Problem 26 - 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. $$ y=\frac{1}{2} \cos \frac{\pi}{2} x $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{2}\), Period: 4, Phase Shift: 0.

Step 1: Identify the Basic Form

The general form of a cosine function is \(y = a \cos(bx - c) + d\), where \(a\) is the amplitude, \(b\) affects the period, \(c\) determines the phase shift, and \(d\) is the vertical shift. In this case, the equation is \(y = \frac{1}{2} \cos \frac{\pi}{2} x\), where \(a = \frac{1}{2}\), \(b = \frac{\pi}{2}\), \(c = 0\), and \(d = 0\).

Step 2: Find the Amplitude

The amplitude of a cosine function is the absolute value of \(a\). In this equation, \(a = \frac{1}{2}\), so the amplitude is \(|\frac{1}{2}| = \frac{1}{2}\).

Step 3: Calculate the Period

The period of a cosine function is calculated by the formula \(\frac{2\pi}{b}\). Here, \(b = \frac{\pi}{2}\) so the period is \(\frac{2\pi}{\frac{\pi}{2}} = 4\). Thus, the graph completes one full cycle every 4 units.

Step 4: Determine the Phase Shift

To find the phase shift, use \(\frac{c}{b}\). Since \(c = 0\), there is no phase shift in this equation. The graph is not shifted left or right from the origin.

Step 5: Graph the Function

Using the amplitude, period, and phase shift information, sketch the cosine graph: - Amplitude is \(\frac{1}{2}\), so peak and trough are at \(\frac{1}{2}\) and \(-\frac{1}{2}\).- The graph completes a cycle every 4 units, oscillating between \(\frac{1}{2}\) and \(-\frac{1}{2}\) with no horizontal or vertical shifts. Therefore, draw one full cycle with these characteristics.

Key Concepts

Amplitude

When dealing with trigonometric functions, the amplitude is an important concept. It represents how far above and below the midline of the graph the function oscillates. In simple terms, amplitude refers to the height of the wave from its center.

For the cosine function in question, given by the equation \(y = \frac{1}{2} \cos \frac{\pi}{2} x\), the amplitude is determined by the coefficient before the cosine term, which is \(\frac{1}{2}\).

To find the amplitude, simply take the absolute value of this coefficient. This means that:

• The maximum value that the graph reaches is \(\frac{1}{2}\).
• The minimum value it dips to is \(-\frac{1}{2}\).

Why is knowing the amplitude important? It tells you how much the function varies around its average value (midline), which in this function is the x-axis \(y = 0\). If amplitude increases, the peaks and valleys of the cosine function become further stretched from the midline.

Period

The period of a trigonometric function is how long it takes to complete one full cycle of its wave. In other words, it's the horizontal length of one complete repeat of the pattern.

For the function \(y = \frac{1}{2} \cos \frac{\pi}{2} x\), we use the formula to calculate the period: \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the cosine function, \(\frac{\pi}{2}\).

Substituting in the value, we get:

• \(\frac{2\pi}{\frac{\pi}{2}} = 4\).

This result shows that this cosine function completes one full cycle every 4 units along the x-axis.

Knowing the period is crucial because it helps in predicting where the function will rise and fall within each cycle, crucial for sketching accurate graphs.

Phase Shift

Phase shift represents the horizontal movement or translation of the graph along the x-axis. It tells us if the function's starting point has shifted left or right from the standard position.

The formula for phase shift is \(\frac{c}{b}\) where \(c\) is the horizontal shift constant bound inside the cosine function, and \(b\) is its coefficient.

In \(y = \frac{1}{2} \cos \frac{\pi}{2} x\), the value of \(c\) is 0, which means:

• \(\frac{0}{\frac{\pi}{2}} = 0\).

This means there is no phase shift; the graph of the cosine wave starts at the origin as usual.

Understanding phase shift helps in properly positioning the graph along the x-axis and is especially helpful when comparing multiple trigonometric functions to one another.