Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=-1+\frac{1}{2} \cos (2 x-3 \pi)$$

Step-by-Step Solution

Answer

Amplitude: 0.5, Period: \(\pi\), Phase Shift: \(\frac{3\pi}{2}\) right, Vertical Translation: 1 down, Range: [-1.5, -0.5].

Step 1: Identify the Amplitude

The given function can be written in the form \( y = a \, ext{cos}(b \, x - c) + d \), where \( a \, = \, \frac{1}{2} \). The amplitude is the absolute value of \( a \), which represents the maximum deviation from the horizontal axis. Therefore, the amplitude is \( \left| \frac{1}{2} \right| = \frac{1}{2} \).

Step 2: Determine the Period

The period of a cosine function \( y = a \, ext{cos}(b \, x) \) is given by \( \frac{2\pi}{b} \). In this function, \( b = 2 \). Therefore, the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate the Phase Shift

The phase shift is given by \( \frac{c}{b} \), where \( c = 3\pi \) and \( b = 2 \). Thus, the phase shift is\( \frac{3\pi}{2} \). Since the form is \( y = a \, ext{cos}(b \, x - c) + d \), the phase shift is to the right by \( \frac{3\pi}{2} \).

Step 4: Find the Vertical Translation

The function before the constant shift is \( y = \frac{1}{2} \, ext{cos}(2x - 3\pi) \). The term \( d = -1 \) indicates a vertical translation downward by 1 unit.

Step 5: Establish the Range

The range of a cosine function \( y = a \, ext{cos}(b \, x) + d \) is \([d-a, d+a]\). Substituting the values \( a = \frac{1}{2} \) and \( d = -1 \), the range is \([-1 - \frac{1}{2}, -1 + \frac{1}{2}] \), or \([-1.5, -0.5]\).

Step 6: Graph the Function

To graph the function over one period: start at the phase shift \( x = \frac{3\pi}{2} \), and calculate the values or plot points periodically such that the function accommodates the full range from \(-1.5\) to \(-0.5\) along with the period \( \pi \). Sketch the cosine wave making sure the wave crest peaks at \(-0.5\) and the troughs reach \(-1.5\), over the interval defined by one complete cycle of the function's period.

Key Concepts

Amplitude

In the study of trigonometric functions such as cosine and sine, the amplitude is a measure of how far the function values deviate from the central horizontal axis. It tells you how high and low the function's peaks and valleys are relative to the center of the wave, usually denoted as the x-axis.
For our exercise, we're dealing with a function of the form\[y = a \cos(bx - c) + d\]Here, the parameter \(a\) represents the amplitude. In essence, amplitude is the absolute value of this coefficient \(a\).

• If \(a\) is positive, the cosine wave maintains its standard upright orientation.
• If \(a\) is negative, the wave inverts, flipping upside down.

For the function \(y = -1 + \frac{1}{2} \cos (2x - 3\pi)\), the amplitude is \(\left| \frac{1}{2} \right|\), which is \(\frac{1}{2}\). This means that the maximum deviation of the wave from its central horizontal axis is \(\frac{1}{2}\).

Period

The period of a trigonometric function describes the distance over which the function repeats itself. For cosine and sine functions, this usually relates to the horizontal stretching or compressing of the wave.
For standard trigonometric functions \(y = a \cos(bx)\), the period is calculated using the formula:\[\text{Period} = \frac{2\pi}{|b|}\]In our function \(y = -1 + \frac{1}{2} \cos(2x - 3\pi)\), the \(b\) coefficient is 2. So the period is\[\frac{2\pi}{2} = \pi\]
This means the cosine function completes one full cycle over a length of \(\pi\) units on the x-axis. Understanding the period helps you determine where the function starts repeating, an important aspect when you sketch or predict future cycles.

Phase Shift

The phase shift of a trigonometric function affects its horizontal displacement along the x-axis. It's like shifting the entire graph left or right to a new starting point.
For a function of the form
\(y = a \cos(bx - c) + d\), the phase shift is calculated using:
\[\text{Phase Shift} = \frac{c}{b}\]In our example function \(y = -1 + \frac{1}{2} \cos (2x - 3\pi)\), we have \(c = 3\pi\) and \(b = 2\). Substituting these values, we get a phase shift of:
\[\frac{3\pi}{2}\]
This result means that the cosine wave is shifted to the right by \(\frac{3\pi}{2}\) units. Phase shifts are particularly useful for aligning waves in periodic phenomena or ensure that they sync with external events or conditions.

Vertical Translation

Vertical translation refers to how a function's central axis shifts up or down along the y-axis. This translation does not affect the shape of the wave but does alter its mean position.
In the general trigonometric form \(y = a \cos(bx - c) + d\), the parameter \(d\) controls vertical translation.

• If \(d\) is positive, the function shifts upward.
• If \(d\) is negative, the function shifts downward.

In the function \(y = -1 + \frac{1}{2} \cos(2x - 3\pi)\), the value of \(d\) is \(-1\). This indicates that the entire graph of the cosine function is shifted downwards by 1 unit.
This kind of transformation is key for altering the average level of the wave, with applications in data normalization and signal processing, among others.