) axis affects the frequency of repetition.

• If \( b = 1 \), the standard sine wave with a period of \( 2\pi \) applies, which is approximately 6.28 radians.
• A larger \( b \) means more cycles are squeezed into the same horizontal space, decreasing the period.
• A smaller \( b \) stretches the cycle, increasing the period.

In our specific case of \( y = \frac{1}{2} \sin \theta \), since \( b = 1 \), the period remains \( 2\pi \). This means that from \( 0 \) to \( 2\pi \), the function completes one full cycle of its waveform, reaching both its maximum and minimum once.

Sine Function Graph

Graphing the sine function provides a visual understanding of its oscillatory nature. A sine graph has a distinct shape like rolling hills, often referred to as a sine wave.To graph the sine function, begin by noting its key characteristics, such as amplitude and period. For the function \( y = \frac{1}{2} \sin \theta \):

• Start at the origin \((0,0)\), indicating that when \( \theta = 0 \), sine is zero.
• The amplitude of \( \frac{1}{2} \) indicates that the peak of one wave hits \( y = \frac{1}{2} \) and the trough falls to \( y = -\frac{1}{2} \).
• It reaches its first peak at \( \frac{\pi}{2} \), crosses zero again at \( \pi \), hits the minimum at \( \frac{3\pi}{2} \), and completes a cycle back to zero at \( 2\pi \).

By plotting these points and drawing a smooth curve through them, you visualize one cycle of the sine wave. If you wish, you can extend this pattern in both directions to represent more cycles.

This knowledge helps in numerous fields, such as physics for studying pendulums, or electronics for analyzing alternating currents. The graph not only represents the function but also conveys how the function changes over its domain.