Question

Determine the amplitude and the period for the function. Sketch the graph of the function over one period. $$ y=-2 \cos 3 x $$

Short answer

The amplitude of the given function \(y = -2 \cos 3x\) is 2, and its period is \(\frac{2\pi}{3}\). To sketch the graph for one period, plot the points (0, -2), \(\left(\frac{\pi}{6}, 0\right)\), \(\left(\frac{\pi}{3}, 2\right)\), \(\left(\frac{\pi}{2}, 0\right)\), and \(\left(\frac{2\pi}{3}, -2\right)\), then connect these points in a sinusoidal manner.

Step-by-step solution

Identify the amplitude and B value from the given function

The given function is: \(y = -2 \cos 3x\) The amplitude is the absolute value of the coefficient of the cosine function, which is |-2| = 2. The B value is the coefficient of x, which is 3.

Calculate the period of the function

Now use the formula for calculating the period of cosine function: \(T = \frac{2\pi}{|B|}\) Substitute the B value we found in step 1: \(T = \frac{2\pi}{|3|} = \frac{2\pi}{3}\)

Sketch the graph of the function y = -2 cos 3x for one period

To sketch this graph, we follow these steps: 1. Draw an x-axis and y-axis on the plane. 2. Label the y-axis with the amplitude, which is 2 and -2. 3. Label the x-axis with the period, which is \(\frac{2\pi}{3}\) (this marks the end of one period). 4. Since this is a cosine function, its starting point at x=0 is the amplitude, i.e., the point (0, -2). 5. As this is a regular cosine function multiplied by -1, the graph will pass through these points (within one period): a. (0, -2), obtained from step 4. b. \(\left(\frac{\pi}{6}, 0\right)\), which is the midpoint of the x-axis where the graph intersects the x-axis. c. \(\left(\frac{\pi}{3}, 2\right)\), which is the maximum value of y (opposite of the regular cosine function). d. \(\left(\frac{\pi}{2}, 0\right)\), which is the second midpoint of the x-axis where the graph intersects the x-axis. e. \(\left(\frac{2\pi}{3}, -2\right)\), which is the endpoint of the period and also the amplitude. The graph then repeats. Now, connect these points in a sinusoidal manner to finish sketching the graph of the function y = -2 cos 3x for one period.

Key concepts

Amplitude of Cosine Function

When learning about trigonometric functions, particularly the cosine function, understanding the concept of amplitude is crucial. Amplitude is a measurement of how tall or short the peaks and troughs of the wave are. Essentially, it tells us the maximum distance the graph deviates from its central axis, both above and below.

In the context of the function \( y = -2 \cos 3x \), the amplitude can be determined by looking at the coefficient of the cosine expression. Normally, the cosine function oscillates between 1 and -1, but when a coefficient is present, in this case -2, the range is scaled accordingly. To find the amplitude, we take the absolute value of -2, which gives us 2.

This means that our graph will reach as high as 2 units above the central axis and as low as 2 units below it. The fact that our coefficient is negative does not affect the amplitude (which is always positive), but it does indicate that the graph has been reflected over the horizontal axis. This reflection causes what would normally be the highest point of the graph to become the lowest and vice versa.

Period of Trigonometric Function

The period of a trigonometric function is the distance over which the function completes one full cycle of its pattern before repeating itself. For the cosine function, this is the horizontal length it takes for the wave to return to the same position.

To calculate the period of the function \( y = -2 \cos 3x \), we consider the formula for the period of a cosine function, which is given by \( T = \frac{2\pi}{|B|} \), where \( B \) is the coefficient before the variable x. This coefficient determines how many cycles will fit into the standard interval of \( 2\pi \), which is the period of a basic cosine function without any transformations.

In our function, \( B = 3 \), so the period is \( T = \frac{2\pi}{3} \). This means that every \( \frac{2\pi}{3} \) units along the x-axis, the cosine curve will have completed a cycle and started the next. It's crucial to correctly identify \( B \) and compute the period, as this influences how the graph is drawn and helps us understand the behavior of the trigonometric function over time.

Sketching Cosine Graphs

Sketching cosine graphs may seem challenging, but with a clear understanding of amplitude and period, it becomes more manageable. Let's apply this understanding to the function \( y = -2 \cos 3x \) and construct its graph step by step over one period.

First, draw a set of perpendicular axes. On the y-axis, mark the amplitude (2 and -2), and on the x-axis, divide it into segments corresponding to the period, here \( \frac{2\pi}{3} \). Since we are dealing with a cosine function that has a negative coefficient, we know it has been reflected on the x-axis. This means that, unlike the standard cosine function which starts at 1, our graph starts at the opposite of our amplitude, specifically at -2 when \( x=0 \).

Then, plot the following characteristic points within one cycle:

• (0, -2): Start of the cycle and minimum point due to negative reflection.
• \(\left(\frac{\pi}{6}, 0\right)\): The graph crosses the x-axis.
• \(\left(\frac{\pi}{3}, 2\right)\): The peak, which is the maximum point due to the negative reflection.
• \(\left(\frac{\pi}{2}, 0\right)\): Another x-axis intersection.
• \(\left(\frac{2\pi}{3}, -2\right)\): End of the cycle and minimum point, matching the start.

By smoothly connecting these points, we form a complete wave or cycle of our cosine graph. Remember, the graph is a smooth, continuous curve exhibiting the characteristic 'wave' pattern of trigonometric functions, and in this case, it should reflect the amplitude's influence on its height and the period's influence on its width.