Problem 21 - Algebra and Trigonometry

Question

\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=10 \sin \frac{1}{2} x $$

Step-by-Step Solution

Answer

Amplitude: 10, Period: \(4\pi\)

Step 1: Identify the form of the sine function

The given function is \( y = 10 \sin \frac{1}{2}x \). This is in the form \( y = a \sin(bx) \), where \( a = 10 \) and \( b = \frac{1}{2} \).

Step 2: Determine the amplitude

The amplitude of a sine function \( y = a \sin(bx) \) is the absolute value of \( a \). Therefore, the amplitude is \( |10| = 10 \).

Step 3: Determine the period

The period of a sine function \( y = a \sin(bx) \) is calculated as \( \frac{2\pi}{b} \). Substituting \( b = \frac{1}{2} \), the period is \[ \frac{2\pi}{\frac{1}{2}} = 4\pi \].

Step 4: Sketch the graph

To sketch the graph of \( y = 10 \sin \frac{1}{2}x \), start by plotting key points:- Start at \((0, 0)\).- Reach the maximum at \(x = 2\pi\) with \( y = 10 \).- Come back to \( y = 0 \) at \(x = 4\pi\).- Reach the minimum at \(x = 6\pi\) with \( y = -10 \).- Return to \( y = 0 \) at \(x = 8\pi\).This completes one full cycle. Repeat these points to show the periodic nature because the period is \(4\pi\).

Key Concepts

Amplitude of Sine Function

In trigonometry, one of the key features of a sine function is its amplitude. The amplitude tells us how tall or short the wave appears on the graph. For the sine function given in the exercise, \(y = 10 \sin \frac{1}{2}x\), we recognize it in the standard form of \(y = a \sin(bx)\). Here, \(a = 10\) is the amplitude.

The amplitude is simply the absolute value of \(a\). Thus, the amplitude of \(y = 10 \sin \frac{1}{2}x\) is \(|10| = 10\). This signifies that the wave reaches a maximum height of 10 units and a minimum height of -10 units from the center position (which is the x-axis in a standard grid).

Amplitude is crucial in understanding how intense a wave is. A larger amplitude means a higher peak and deeper trough, whilst a smaller amplitude indicates a flatter wave.

Period of Sine Function

The period of a sine function is the length of one complete cycle of the wave. It tells us how often the wave repeats as we move along the x-axis. In our function \(y = 10 \sin \frac{1}{2}x\), the general form \(y = a \sin(bx)\) gives us a way to find the period by focusing on \(b\), which is \(\frac{1}{2}\) in this case.

To find the period, we use the formula \( \frac{2\pi}{b} \). Plugging in \( b = \frac{1}{2} \), we find the period to be:

• \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)

This means the sine wave takes a total of \(4\pi\) units along the x-axis to complete one full cycle from any starting point to that exact point again as it continues onwards. Understanding the period helps us determine the spacing of the wave's peaks and valleys.

Graphing Sinusoidal Functions

Graphing sinusoidal functions like sine can initially seem challenging, but it becomes systematic once key properties like amplitude and period are understood. Let's dive into graphing the example \(y = 10 \sin \frac{1}{2}x\).

To start, identify key points within one period, which is determined earlier to be \(4\pi\). These points are essential:

• Starting Point: (0, 0) where the wave begins.
• Maximum: At \(x = 2\pi\), the wave will reach a peak of \(y = 10\).
• Return to Zero: At \(x = 4\pi\), the wave returns to cutting the x-axis.
• Minimum: At \(x = 6\pi\), reaching a low point of \(y = -10\).
• Ending Point: At \(x = 8\pi\), it returns to z zero and a complete cycle ends.

Once plotted, repeat these points to show that this wave shape will continue indefinitely both to the left and right. Breaking it down into such sections makes sketching a breeze and highlights the periodic, repeating nature of sine waves effectively.