Problem 67 - Algebra 2

Question

Graph each function in the interval from 0 to 2\(\pi\). \(y=-2 \cos \left(x-\frac{\pi}{3}\right)-4\)

Step-by-Step Solution

Answer

The graph of the function \(y=-2 \cos \left(x-\frac{\pi}{3}\right)-4\) from 0 to \(2\pi\) starts at a minimum value, increases to a maximum at a value rightwards \(\frac{\pi}{3}\) from the y-axis, and decreases back to a minimum by \(2\pi\), all while keeping a reflection over the x-axis. The highest and lowest points are 2 units more than and less than -4, respectively.

Step 1: Identify Characteristics of the Function

The given function \(y=-2 \cos \left(x-\frac{\pi}{3}\right)-4\) is a cosine function with an amplitude of 2 (due to the factor -2), a phase shift of \(\frac{\pi}{3}\) to the right (observed in the parentheses) and a vertical displacement of 4 units downwards (observed by the -4 at the end of the function). The negative sign in front of the 2 means that the function is also reflected over the x-axis.

Step 2: Draw the Reference Cosine Curve

Begin by drawing the 'regular' cosine function in the interval from 0 to \(2\pi\). This graph starts at a maximum (1), descends to a minimum at \(\pi\), and returns to a maximum at \(2\pi\).

Step 3: Apply the Amplitude, Phase Shift, and Vertical Displacement

Next, modify this graph according to the characteristics identified in Step 1. Increase the height and depth of the function by a factor of 2 to incorporate the amplitude, shift the entire graph \(\frac{\pi}{3}\) to the right to include the phase shift, then move the graph vertically downwards by 4 units to account for the vertical displacement.

Step 4: Apply the Reflection

Lastly, reflect the entire curve over the x-axis because the function is negative. This means that the modified cosine curve will start with a minimum, ascend to a maximum, then return to a minimum in the provided interval.

Key Concepts

Amplitude

When discussing the amplitude of a trigonometric function, we refer to the height of the wave from its middle point to its peak. For the function given, the amplitude is 2. This is determined by the absolute value of the coefficient placed in front of the cosine function, \(-2\), which is simply \(|-2| = 2\).

The amplitude tells us how much the curve stretches or compresses vertically. In essence, it is how tall the wave appears relative to the x-axis. So, in the graph of \(y = -2 \cos\left(x-\frac{\pi}{3}\right) - 4\), the peaks of the wave are 2 units above and below the central line of the wave when disregarding other transformations.

• A large amplitude will make the wave appear taller.
• A small amplitude compresses the wave, making it shorter.

Understanding amplitude is crucial as it directly affects the sharpness or flatness of the curves that make up our graph.

Phase Shift

Phase shift in trigonometric functions indicates the horizontal movement of the graph along the x-axis. It essentially tells us how much the wave is moved from its usual position. In our exercise, the phase shift is \(\frac{\pi}{3}\) to the right. This is due to the \(x - \frac{\pi}{3}\) term in the cosine function.

• If we see \(x - C\) in the function, the phase shift is \(C\) units to the right.
• If it is \(x + C\), the shift would be \(C\) units to the left.

A phase shift alters where the wave begins its cycle, which means that the usual starting point of the cosine wave, which is at a maximum, will now be moved to the right by \(\frac{\pi}{3}\). Therefore, every point on the graph shifts accordingly, allowing you to adjust and predict how the function behaves over its given interval.

Vertical Displacement

The vertical displacement of a trigonometric graph describes the movement of the entire wave up or down along the y-axis. It is the constant added or subtracted from the function. In our case, the function \(y = -2 \cos\left(x-\frac{\pi}{3}\right) - 4\) has a vertical displacement of 4 units downwards, indicated by \(-4\).

• A positive vertical displacement moves the graph upwards.
• A negative displacement moves it downwards.

This aspect of the graph changes the baseline or midline of the wave. In this scenario, instead of oscillating with peaks and valleys around the x-axis, the wave now centers at \-4\; thus, the maximum and minimum values of the curve are accordingly shifted downwards by 4 units, creating a new resting point below the horizontal axis. Understanding this helps in visualizing how the curve’s position in the vertical space changes.

Reflection of Functions

Reflection in functions occurs when the graph is flipped over a given axis. In trigonometric functions like our example, a reflection occurs over the x-axis due to the negative sign in front of the amplitude \(-2\).

When a function is reflected over the x-axis, each point of the original graph is mirrored. In a cosine graph that normally starts at a maximum above the x-axis, this reflection means starting at a minimum below the x-axis instead. The wave will now reach its maximum in the negative direction first, followed by ascending and descending in reverse order.

• A negative coefficient before the trigonometric function indicates reflection.
• For example, \(-f(x)\) is a reflection of \(+f(x)\).

Such transformations are vital in accurately modeling real-life scenarios with trigonometric functions, as they dictate how the entire graph is inverted across its horizontal plane, affecting the order of peaks and troughs in periodic waves.