Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=\sqrt{3} \cos \left(\frac{\pi}{4} x-\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Amplitude: \(\sqrt{3}\), Period: 8, Phase Shift: 2 units right.

Step 1: Identify the Amplitude

The equation is in the form of a cosine function: \[y = a \cos(bx + c)\].For this specific equation, \(a = \sqrt{3} \). The amplitude is the absolute value of \(a\), so:\[\text{Amplitude} = |\sqrt{3}| = \sqrt{3}.\]

Step 2: Determine the Period

The period of a cosine function is calculated using the formula \(\frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(x\) within the cosine function. Here, \(b = \frac{\pi}{4}\). Thus, the period is:\[\text{Period} = \frac{2\pi}{\left|\frac{\pi}{4}\right|} = \frac{2\pi}{\pi/4} = 8.\]

Step 3: Calculate the Phase Shift

The phase shift is calculated using \(\frac{-c}{b}\), where \(c\) is the constant inside the cosine function. In this equation, \(c = -\frac{\pi}{2}\) and \(b = \frac{\pi}{4}\). Substituting these into the phase shift formula gives us: \[\text{Phase Shift} = \frac{-(-\frac{\pi}{2})}{\frac{\pi}{4}} = \frac{\pi/2}{\pi/4} = 2.\]The phase shift is 2 units to the right.

Step 4: Sketch the Graph

To sketch the graph, start by plotting the standard cosine wave, noting the amplitude, period, and phase shift. The amplitude \(\sqrt{3}\) means the wave will reach a maximum of \(\sqrt{3}\) and a minimum of \(-\sqrt{3}\). The period of 8 means the pattern repeats every 8 units along the x-axis. The phase shift of 2 moves the entire graph 2 units to the right. Begin plotting at x = 2 and complete one cycle by x = 10 to reflect these characteristics.

Key Concepts

Amplitude

In trigonometric functions like cosine, the amplitude refers to how "tall" or "short" the wave appears on a graph. It measures the vertical stretch of the wave.
For the function given, the amplitude is derived from the coefficient in front of the cosine function, denoted as \(a\) in the general form \(y = a \cos(bx + c)\). Here, \(a = \sqrt{3}\). Thus, the amplitude is \(|\sqrt{3}|\), which simply gives us \(\sqrt{3}\).

The amplitude essentially tells us:

• The wave reaches a maximum height of \(\sqrt{3}\) in the positive direction.
• The wave dips to a minimum of \(-\sqrt{3}\) below the x-axis.

This value doesn't affect the x-values but instead stretches or compresses the wave vertically in the y-axis direction. Therefore, when sketching, remember that the highest point the wave reaches will be \(\sqrt{3}\) and the lowest will be \(-\sqrt{3}\).

Period

The period of a trigonometric function describes how long it takes for the wave to complete one full cycle before repeating itself.
In the trigonometric function \(y = a \cos(bx + c)\), the formula for finding the period is \(\frac{2\pi}{|b|}\).

For the exercise at hand, \(b = \frac{\pi}{4}\).

• Substituting into the formula, we calculate \(\frac{2\pi}{\left|\frac{\pi}{4}\right|} = 8 \).

Hence, the period is 8.

This means the pattern of the wave repeats every 8 units along the x-axis. So when you plot, start from a chosen x-value and then mark the completion of a cycle after 8 units. This helps in understanding where the wave starts to repeat itself and is crucial for sketching and interpreting trigonometric graphs accurately.

Phase Shift

The phase shift of a trigonometric function refers to the horizontal movement of the wave on the graph.
In our function \(y = a \cos(bx + c)\), we determine the phase shift using the formula \(\frac{-c}{b}\).

Using the given function:
\(c = -\frac{\pi}{2}\) and \(b = \frac{\pi}{4}\).

• Plug these into the formula to get \(\frac{-(-\frac{\pi}{2})}{\frac{\pi}{4}} = \frac{\pi/2}{\pi/4} = 2\).

The result is a phase shift of 2 units to the right.

This horizontal shift means that the entire wave graph is moved 2 units to the right along the x-axis. Thus, when plotting or analyzing, start the wave graph at \(x = 2\) instead of the usual \(x = 0\). This adjustment is crucial to ensure the graph accurately represents the function in question.