Problem 18 - Algebra 2

Question

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=\sin 4 \theta\)

Step-by-Step Solution

Answer

Amplitude is 1; Period is \(\frac{\pi}{2}\).

Step 1: Identify the Basic Form

The function given is \(y = \sin 4\theta\). The standard form of a sine function is \(y = a \sin(b\theta) + c\), where \(a\) determines the amplitude, \(b\) affects the period, and \(c\) shifts the function vertically. Here \(a = 1\) and \(b = 4\).

Step 2: Determine the Amplitude

For a sine function, the amplitude is the absolute value of \(a\). In this function, \(a = 1\). So, the amplitude is \(|1| = 1\).

Step 3: Calculate the Period

The period of a sine function is calculated using the formula \(\frac{2\pi}{|b|}\). Here, \(b = 4\). Substituting the value of \(b\) gives the period as \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 4: Graph the Function

To graph \(y = \sin 4\theta\), utilize the amplitude and period information. The amplitude determines the maximum and minimum values of the function (-1 and 1), and the period tells us that the function completes one cycle every \(\frac{\pi}{2}\) units on the \(\theta\)-axis. Hence, plot points for one cycle at key values: 0, \(\frac{\pi}{8}\), \(\frac{\pi}{4}\), \(\frac{3\pi}{8}\), \(\frac{\pi}{2}\) and repeat the pattern for additional cycles.

Key Concepts

Amplitude

Amplitude is a crucial aspect of trigonometric functions like sine and cosine. It indicates how much a function deviates above and below its center line. Specifically, amplitude is the vertical distance from the center line to the peak (or trough) of the wave. This means it's always a positive distance.
For a function in the form of \(y = a \sin(b\theta) + c \), the amplitude is given by the absolute value of \(a\). Thus, regardless of whether \(a\) is positive or negative, the amplitude remains positive.

• For example, if you have \(y = \sin 4\theta\), since \(a = 1\), the amplitude is \( |1| = 1\).
• This tells us that as the sine function oscillates, it reaches up to +1 and down to -1, centered around the horizontal axis (\(y = 0\)).

Periodicity

Periodicity refers to how often a function repeats its pattern over a set distance or interval. For trigonometric functions, this periodic behavior means they create repeating cycles. The standard sine wave repeats every \(2\pi\).
To determine the period of a sine function \(y = a \sin(b\theta) + c\), use the formula \(\frac{2\pi}{|b|}\). Here, \(b\) indicates how much the wave is stretched or compressed horizontally.

• In our example, \(y = \sin 4\theta\), \(b\) is 4. So, the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).
• This means instead of completing a cycle every \(2\pi\), this function finishes one complete wave cycle in just \(\frac{\pi}{2}\). Therefore, it's much quicker to repeat its pattern.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their amplitude and period, along with identifying specific points that complete one cycle. Through these points, we can sketch the characteristic wave-like pattern.
For graphing \(y = \sin 4\theta\), follow these steps:

• First, note the amplitude of the function. As previously determined, it's 1, so the peaks and troughs will be at \(y = 1\) and \(y = -1\), respectively.
• Next, identify the period, which is \(\frac{\pi}{2}\). This tells us that the function completes one wave cycle every \(\frac{\pi}{2}\) units on the \(\theta\)-axis.
• Plot key points: start at (0, 0), peak at \((\frac{\pi}{8}, 1)\), back to (\(\frac{\pi}{4}, 0\)), trough at \((\frac{3\pi}{8}, -1)\), and finish the cycle at \((\frac{\pi}{2}, 0)\).
• Repeat these points for additional cycles to extend the graph.

With these steps, you create a smooth, continuous wave that reflects both the amplitude and periodicity of the function.