Problem 91 - Algebra 2

Question

Write a cosine function for each description. amplitude \(\frac{\pi}{4},\) period 3\(\pi\)

Step-by-Step Solution

Answer

The cosine function with an amplitude of \(\frac{\pi}{4}\) and a period of \(3\pi\) is \(y=\frac{\pi}{4} \cdot \cos(\frac{2}{3}x)\).

Step 1: Identify the given values for amplitude and period

According to the task, the amplitude A is \(\frac{\pi}{4}\) and the period P is \(3\pi\).

Step 2: Apply the amplitude to the general form of the equation

In the general form \(y=A \cdot \cos(Bx+C)+D\), substitute A with the given amplitude \(\frac{\pi}{4}\). This gives us \(y=\frac{\pi}{4} \cdot \cos(Bx+C)+D\).

Step 3: calculate the value of B using the period

The formula to determine the coefficient B based on the period is \(B=\frac{2\pi}{P}\). Substituting P with \(3\pi\), we get \(B=\frac{2}{3}\).

Step 4: Apply B to the equation derived in Step 2

Substitute B in the equation \(y=\frac{\pi}{4} \cdot \cos(Bx+C)+D\) with the value calculated in Step 3. The resulting cosine function is \(y=\frac{\pi}{4} \cdot \cos(\frac{2}{3}x+C)+D\). Since the task doesn't provide specific phase or vertical shifts, we can assume them to be zero (C=D=0).

Step 5: Final cosine function

Plugging C=0 and D=0 into \(y=\frac{\pi}{4} \cdot \cos(\frac{2}{3}x+C)+D\) results in the final cosine function: \(y=\frac{\pi}{4} \cdot \cos(\frac{2}{3}x)\).

Key Concepts

Amplitude in Trigonometric Functions

In trigonometric functions, the amplitude represents the height of the wave. It is the absolute value of the maximum deviation of the wave from its central position on the graph. For a cosine function, this is the distance from the midline to the peak or trough of the wave.

To understand amplitude better, visualize it as controlling how tall or short the wave is. The larger the amplitude, the taller the peaks and deeper the troughs of the function will be.

For the given exercise, the amplitude is \(\frac{\pi}{4}\). This indicates that the highest point of the wave is at \(\frac{\pi}{4}\) and the lowest at \(-\frac{\pi}{4}\) when it's centered around the x-axis. Always remember, amplitude is a positive value reflecting how far the wave goes from the middle, regardless of direction (above or below).

Period of Trigonometric Functions

The period of a trigonometric function dictates how often the wave repeats itself. For standard sine and cosine functions, this is typically \(2\pi\), as these functions complete one full cycle within this interval. However, when coefficients are introduced, the period can change.

The formula to find the period in the context of a transformed cosine function is \(\frac{2\pi}{B}\), where \(B\) is the frequency multiplier in the function. Altering \(B\) affects how quickly the function goes through its cycles. A larger \(B\) results in a shorter period, meaning the wave repeats more frequently.

In our exercise, with a period of \(3\pi\), the wave completes one cycle over \(3\pi\) units along the x-axis. To find \(B\), we rearrange the formula for the period, solving \(B\) as \(\frac{2\pi}{3\pi} = \frac{2}{3}\). This reflects a longer cycle compared to the normal \(2\pi\), meaning the wave stretches out over a wider interval on the x-axis.

Trigonometric Function Equations

Understanding how to write trigonometric function equations involves recognizing the components of the function. The typical form for a cosine function is \(y = A \cos(Bx + C) + D\). Each part of this expression controls a different aspect of the wave.

1. **Amplitude \(A\):** As discussed, \(A\) determines the height of the wave.
2. **Frequency multiplier \(B\):** Affects the period of the cycle, as seen with \(\frac{2}{3}\) in our exercise for a period of \(3\pi\).
3. **Phase shift \(C\):** Adjusts the wave left or right on the graph, depending on its positive or negative value. In our exercise, \(C\) is assumed to be zero, so no horizontal shift occurs.
4. **Vertical shift \(D\):** Moves the wave up or down. Again here, \(D\) is zero, keeping the wave centered around the x-axis.

Writing trigonometric equations involves substituting the known values for these parameters into the formula, as we did to achieve the final form \(y = \frac{\pi}{4} \cos(\frac{2}{3}x)\). This approach allows for precise representation of waves based on specific characteristics like amplitude and period.