Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-14,\) sketch one cycle of the graph. $$ y=\cos 3 x $$

Step-by-Step Solution

Answer

The graph of \( y = \cos 3x \) has a period of \( \frac{2\pi}{3} \) and completes one cycle in this interval.

Step 1: Understanding the Basic Cosine Function

The standard cosine function is given by \( y = \cos x \). The graph of \( y = \cos x \) is periodic, with a period of \( 2\pi \). It starts at \( y = 1 \) when \( x = 0 \), goes down to \( y = -1 \) at \( x = \pi \), and returns to \( y = 1 \) at \( x = 2\pi \).

Step 2: Identify Amplitude and Phase Shift

The amplitude of the cosine function \( y = \cos 3x \) is 1, as there is no coefficient altering it in front of the \( \cos \) function. There is also no horizontal or vertical phase shift because the cosine function is not added to or subtracted by any values.

Step 3: Determine the Period

For a function \( y = \cos kx \), the period is given by \( \frac{2\pi}{k} \). In this case, \( k = 3 \), so the period of the function \( y = \cos 3x \) is \( \frac{2\pi}{3} \). This means one complete cycle of the graph spans an interval of \( \frac{2\pi}{3} \) on the x-axis.

Step 4: Sketching One Cycle

To sketch one cycle of \( y = \cos 3x \), plot the key points: Starting point at \( x = 0 \) with \( y = 1 \), descend to \( y = -1 \) at \( x = \frac{\pi}{3} \), and return to \( y = 1 \) at \( x = \frac{2\pi}{3} \). This cycle follows the basic shape and key points of the cosine function, compressed horizontally to fit within \( \frac{2\pi}{3} \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function often represented as \( y = \cos x \). It plays a key role in describing oscillatory behavior, like waves and circular motion.
The graph of the standard cosine function starts at its maximum value of 1 when \( x = 0 \). It then decreases to reach its minimum value of -1 at \( x = \pi \). Finally, it rises back to 1 at \( x = 2\pi \). This repetition is what makes it periodic.
The shape of the cosine graph is a smooth curve, known for its symmetrical peaks and troughs. Learning to recognize this waveform is crucial because it forms the basis for understanding more complex trigonometric functions.

Period of a Trigonometric Function

The period of a trigonometric function is the distance one complete cycle takes along the x-axis. For the basic cosine function \( y = \cos x \), the period is \( 2\pi \).
To find the period of more complex cosine functions like \( y = \cos kx \), we use the formula \( \frac{2\pi}{k} \). Here, \( k \) represents the frequency, or how many cycles fit in \( 2\pi \).

• If \( k = 1 \), the period is \( 2\pi \).
• If \( k = 3 \), the period shortens to \( \frac{2\pi}{3} \).

This tells us how the graph is compressed or stretched horizontally on the x-axis. Understanding period helps us predict the behavior of the function as x-values change.

Amplitude

Amplitude measures the height of the wave from its center line to a peak or trough. For the function \( y = \cos x \), the amplitude is 1 since the maximum value is 1 and the minimum is -1, representing a uniform wave height.
The formula for amplitude is often just the coefficient in front of the cosine function. If the function is \( y = a\cos x \), then the amplitude is \(|a|\).

• For \( y = \cos 3x \), the amplitude remains 1 because there is no coefficient other than 1.
• A larger amplitude would stretch the graph vertically, while a smaller amplitude compresses it.

Recognizing amplitude is essential in understanding how intense or subdued the wave appears in any trigonometric function.

Phase Shift

A phase shift occurs when a trigonometric graph is shifted horizontally across the x-axis. It gives insight into where the function starts compared to the typical starting point shown by its graph.
For a cosine function expressed as \( y = \cos(kx + c) \), the phase shift is determined by solving \( -\frac{c}{k} \). This calculation shows how far and in which direction the function shifts from the origin.

• If there is no \( c \) value, like in \( y = \cos 3x \), there is no phase shift.
• A positive c would shift the graph to the left, while a negative value would move it right.

Phase shifts are useful when aligning or comparing functions as they transition across cycles.