Problem 18 - 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. $$ y=3 \cos (3 x-\pi) $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \( \frac{2\pi}{3} \), Phase Shift: \( \frac{\pi}{3} \) (right).

Step 1: Identify the amplitude

The amplitude of a cosine function of the form \( y = a \cos(bx - c) \) is given by the absolute value of \( a \). In this equation, \( a = 3 \). Therefore, the amplitude is \( |3| = 3 \).

Step 2: Determine the period

The period of a cosine function is calculated by the formula \( \frac{2\pi}{b} \). For this equation, \( b = 3 \). Hence, the period is \( \frac{2\pi}{3} \).

Step 3: Find the phase shift

The phase shift is found by solving \( bx - c = 0 \) for \( x \). Set \( 3x - \pi = 0 \), then \( 3x = \pi \). Therefore, the phase shift is \( \frac{\pi}{3} \). Since the expression is \( (bx - c) \), it's a shift to the right.

Step 4: Sketch the graph

To sketch the graph, start from the phase shift \( x = \frac{\pi}{3} \). The curve begins at this point with a maximum at \( y=3 \) due to the amplitude. As the function has a period of \( \frac{2\pi}{3} \), the graph makes one full wave cycle in this interval. Mark key points: The maximum is at the phase shift, the minimum at \( x = \frac{5\pi}{6} \), and it returns to its initial value at \( x = \frac{\pi}{3} + \frac{2\pi}{3} = \pi \). Record the main points for accuracy in sketching.

Key Concepts

Amplitude

Understanding the amplitude of a cosine function is crucial, as it indicates how much the function oscillates from its central position. The standard form of a cosine function is given by:

• \( y = a \cos(bx - c) \)

The amplitude is determined by the absolute value of the coefficient \( a \).
In the equation \( y = 3 \cos(3x - \pi) \), the value of \( a \) is 3.
Thus, the amplitude is \( |3| = 3 \).
This means the cosine curve will reach a maximum of 3 units above and a minimum of 3 units below the central axis. This vertical distance from the middle to the peak (or trough) characterizes how "tall" the waves of the graph appear.

Period

The period of a cosine function helps in identifying how long it can take for the function to complete one full cycle. For a cosine function in the form:

• \( y = a \cos(bx - c) \)

The period is calculated using the formula \( \frac{2\pi}{b} \).
In our case, \( b = 3 \).
This makes the period \( \frac{2\pi}{3} \).
This fraction of \( 2\pi \) tells you that the function will complete one wave cycle—a crest and a trough—every \( \frac{2\pi}{3} \) units along the x-axis.
It means that if you take a length on the x-axis equal to \( \frac{2\pi}{3} \), you'll see the entire rising and falling pattern of the wave.

Phase Shift

Phase shift refers to the horizontal movement of the graph of the function along the x-axis. It tells us where the wave starts compared to usual. In the equation:

• \( y = a \cos(bx - c) \)

The phase shift can be determined by solving \( bx - c = 0 \) for \( x \).
Here we have:\( 3x - \pi = 0 \), leading to \( x = \frac{\pi}{3} \).
This solution shows the graph is shifted \( \frac{\pi}{3} \) units to the right.
So, instead of starting at \( x = 0 \), the cosine wave starts its cycle at \( x = \frac{\pi}{3} \).
This shift is crucial when sketching the graph, as it defines where the cycle will commence, affecting the placement of critical points such as peaks and troughs.