Problem 25 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\) $$ y=3 \sin \frac{1}{2} x $$

Step-by-Step Solution

Answer

The graph of the function \( y = 3 \sin \frac{1}{2}x \) is a sine wave with an amplitude of 3 and a period of \( 4\pi \). Within the interval from 0 to \( 2\pi \), half of the wave cycle is completed, starting from the origin, reaching maximum at \( (\pi,3) \), returning to zero at \( (2\pi,0) \).

Step 1: Identify key characteristics

The function \( y = 3 \sin \frac{1}{2}x \) is a sine function with an amplitude of 3 (the factor in front of the sine) and a period of \( 4\pi \) (given by \( 2\pi \) divided by the absolute value of the factor in front of \( x \) within the sine function, in this case \( \frac{1}{2} \). The graph will fluctuate between -3 and 3 and complete one full oscillation every \( 4\pi \) units of \( x \).

Step 2: Compute key points

Plot the points for \( x \) at multiples of \( \pi \). Starting from \( x = 0 \), calculate the corresponding \( y \) value in the function and mark the point on the graph. For sine function, these points will provide the peaks, troughs and zero crossings of the wave.

Step 3: Graph the Function

Draw a smooth curve through the points plotted in step 2 to form a complete sine wave within the interval from 0 to \( 2\pi \). The key to get it right is to note that the wave should be symmetric and smooth. The graph should start from the origin (0,0), reach its peak at \( (\pi,3) \), come back to zero at \( (2\pi,0) \), reach its trough at \( (3\pi,-3) \), and come back to zero at \( (4\pi,0) \).

Step 4: Verify the Result

After completing the graph, it's sensible to verify that it meets the fundamental characteristics identified in step 1. It should have a maximum height of 3, a minimum height of -3, start from the origin and complete one full oscillation over the interval \( 4\pi \), in which half of the graph falls in the range of \( 0 \ to \ 2\pi \)

Key Concepts

Amplitude in Trigonometric Functions

Amplitude is a crucial concept when graphing trigonometric functions, as it determines the height of the wave's peaks and troughs. In the sine function given by \( y = 3 \sin \frac{1}{2}x \), the amplitude is 3. This means that the wave will oscillate between 3 and -3. Amplitude can be found by looking at the coefficient in front of the sine or cosine in the function. This coefficient directly influences the vertical stretch of the wave. When graphing:

• The maximum value on the graph of y will be the amplitude, which is 3 here.
• The minimum value will be the negative amplitude, which is -3.
• The graph will pass through the equilibrium position, or midline, at y = 0.

Understanding amplitude helps you sketch trigonometric graphs accurately, ensuring you don't miss out on the range of wave oscillations.

Understanding Periodicity

Periodicity refers to the repeating nature of trigonometric functions over specific intervals. For the function \( y = 3 \sin \frac{1}{2}x \), the period is determined by dividing \( 2\pi \) by the coefficient of \( x \) in the sine function. In this case, the coefficient is \( \frac{1}{2} \), so the period is \( 4\pi \).This means:

• The sine wave will complete one full cycle from start to end over every \( 4\pi \) units of \( x \).
• Within the graph's section from 0 to \( 2\pi \), which is half of the full period, we will see parts of a wave that includes a peak and returns to zero.

Periodic functions are key in understanding how these waves behave over time, which is useful in a wide array of applications, from physics to engineering.

Types of Trigonometric Functions

Trigonometric functions, such as sine, describe relationships between the angles and sides of a right triangle. These functions are fundamental in various scientific fields because they model periodic phenomena, such as sound waves or the motion of pendulums. Key Trigonometric Functions:

• Sine (sin): Represents the vertical position of a rotating point around a circle over time.
• Cosine (cos): Indicates the horizontal position in a similar scenario.
• Tangent (tan): Depicts the ratio of sine to cosine, describing slope or elevation angles frequently.

The sine function is especially important, as demonstrated in our example, because it showcases smooth, repetitive oscillations characteristic of many natural processes.

Graphing Techniques for Sine Functions

Graphing sine functions involves identifying key characteristics such as amplitude and period and plotting points based on these attributes. With the function \( y = 3 \sin \frac{1}{2}x \), we apply these principles for accurate plotting. Here's a straightforward approach:

• Start by plotting at key points like \( x = 0 \), where \( y \) is zero. This is your starting point.
• Move to \( x = \pi \), where \( y \) reaches the amplitude of 3. This is the peak of the sine wave.
• Continue to \( x = 2\pi \), bringing \( y \) back to zero.
• Check the symmetry by ensuring \( x = 3\pi \) reaches the trough at \( y = -3 \), and \( x = 4\pi \) returns \( y \) to zero.

As you sketch, ensure the curve is smooth and symmetrical around these points. Graphing these points provides a visual interpretation of mathematical principles and helps in understanding how changes in function parameters alter the graph's shape.