Problem 58 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=3 \cos (x+\pi)-3 $$

Step-by-Step Solution

Answer

The function has amplitude of 3, period 2π, phase shift π to the left and a vertical shift 3 units down.

Step 1: Identify the amplitude A

The amplitude (A) of the function is the absolute value of the coefficient of the cosine function which here equals 3. It determines the 'height' of the wave; the wave oscillates from A to -A which is from 3 to -3 caused by y=A cos(Bx-C).

Step 2: Identify the period B

The coefficient of x inside the cosine function determines the period of the function. The period is given by 2π/B. Here, B = 1(single 'x' inside the function) hence the period 2π/1 = 2π.

Step 3: Identify the phase shift C

The phase shift measures any horizontal shift of the function and is represented by the term 'C' inside the cosine function, cos(B(x - C)). Here, C = -π (which makes it (x+π)), so there's a shift of π to the left.

Step 4: Identify the vertical shift D

The vertical shift is the constant term at the end of the function (D), which moves the function up or down on the coordinate plane. Here, D = -3 so the function is shifted -3 units down.

Step 5: Plot the function

Now we can sketch the function using all the information gathered. Start by rerunning the trigonometric function along the x-axis, and show a shift π units to the left. Then bring in the amplitude and show the function oscillating from 3 units above the x-axis to 3 units below it over a period of 2π. Finally, lower the entire function by 3 units to account for the vertical shift. Repeat for two full periods.

Key Concepts

Amplitude

When graphing a trigonometric function, the amplitude is an important feature to identify. The amplitude refers to the maximum height the function reaches from its average value or the x-axis. Consider it as the 'peak' height on a wave. This means for our function, \(y = 3 \cos(x + \pi) - 3\), the amplitude is \(3\).

• The formula to find the amplitude for a cosine or sine function is the absolute value of the coefficient before the trigonometric function. Thus, it's \(|A|\).
• In the given function, the amplitude is \(3\), meaning the cosine wave rises \(3\) units above and falls \(3\) units below its midline.

This 'height' will affect how tall the waves are, dictating how the graph looks visually.

Period of Trigonometric Functions

In trigonometric graphs, the period is the horizontal length required for the function to complete one cycle of its pattern.

• Generally, the period for sine and cosine functions is calculated using the formula \(2\pi / B\), where \(B\) is the coefficient of \(x\) inside the function.
• For our expression \(y = 3 \cos(x + \pi) - 3\), the \(B\) value is \(1\).

Therefore, the period of the function is \(2\pi\). This means it takes \(2\pi\) units along the x-axis for the wave pattern to start repeating again.

Phase Shift

The phase shift in a trigonometric function measures the horizontal shift along the x-axis. It determines where the wave pattern starts compared to the standard cosine or sine graph.

• Phase shift is identified by the term \(C\) in \(y = A \cos(B(x - C)) + D\). A shift right occurs if \(C\) is positive and a shift left if \(C\) is negative.
• In our function, because of the expression \((x + \pi)\), \(C = -\pi\). This alteration in signs indicates a shift \(\pi\) units to the left.
• Graphically, this means the wave starts its cycle sooner than it typically would.

Vertical Shift

A vertical shift in trigonometric functions indicates how the entire graph of the function is moved up or down the y-axis.

• This is marked by the constant \(D\) in the equation \(y = A \cos(Bx - C) + D\).
• For the given function, \(D = -3\) which means the entire wave is shifted \(3\) units downward.

This movement adjusts the midline or the average height of the wave. It changes where the waveoscillates around on the graph, without altering the amplitude or the shape itself.