Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=-\frac{1}{2} \sin \frac{x}{2}\\\ &g(x)=2 \sin \frac{x}{4} \end{aligned}$$

Step-by-Step Solution

Answer

The graphs of \(f(x) = - \frac{1}{2} \sin \frac{x}{2}\) and \(g(x) = 2 \sin \frac{x}{4}\) have been transformed from the basic \(\sin x\) graph. The graph of \(f(x)\) starts from zero and oscillates from -1/2 to 1/2 with a period of \(4\pi\). The graph of \(g(x)\) also starts from zero but it oscillates from -2 to 2 with a period of \(8\pi\).

Step 1: Recall \(\sin x\) characteristics

The graph of the function \(\sin x\) in a period from 0 to \(2\pi\) is a wave that start at 0, reaches a maximum of 1 at \(\pi /2\), back to 0 at \(\pi\) , reaches a minimum of -1 at \(3\pi /2\) and back to 0 at \(2\pi\) . The period of the function is \(2\pi\).

Step 2: Transformations on \(f(x)=-\frac{1}{2} \sin \frac{x}{2}\)

The negative sign in \(f(x)=-\frac{1}{2} \sin \frac{x}{2}\) reflects the graph in the x-axis, the 1/2 factor compresses it vertically by half, and the x/2 stretches horizontally by factor of 2. Therefore, the graph starts at 0, goes to -1/2 at \(\pi\), back to 0 at \(2\pi\), goes to 1/2 at \(3\pi\) and back to 0 at \(4\pi\). The period is \(4\pi\).

Step 3: Transformations on \(g(x)=2 \sin \frac{x}{4}\)

In \(g(x)=2 \sin \frac{x}{4}\), the 2 stretches the graph vertically by a factor of 2, and x/4 stretches it horizontally by a factor of 4. Therefore, the graph starts at 0, reaches a maximum of 2 at \(2\pi\) , back to 0 at \(4\pi\) , reaches a minimum of -2 at \(6\pi\) and back to 0 at \(8\pi\) . The period is \(8\pi\).

Key Concepts

Sine Function

The sine function is one of the basic trigonometric functions. Its graph is known as a sine wave, characterized by its smooth and continuous oscillation. The function, expressed as \(y = \sin(x)\), has some distinct features that make it easy to recognize:

• It is periodic, which means it repeats its pattern over regular intervals.
• The standard period of a sine wave is from 0 to \(2\pi\), where it completes one full cycle.
• It begins at 0, rises to a peak of 1 at \(\pi/2\), returns to 0 at \(\pi\), dips to a trough of -1 at \(3\pi/2\), and returns to 0 at \(2\pi\).
• The amplitude of the sine function is 1, which is the distance from the midline to its peak or trough.

Understanding the basic pattern of the sine function helps when applying transformations, a topic we'll explore further below.

Periodic Functions

A periodic function is one that repeats its values at regular intervals. In trigonometry, the sine function is a classic example. Its periodic nature means that, after a certain interval, the function's values start to repeat. This interval is known as the period.

• The period of a sine function, \(y=\sin(x)\), is \(2\pi\). This indicates that the wave pattern repeats every \(2\pi\) units along the x-axis.
• When dealing with variations of sine functions such as \(f(x)=-\frac{1}{2} \sin\left(\frac{x}{2}\right)\) and \(g(x)=2 \sin\left(\frac{x}{4}\right)\), the periods change due to horizontal transformations.
• For instance, dividing x by a number within the sine function changes the period. In \(f(x)\), the x is divided by 2, doubling the period to \(4\pi\). While in \(g(x)\), it is divided by 4, quadrupling the period to \(8\pi\).

This predictable repetition allows for easy graphing over long intervals and is a key feature of trigonometric functions.

Graph Transformations

Transformations manipulate the graph of a function, altering its shape and position without changing its basic characteristics. For sine functions, transformations include shifting, stretching, compressing, and reflecting the graph.

• Vertical Stretching and Compressing: Multiplying the sine function by a coefficient (like 2 in \(g(x) = 2\sin\left(\frac{x}{4}\right)\)) changes the amplitude, stretching it vertically. A multiplier less than 1 compresses it.
• Horizontal Stretching and Compressing: Dividing the input \(x\) inside the sine function by a number modifies the period. Larger divisors stretch it horizontally, as seen in both \(f(x)=-\frac{1}{2} \sin\left(\frac{x}{2}\right)\) and \(g(x)=2 \sin\left(\frac{x}{4}\right)\).
• Reflection: A negative sign in front of the sine, as in \(f(x)\), reflects the graph over the x-axis, flipping its peaks and troughs.

These transformations are vital in graphing customized sine functions and interpreting their behavior over different intervals of the x-axis.