Problem 44 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=\sin 4 x $$

Step-by-Step Solution

Answer

To sketch the graph of a function like \(y = \sin 4x\), the steps in summary include determining the period (\(\pi/2\) in this case), constructing a corresponding value table for two periods, and plotting these as a sinusoidal curve on the x-y plane.

Step 1: Determining the period

Given that the function is \(y = \sin 4x\), the key attribute influencing the period of the function is the value of '4'. Frequencies for such functions are given as \(\frac{2\pi}{|n|}\), where 'n' refers to the coefficient of x in the sin function - in this case, n=4. The period for this function is thus \(\frac{2\pi}{4}\) or \(\frac{\pi}{2}\).

Step 2: Constructing a value table

The table allows one to obtain values for the function over two periods, and thus to draw the graph accurately. When x ranges from 0 to \(2\pi\), the function, \(y=\sin 4x\), varies between -1 and 1. To form two full periods, values of x should be selected from 0 to 2 times the period.

Step 3: Sketching the graph

Based on the values from the table, plot these value pairs (x, y) on a graph. Follow the typical sinusoidal patterns of cresting at 1 and troughing at -1, while crossing the x-axis at each quarter-period (derived from the key values). Thus draw the graph for two periods of the function \(\sin 4x\). The graph starts and ends at the origin due to the symmetry of sine curves in such cases.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function represented as \( y = \sin x \). It describes a continuous wave, often referred to as a sinusoidal wave, that oscillates above and below the horizontal axis. This function is periodic, repeating its wave-like pattern at regular intervals, known as periods. A few characteristics of the sine function include:

• Range: The sine function values oscillate between -1 and 1. This means that for any input \( x \), the output \( y \) will always lie within this range.
• Period: A complete cycle of \(\sin x\) repeats every \(2\pi\).
• Symmetry: The sine graph is symmetric about the origin, making it an odd function, meaning \( \sin(-x) = -\sin(x) \).

Understanding the basic properties of the sine function lays the groundwork for effectively graphing more complex sine transformations, such as changes in amplitude and period as seen in functions like \( \sin 4x \). The transformation often affects the frequency and phase shifting of the wave.

Period of a Function

The period of a trigonometric function is the horizontal distance required for the function to complete one full cycle of its pattern. For the basic sine function, \( y = \sin x \), the period is \( 2\pi \).When working with functions like \( y = \sin nx \), where \( n \) is a constant factor in front of \( x \), the formula to determine the new period is:\[\text{Period} = \frac{2\pi}{|n|}\]

• In \( y = \sin 4x \), the coefficient is 4, leading to a shorter period of \( \frac{\pi}{2} \). This means the function completes its cycle more quickly than the standard sine function.
• A small period implies more cycles or oscillations over the same interval on the horizontal axis.

Graphically, this leads to more frequent crossings of the x-axis, peaks, and troughs, giving rise to denser wave patterns within the same horizontal span.

Sinusoidal Graph

A sinusoidal graph is generated by the sine function and its transformations. It visually represents the oscillations of the sine wave, which alternate between maximum (crest) and minimum (trough) values.When sketching the sinusoidal graph of a function like \( y = \sin 4x \):

• Start by determining the key points over one period: the crest, trough, and x-axis crossings. In this function, each period is \( \frac{\pi}{2} \).
• Mark these on the graph: the sine function starts at the origin, rises to 1 at \( \frac{\pi}{8} \), returns to 0 at \( \frac{\pi}{4} \), drops to -1 at \( \frac{3\pi}{8} \), and back again to 0 at \( \frac{\pi}{2} \).

Repeat this pattern to sketch additional periods, illustrating the continuous nature of sinusoidal waves. It helps to use either dotted lines to denote each quarter-period and maintain the symmetry of the wave. Identifying these points and understanding symmetry ensures the graph remains accurate and true to its mathematical description. Doing this effectively depicts the transformation brought by modifying the period of the basic sine wave.