Problem 31 - 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. $$ y=-5 \cos \left(\frac{1}{3} x+\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(6\pi\), Phase Shift: \(-\frac{\pi}{2}\).

Step 1: Identify Amplitude

The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine. In the equation \( y = -5 \cos \left( \frac{1}{3}x + \frac{\pi}{6} \right) \), the coefficient is \(-5\). Thus, the amplitude is \( | -5 | = 5 \).

Step 2: Determine the Period

The formula to find the period of a cosine function is \( \frac{2\pi}{b} \), where \(b\) is the coefficient in front of \(x\) inside the cosine function. In this case, \(b = \frac{1}{3}\), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).

Step 3: Calculate Phase Shift

The phase shift for a cosine function is given by \( -\frac{c}{b} \), where \(c\) is the constant added inside the function. Here, \(c = \frac{\pi}{6}\) and \(b = \frac{1}{3}\), so the phase shift is \(-\frac{\pi}{6} \div \frac{1}{3} = -\frac{\pi}{6} \times 3 = -\frac{\pi}{2}\).

Step 4: Sketch the Graph

To sketch the graph of \(y = -5 \cos \left(\frac{1}{3}x + \frac{\pi}{6}\right)\), note the following changes from the standard \(y = \cos(x)\): amplitude is 5 (stretched vertically and reflected), period is \(6\pi\) (stretched horizontally), and phase shift is \(-\frac{\pi}{2}\) to the left. Start from the point \(x = -\frac{\pi}{2}\) and plot one period length of \(6\pi\), noting the maximum \(5\) and minimum \(-5\) points on the y-axis, then repeat this pattern.

Key Concepts

Amplitude

The amplitude of a cosine function helps us understand how tall the waves of the graph are. It's essentially the height from the middle of the wave to a peak or a trough. For any cosine function in the form of \( y = a \cos(bx + c) \), the amplitude is simply the absolute value of \( a \).

In our exercise with the equation \( y = -5 \cos \left( \frac{1}{3} x + \frac{\pi}{6} \right) \), the amplitude, \( a \), is \(-5\). However, because amplitude is always positive, we take the absolute value, giving us an amplitude of \( 5 \).

The negative sign in \(-5\) indicates that the entire cosine wave is flipped upside down, but it doesn't affect the amplitude. The flip simply means that what would ordinarily be the upward slope starts by moving down.

Period

The period of a cosine function determines how long it takes for the wave to complete one full cycle and return to its starting point. This concept tells us how stretched or compressed the graph is horizontally.

For a function \( y = a \cos(bx + c) \), the period is calculated using the formula \( \frac{2\pi}{|b|} \). Here, \( b \) is the coefficient of \( x \). In our given function, \( b \) is \( \frac{1}{3} \), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).

• This result tells us that the wave starts repeating its pattern every \( 6\pi \) units along the x-axis.
• This is a stretch compared to the standard \( 2\pi \) period of the regular cosine function \( \cos(x) \).

Phase Shift

The phase shift in a cosine function indicates how the entire graph shifts horizontally from its standard position. It represents the horizontal movement of the graph towards the left or right, depending on the sign.

Phase shift is calculated using the formula \(-\frac{c}{b}\), where \( c \) is the constant term inside the parentheses, and \( b \) is the coefficient of \( x \).

• In our exercise equation, \( c \) is \( \frac{\pi}{6} \) and \( b \) is \( \frac{1}{3} \).
• By applying the formula, the phase shift is \(-\left(\frac{\pi}{6} \div \frac{1}{3}\right) = -\frac{\pi}{2}\).

This value means the graph starts \( \frac{\pi}{2} \) units to the left of where a standard cosine wave would typically begin its cycle. This leftward shift impacts where you plot the start of the graph, changing how it aligns with the x-axis.