Problem 7 - Precalculus

Question

Graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function. \(y=-\frac{1}{3} \cos \left(\frac{1}{2} x+\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

Period: \(4\pi\), Amplitude: \(\frac{1}{3}\), Phase Shift: \(-\frac{2\pi}{3}\), Vertical Shift: 0.

Step 1: Identify the amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient in front of the cosine. In this case, the coefficient is \(-\frac{1}{3}\), so the amplitude is \(\left| -\frac{1}{3} \right| = \frac{1}{3}\).

Step 2: Determine the period

The period of a cosine function is given by \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the cosine function. Here, \(b = \frac{1}{2}\), so the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

Step 3: Calculate the phase shift

The phase shift is determined by solving \(\frac{bx}{2} + c = 0\). In this case, \(b = \frac{1}{2}\) and \(c = \frac{\pi}{3}\), so the phase shift is \(-\frac{\pi}{3} \div \frac{1}{2}\) which simplifies to \(-\frac{2\pi}{3}\).

Step 4: Identify the vertical shift

The function as given has no constant term added or subtracted outside of the cosine, which means the vertical shift is \(0\).

Step 5: Graph the function

Start by plotting the cosine curve, reflecting it over the x-axis due to the negative sign, scaling it vertically by the amplitude \(\frac{1}{3}\), stretching it horizontally to have a period of \(4\pi\), and finally shifting it horizontally by \(-\frac{2\pi}{3}\). Since the vertical shift is 0, the midline is on the x-axis.

Key Concepts

Amplitude

The concept of amplitude in a cosine function represents how much the function stretches or compresses vertically. For any cosine function described by the formula \( y = a \cdot \cos(bx + c) + d \), the amplitude is determined by the absolute value of \( a \). In simpler terms, the amplitude is how high or low the wave reaches from its central axis. In the given function \( y = -\frac{1}{3} \cos \left(\frac{1}{2} x + \frac{\pi}{3}\right) \), the coefficient \( a \) is \(-\frac{1}{3}\). Therefore, the amplitude is \( \left| -\frac{1}{3} \right| = \frac{1}{3} \).

In plotting terms, this means every peak and trough of your cosine wave will be located at \( 1/3 \) units above and \( 1/3 \) units below the midline, which is the x-axis in this case. This scaled version of the wave demonstrates how compression affects the vertical distance from the axis, influencing the wave's peaks and troughs.

Period of Function

The period of a cosine function determines how long it takes for the function to complete one full cycle before it starts repeating itself. This period is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the multiplier of the variable \( x \) inside the cosine function.

For the function \( y = -\frac{1}{3} \cos \left(\frac{1}{2} x + \frac{\pi}{3}\right) \), \( b \) is \( \frac{1}{2} \). Plugging this into our formula gives:

• \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)

This result means the function needs a \( 4\pi \) interval to complete a full cycle. It represents a stretch in the direction of the x-axis compared to the standard cosine wave, which has a period of \( 2\pi \). By understanding this property, we can rightly determine how wide each cycle needs to be when graphing.

Phase Shift

The phase shift in a cosine function defines how the starting point of the wave is horizontally translated left or right along the x-axis. This shift is calculated using the elements within the cosine function's parenthesis of the form \( bx + c \). Here, you solve \( bx + c = 0 \) to find the value of the phase shift.

In our cosine function \( y = -\frac{1}{3} \cos \left(\frac{1}{2} x + \frac{\pi}{3}\right) \), we have:

• \( \left(\frac{1}{2}x + \frac{\pi}{3} = 0\right) \)

Solving for \( x \), you get:

• \( x = -\frac{2\pi}{3} \)

This negative phase shift of \(-\frac{2\pi}{3}\) indicates that the entire graph of the cosine function is shifted towards the left by \( \frac{2\pi}{3} \) units on the x-axis. Phase shifts are crucial for positioning the starting point of the wave correctly on the graph.

Vertical Shift

A vertical shift in the cosine function indicates how the entire graph of the function moves up or down along the y-axis relative to its original position. This shift is determined by the constant \( d \) in the function formula \( y = a \cdot \cos(bx + c) + d \). If \( d \) is positive, the graph moves upward; if negative, it shifts downward.

For the function \( y = -\frac{1}{3} \cos \left(\frac{1}{2} x + \frac{\pi}{3}\right) \), there is no \( d \), meaning it's 0. As such, there is no vertical shift, and the midline of the cosine function remains on the x-axis.

• Midline remains at \( y = 0 \)
• Graph remains centered around the x-axis

This lack of vertical shift ensures the wave's central axis stays aligned with the x-axis, simplifying graphing.