Problem 12 - 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. $$ y=3 \cos \left(x+\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(2\pi\), Phase Shift: \(-\frac{\pi}{6}\), graph is left by \(\frac{\pi}{6}\).

Step 1: Identify the Amplitude

The cosine function given is in the form \( y = a \cos(bx + c) \). In this equation, \( a = 3 \), which represents the amplitude. Therefore, the amplitude is 3.

Step 2: Determine the Period

The period of a cosine function in the standard form \( y = a \cos(bx + c) \) is given by \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period of the function is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Calculate the Phase Shift

The phase shift of a function \( y = a \cos(bx + c) \) is given by \( -\frac{c}{b} \). In the given equation, \( c = \frac{\pi}{6} \) and \( b = 1 \), so the phase shift is \( -\frac{\pi}{6} \). This indicates a shift to the left by \( \frac{\pi}{6} \) units.

Step 4: Sketch the Graph

With the amplitude, period, and phase shift identified, sketch the graph of the function. The graph of \( y = 3\cos(x + \frac{\pi}{6}) \) will have a maximum value of 3, a minimum value of -3, a period of \( 2\pi \), and will start \( \frac{\pi}{6} \) units to the left of the y-axis compared to the standard cosine graph.

Key Concepts

Amplitude

When dealing with trigonometric functions like cosine, the amplitude is a crucial component that determines how tall or short the graph appears. The amplitude is the distance from the middle of the wave to its peak or trough.

• In the standard cosine function formula, \(y = a \cos(bx + c)\), the amplitude is represented by \(a\).
• For the equation \(y = 3\cos\left(x+\frac{\pi}{6}\right)\), the amplitude is \(3\). This means that the graph will reach 3 units above and 3 units below the center line, or midline, of the graph.

Understanding amplitude is like knowing how loud sound can be when you turn up the volume. Similarly, an amplitude of 3 turns up our wave to be 3 times 'tall'. The maximum value of the graph is the amplitude itself, positive and negative, which in this case is 3 and -3.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle of its pattern. Essentially, it tells us how 'stretched' or 'compressed' the graph of the function is.

• For the cosine function \(y = a \cos(bx + c)\), the period is calculated using the formula \(\frac{2\pi}{b}\).
• In our equation \(y = 3 \cos\left(x+\frac{\pi}{6}\right)\), the value of \(b\) is 1, giving us a period of \(\frac{2\pi}{1} = 2\pi\).

This means that the cosine wave repeats its shape every \(2\pi\) units along the x-axis. Imagine a roller coaster track that repeats its ups and downs at consistent intervals. Here, every \(2\pi\) units, the roller coaster comes back to the same point in the cycle.

Phase Shift

The phase shift of a trigonometric function tells us how much the graph is shifted to the left or right compared to its standard position. This happens when we adjust where the function 'starts' its cycle on the x-axis.

• The phase shift is found using the formula \(-\frac{c}{b}\) in the function \(y = a \cos(bx + c)\).
• For \(y = 3 \cos\left(x+\frac{\pi}{6}\right)\), \(b = 1\) and \(c = \frac{\pi}{6}\). Hence, the phase shift is \(-\frac{\pi}{6}\).

A phase shift of \(-\frac{\pi}{6}\) means the entire graph is shifted to the left by \(\frac{\pi}{6}\) units. Think of the phase shift as pushing the start of the function's cycle to begin earlier than it usually would. In this case, our cosine wave starts \(\frac{\pi}{6}\) units to the left, making its first peak appear sooner on the x-axis.