Problem 28 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y=\sin{3}(x+\frac{\pi}{3})\) from 0 to 2\(\pi\) is a sine curve with a period of \(\frac{2\pi}{3}\), shifted left by \(\frac{\pi}{3}\), covering the entire interval from 0 to \(2\pi\).

Step 1: Determine the period of the function

Given that the frequency multiplier is 3, the period of this sine function will not be \(\pi\). In general, the period of a sine function is given by \(\frac{2\pi}{|frequency|}\). Therefore, in this case, the period is calculated as \(T=\frac{2\pi}{3}\)

Step 2: Identify the phase shift

The phase shift of the function is found in the variable portion of the argument, \(x + \frac{\pi}{3}\). The value accompanying the \(x\), \(\frac{\pi}{3}\), represents the phase shift. Since it is positive, the graph will shift to the left by \(\frac{\pi}{3}\).

Step 3: Plot key points within the interval and sketch the graph

Within the interval from 0 to \(2\pi\), four key points that define the shape of one period of a sine function can be plotted: the starting point, peak, mid-point (or 'trough'), and end-point. Given the period and phase shift calculated in earlier steps, these key points for this specific function are: Start - \(\frac{\pi}{3}\), Peak - \(\frac{\pi}{3}+\frac{T}{4}\), Trough - \(\frac{\pi}{3}+\frac{T}{2}\), End - \(\frac{\pi}{3}+T\). Now, these points can be plotted on an \(x-y\) graph. Link the points smoothly to form a sine curve, making sure to continue the pattern outside of one period to fill the entire interval from 0 to \(2\pi\).

Key Concepts

Graphing Sine Functions

Graphing sine functions can be an exciting journey through the rhythm of waves. Understanding how to graph these functions involves recognizing the basic structure of the sine curve and how it shifts or scales with various parameters.
The sine function, generally denoted as \(y = a \sin(bx + c) + d\), has a few key parts that affect its graph:

• \(a\): Amplitude - Determines the height of the wave.
• \(b\): Frequency - Affects the period, which is the length of one wave cycle.
• \(c\): Phase Shift - Moves the graph horizontally.
• \(d\): Vertical Shift - Moves the graph up or down.

To graph a specific sine function, plot key points across one complete cycle. These points include the start, peak, trough, and end of the cycle. A sine function typically oscillates between its maximum and minimum values following a smooth curve. You shift, stretch, or compress this basic wave according to the parameters above to create a graph that fits your specific function.

Period of Trigonometric Functions

The concept of the period is essential when it comes to trigonometric functions like sine. The period refers to the length of one complete cycle of the function. For a basic sine function \(y = \sin x\), this period is \(2\pi\).
However, when the frequency multiplier \(b\) in \(y = \sin(bx)\) changes, so does the period. The period \(T\) of a sine function is calculated as:\[T = \frac{2\pi}{|b|}\]Thus, increasing \(b\) reduces the period, meaning the function completes its cycle faster, and the waves appear more frequent. Conversely, a smaller \(b\) results in a longer period with more stretched waves.
In our example, the function \(y = \sin 3(x + \frac{\pi}{3})\) has a frequency of 3, leading to a period of \(\frac{2\pi}{3}\). This means that within the interval \([0, 2\pi]\), it completes multiple cycles, making the waves appear more congested.

Phase Shift in Trigonometric Functions

Phase shift is another crucial factor that affects the positioning of the sine function on the graph. It involves moving the sine wave left or right along the x-axis.
In a sine function expressed as \(y = \sin(bx + c)\), the term \(c\) determines the phase shift. The rule is:

• If \(c\) is positive, the graph shifts to the left.
• If \(c\) is negative, the graph shifts to the right.

Determine the phase shift by calculating \(-\frac{c}{b}\). For the example \(y = \sin 3(x + \frac{\pi}{3})\), the shift is given by:\[-\frac{\frac{\pi}{3}}{3} = -\frac{\pi}{9}\]Thus, the graph shifts by \(\frac{\pi}{3}\) units to the left.
This horizontal movement does not affect the height or frequency of the wave but changes where the cycle starts on the x-axis. Understanding phase shifts helps in predicting the graph's starting point, crucial for achieving accuracy in trigonometric graphing.