Problem 18 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=3 \cos (3 x-\pi)\)

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(\frac{2\pi}{3}\), Phase shift: Right \(\frac{\pi}{3}\).

Step 1: Identify the Amplitude

The amplitude of a trigonometric function like cosine is given by the absolute value of the coefficient in front of the cosine function. For the equation \(y = 3 \cos(3x - \pi)\), the coefficient is 3. Thus, the amplitude is \(|3| = 3\).

Step 2: Determine the Period

The period of a cosine function is found using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the cosine function. In the equation \(y = 3 \cos(3x - \pi)\), \(b = 3\). Therefore, the period is \(\frac{2\pi}{3}\).

Step 3: Calculate the Phase Shift

The phase shift of a trigonometric function in the form \(y = a \cos(bx - c)\) is calculated as \(\frac{c}{b}\). For our equation, \(c = \pi\) and \(b = 3\). The phase shift is \(\frac{\pi}{3}\). Since the \(c\) term is \(3x - \pi\), which is negative, it indicates a shift to the right by \(\frac{\pi}{3}\).

Step 4: Sketch the Graph

To sketch the graph, start by plotting the basic cosine graph. Adjust for the amplitude by scaling the height to 3. Then, compress the wave horizontally due to the shorter period \(\frac{2\pi}{3}\). Finally, shift the entire graph to the right by \(\frac{\pi}{3}\). The wave will repeat every \(\frac{2\pi}{3}\) units along the x-axis, with peaks reaching a height of 3 and troughs at -3.

Key Concepts

Understanding Amplitude

Amplitude in trigonometric functions refers to the height from the centerline to the peak (or trough) of the wave. In mathematical terms, it is the

• "maximum vertical distance"
• from the equilibrium position of the wave.

For a cosine function represented by the general form \(y = a \cos(bx - c)\), amplitude is determined by the absolute value of \(a\).
So, if you have \(y = 3 \cos(3x - \pi)\), you simply take the coefficient in front of the cosine, which is 3, thus the amplitude is

• \(|3| = 3\).

An easy way to visualize this is: Imagine a straight line running through the middle of your sine or cosine wave; the amplitude is how much the wave bounces away from this line, both upwards and downwards. The greater the amplitude, the taller the wave crest and the deeper the trough, but

• it doesn't affect the length of one complete cycle.

Understanding amplitude is crucial when sketching graphs of trigonometric functions as it helps you determine how tall the wave will be.

Determining the Period

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle before repeating. It determines how stretched or compressed a wave looks along the x-axis.
This is crucial for illustrating the function accurately on a coordinate plane.
For functions of the form \(y = a \cos(bx - c)\), the period is calculated using the formula:

• \(\frac{2\pi}{b}\)

In our example, \(b = 3\),
so the period is \(\frac{2\pi}{3}\).
A smaller period indicates that the graph repeats more frequently within a given range on the x-axis, making the wave look more

• "squished"
• or "compressed"

than usual. The period directly affects the width of each cycle, altering the x-distance between repeating features in successive cycles, such as peaks and troughs.
When plotting, knowing the period helps you allocate the right amount of space on the x-axis for each cycle, allowing for a visually accurate representation of how the wave behaves over time.

Grasping Phase Shift

Phase shift involves the horizontal translation of a wave along the x-axis. This concept comes into play in equations like \(y = a \cos(bx - c)\), where the phase shift is determined by the part of the function \(c\).
To find out the exact shift, use the formula:

• \(\frac{c}{b}\)

For \(y = 3 \cos(3x - \pi)\), \(c = \pi\) and \(b = 3\). Thus, the phase shift equals \(\frac{\pi}{3}\).
Because it’s negative \(3x - \pi\), the graph will shift to the right.
Consider this: if you're looking at a cosine wave, usually, it starts at a peak when \(x = 0\). With a phase shift, this familiar starting point moves left or right, altering the x-value that the wave appears to start its cycle.
In our example, the graph's initial peaks, troughs, and midpoints all move \(\frac{\pi}{3}\) units to the right on the x-axis.
Phase shift changes when and where the wave-like pattern begins without affecting the actual shape or height of the wave itself; it merely "slides" the whole function horizontally.