Problem 63 - Precalculus: Functions and Graphs

Question

Exer. \(63-66:\) (a) Use the formula from Example 6 to express \(f\) in terms of the cosine function. (b) Determine the amplitude, period, and phase shift of \(f\). (c) Sketch the graph of \(f\). $$f(x)=\sqrt{3} \cos 2 x+\sin 2 x$$

Step-by-Step Solution

Answer

The function is \( f(x) = 2 \cos(2x - \frac{\pi}{6}) \) with amplitude 2, period \( \pi \), and phase shift \( \frac{\pi}{12} \) to the right.

Step 1: Conversion to a Single Cosine Function

To express the trigonometric function \( f(x) = \sqrt{3} \cos 2x + \sin 2x \) as a single cosine function, we use the identity \( a \cos x + b \sin x = R \cos(x - \alpha) \). First, calculate \( R \): \[ R = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = 2 \]Now determine \( \alpha \) using \( \cos \alpha = \frac{\sqrt{3}}{2} \) and \( \sin \alpha = \frac{1}{2} \), which are the standard values for \( \alpha = \frac{\pi}{6} \). Thus, we have:\( f(x) = 2 \cos(2x - \frac{\pi}{6}) \).

Step 2: Determine Amplitude

The amplitude of the function is the coefficient \( R \) in the expression \( R \cos(x - \alpha) \). Therefore, the amplitude is \( 2 \).

Step 3: Determine Period

The period of a cosine function \( \cos(Bx) \) is given by \( \frac{2\pi}{B} \). For the function \( \cos(2x) \), the period is \( \frac{2\pi}{2} = \pi \).

Step 4: Determine Phase Shift

The phase shift for a function in the form \( \cos(Bx - C) \) is determined by \( \frac{C}{B} \). Here, substitute \( C = \frac{\pi}{6} \) and \( B = 2 \), giving a phase shift of \( \frac{\frac{\pi}{6}}{2} = \frac{\pi}{12} \). The shift is to the right.

Step 5: Sketch the Graph

To sketch the graph of \( f(x) = 2 \cos(2x - \frac{\pi}{6}) \), start by marking the amplitude at 2. The period is \( \pi \), meaning the curve completes one full cycle over each period. Since the phase shift is \( \frac{\pi}{12} \) to the right, the graph starts \( \frac{\pi}{12} \) units to the right of the y-axis.

Key Concepts

Amplitude

Amplitude is an essential concept when it comes to understanding trigonometric functions. It refers to the height from the centerline of a graph to its peak or trough. For a cosine function, the amplitude determines how "tall" the graph will appear.

In this exercise, the original function is expressed in the form \( R \cos(x - \alpha) \). The amplitude is simply the leading coefficient \( R \) of this equation. In our solution, we found \( R = 2 \). This tells us that, from the central axis (or the mean line of the graph), the curve will rise and fall by 2 units.

Understanding amplitude helps in predicting how the waveform behaves visually. Larger amplitudes create more pronounced waves, while smaller amplitudes reduce the height of these waves.

Period

The period of a trigonometric function is the interval over which the function repeats its values. It's like a "reset" point for the function, where it begins to repeat its pattern.

For basic cosine functions \( \cos(Bx) \), the formula to determine the period is \( \frac{2\pi}{B} \). In our problem, the term \( 2x \) indicates the stretching or compressing of the basic cosine wave due to the coefficient 2. Plugging in the value gives us the period as \( \frac{2\pi}{2} = \pi \).

This means the function completes one full oscillation from start to finish over an interval of \( \pi \) on the x-axis. Learning about a function's period helps us understand how frequently the waves occur along the horizontal axis.

Phase Shift

Phase shift refers to the horizontal shift of a function on its graph. It is the offset from the origin along the x-axis. This parameter allows you to move the entire graph left or right along the x-axis, which alters the starting position of the wave.

In a cosine function expressed as \( \cos(Bx - C) \), the phase shift can be calculated using \( \frac{C}{B} \). For our exercise, the function becomes \( 2 \cos(2x - \frac{\pi}{6}) \), with \( B = 2 \) and \( C = \frac{\pi}{6} \). Plugging in these values gives a phase shift of \( \frac{\pi}{12} \).

This shift is positive, meaning the entire curve shifts to the right. Understanding phase shift helps in accurately determining where the wave cycle begins, especially when visualizing or sketching graphs.