Question

Make a complete graph of each function. Find the amplitude, period, and phase shift. $$y=-2 \cos x$$

Short answer

The amplitude is 2, the period is \(2\pi\), and the phase shift is 0. The graph starts at a minimum value due to the negative sign in front of the amplitude.

Step-by-step solution

Identifying the Function Type

First, recognize that the function provided is a cosine function, which is periodic and has the general form of \(y = A\cos(Bx - C) + D\), where A is the amplitude, \(\frac{2\pi}{B}\) is the period, C is the phase shift, and D is the vertical shift.

Finding the Amplitude

The amplitude is the coefficient in front of the cosine function, which determines the height of the peaks and the depth of the troughs from the midline of the graph. For the function \(y = -2\cos x\), the amplitude is the absolute value of -2, which is 2.

Determining the Period

The period is found by the formula \(\frac{2\pi}{B}\), where B is the coefficient of x inside the cosine function. Since there is no coefficient other than 1 (implicit), B is 1, making the period \(\frac{2\pi}{1} = 2\pi\).

Calculating the Phase Shift

The phase shift is calculated by finding the value of C in the general form, which is the horizontal shift of the function. Since there is no horizontal shift in the equation \(y = -2\cos x\), the phase shift is 0.

Sketching the Graph

Using the information from previous steps, the graph of the function can now be sketched. Since the amplitude is 2, the graph will oscillate 2 units above and below the x-axis. The period is \(2\pi\), so the graph will repeat every \(2\pi\) units. The phase shift is 0, so the graph starts at the maximum value when \(x=0\). The negative sign in front of the 2 suggests that the graph is reflected over the x-axis, so it starts at the minimum value instead. Mark key points such as the maximum, minimum, and x-intercepts every \(\frac{\pi}{2}\) and draw the smooth, periodic wave.

Key concepts

Amplitude of Trigonometric Functions

In trigonometry, amplitude refers to the maximum displacement of a periodic wave from its central position, which correlates to the height of its peaks and depths of its troughs. For the cosine function, it is given by the coefficient that multiplies the cosine term.

For instance, in the function \(y = -2 \cos x\), the amplitude is the absolute value of the coefficient in front of the cosine, which is \(|-2| = 2\). This means the graph will oscillate 2 units above and below the midline (typically the x-axis for a cosine function). Understanding amplitude is crucial as it influences the 'loudness' or 'intensity' of the wave. Additionally, if the amplitude is negative, like in our example, it indicates the graph is a reflection over the midline, starting at a minimum point rather than a maximum.

Period of Trigonometric Functions

The period of a trigonometric function describes the length of one full cycle of the wave, after which the pattern repeats. For cosine and sine functions, this period is generally \(2\pi\) for one complete oscillation from start to finish.

However, when we have a coefficient B in the function as in \(y = A\cos(Bx - C)\), the period changes and is calculated with the formula \(\frac{2\pi}{B}\). If no coefficient is altering the x-value (like in \(y = -2\cos x\), where B equals 1), the period remains the traditional \(2\pi\). It's critical to recognize how the period affects the frequency of the trigonometric function; the larger the value of B, the more compressed the wave becomes, essentially increasing the frequency.

Phase Shift of Trigonometric Functions

The phase shift is the horizontal displacement that shifts the basic cosine curve left or right on a graph. Determined by the C value in the function's general form \(y = A\cos(Bx - C)\), it tells us where the wave starts horizontally.

A positive phase shift means the graph shifts to the right, while a negative one shifts it to the left. In \(y = -2\cos x\), there's no addition or subtraction inside the cosine, so the phase shift is 0, denoting that the function starts at its regular initial point on the graph. Notably, understanding phase shift is essential when trying to align or compare trigonometric functions for their synchronous behavior.

Sketching Trigonometric Graphs

To sketch a trigonometric graph, particularly for the cosine function, we need to pinpoint critical elements such as amplitude, period, and phase shift, which were identified in the previous steps. Begin by marking the central axis, typically the x-axis, and plot the maximum and minimum values based on the amplitude.

Next, divide the period into equal sections to place the characteristic points of the cosine wave (maximum, midpoint, minimum, midpoint). With the absence of a phase shift, the graph of \(y = -2\cos x\) will begin at the minimum point because of the negative amplitude, indicating reflection. Connect these points with a smooth, wave-like curve to complete one cycle. Repeat the pattern for the length of the graph you need, ensuring consistency in wave structure for accuracy.