Problem 42 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=4 \cos x $$

Step-by-Step Solution

Answer

The graph of \(y = 4 \cos x\) is a cosine wave with amplitude 4 and period \(2\pi\), without any phase shift. It spans across two full periods from \(x = 0\) to \(x = 4\pi\).

Step 1: Identify the Amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient of the function. For the function \(y = 4 \cos x\), the amplitude is \(|4|\), which is equal to 4.

Step 2: Determine the Period

The period of a basic cosine function, \(\cos x\), is \(2\pi\). For the given function, there's no coefficient attached to \(x\), hence the period of this function remains \(2\pi\).

Step 3: Identify the Phase Shift

Phase shift refers to the horizontal shift of the function. It would be the value of \(c\) in the cosine function in the format \(\cos(x - c)\). The given function is \(y = 4 \cos x\), hence there is no phase shift, indicating that the graph starts from the origin.

Step 4: Plot the Graph

Plot the cosine function starting from the origin. At \(x = 0\), \(\cos 0 = 1\), hence point on the graph is at the origin. The graph will reach its peak at the amplitude of 4 (at \(x=0\)), then drop to 0 (at \(x=\frac{\pi}{2}\)), descend to -4 (at \(x=\pi\)), rise back to 0 (at \(x=\frac{3\pi}{2}\)), and finally return to 4 at \(x = 2\pi\), completing one period. Repeat the pattern for the second period from \(x = 2\pi\) to \(x = 4\pi\).

Key Concepts

Amplitude of Trigonometric Functions

Understanding the amplitude of a trigonometric function, such as the cosine function, is crucial for correctly graphing it. The amplitude represents the maximum distance from the horizontal axis to the peak or trough of the wave. In simple terms, it determines how 'tall' or 'short' the waves of the graph will be.

For the function given in the exercise,  \(y = 4 \cos x\), the amplitude is the absolute value of the coefficient before the cosine function, which in this case is \(|4|\), leading to an amplitude of 4. This tells us that the graph will have peaks at 4 units above the horizontal axis and troughs at 4 units below. It’s vital to note that amplitude is always a positive value, reflecting the 'height' of waves, not their direction.

Period of Cosine Function

The period of a cosine function is the distance along the x-axis it takes for the function to complete one full cycle before repeating itself. The standard cosine function, \(\cos x\), has a period of \(2\pi\) radians.

For a modified cosine function like \(y = A \cos(Bx - C) + D\), the period can be found by calculating \(\frac{2\pi}{|B|}\). Since the exercise features the function \(y = 4 \cos x\), with no coefficient multiplying the variable x, the period remains unchanged as \(2\pi\). This period informs us that every \(2\pi\) units along the x-axis, the cosine function will have gone through a full cycle of its wave-like pattern.

Phase Shift in Trigonometry

Phase shift in trigonometry refers to the horizontal shifting of the graph of a trigonometric function. This shift can either be to the right or left of the standard position. It is determined by the value of \(C\) in the function's equation written in the form \(y = A \cos(Bx - C) + D\). A positive value of \(C\) indicates a shift to the right, while a negative value indicates a shift to the left.

In the given function from the exercise, \(y = 4 \cos x\), there is no subtractive or additive constant accompanying the variable x inside the cosine function. Therefore, it lacks a phase shift and the graph of this function starts at the origin. This means that the initial point of the cosine function, for \(x = 0\), is not displaced sideways and remains at the coordinates (0, 4), which is an important anchor point when graphing the function.