Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=-1-2 \cos 5 x$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(\frac{2\pi}{5}\), No phase shift, Vertical translation: -1, Range: [-3, 1]

Step 1: Identify the amplitude

The function is \(y = -1 - 2\cos(5x)\). Rewriting: \(y = -2\cos(5x) - 1\).

The amplitude is \(|{-2}| = \boxed{2}\).

Step 2: Find the period

For \(y = A\cos(Bx)\), the period is \(\frac{2\pi}{|B|}\).

Here \(B = 5\), so the period is \(\frac{2\pi}{5}\).

Step 3: Determine phase shift and vertical translation

There is no horizontal shift (phase shift = 0) since the argument is simply \(5x\) with no added constant.

The vertical translation is \(-1\) (the function is shifted down by 1 unit).

Step 4: Find the range

Since \(-1 \leq \cos(5x) \leq 1\):
\(-2(1) - 1 \leq y \leq -2(-1) - 1\)
\(-3 \leq y \leq 1\)

The range is \(\boxed{[-3, 1]}\).

Key Concepts

Amplitude

Amplitude is a key attribute of trigonometric functions like sine and cosine. In the context of our function, amplitude describes how tall the "waves" in a cosine graph are. It's essentially the maximum displacement from the average or midline of the graph.
To find the amplitude of the given function, we focus on the coefficient of the cosine term. The function provided is \( y = -1 - 2 \cos(5x) \). Here, the coefficient of \( \cos \) is \(-2\).
The amplitude is the absolute value of this coefficient because the amplitude cannot be negative.

• Amplitude = \( |a| = |-2| = 2 \)

This value tells us that the peaks and troughs of our cosine wave will be 2 units away from the midline of the graph.

Period

The period of a trigonometric function determines the length of one complete cycle of the pattern. For the cosine and sine functions, the period is related to the coefficient \(b\) in front of the \(x\) variable. It affects how "stretched" or "compressed" the graph appears horizontally.
For cosine functions, the period can be determined using the formula:

• \( \text{Period} = \frac{2\pi}{|b|} \)

In our function \( y = -1 - 2\cos(5x) \), the value of \(b\) is 5.

• Therefore, the period is \( \frac{2\pi}{5} \).

This means that the function completes its cycle every \( \frac{2\pi}{5} \) units along the x-axis.

Phase Shift

The phase shift of a trigonometric function indicates how the function is horizontally shifted along the x-axis. This is determined by the value of \(c\) in the standard form
\( y = a \cos(bx - c) + d \).
For our function, \( y = -1 - 2\cos(5x) \), the term \(bx - c\) simplifies to \(5x\) which implies \(c = 0\).
The phase shift is calculated as follows:

• \( \text{Phase Shift} = \frac{c}{b} = \frac{0}{5} = 0 \)

This calculation means there is no horizontal shift in our function. The wave begins its cycle at the origin, just as a standard cosine wave would without any modifications.

Vertical Translation

Vertical translation involves shifting the entire graph of the function up or down along the y-axis. This translation is determined by the constant \(d\) in the equation \( y = a \cos(bx - c) + d \).
In the given function \( y = -1 - 2\cos(5x) \), \(d\) is \(-1\).
Hence, our graph will move 1 unit downwards from the usual midline, which typically occurs at \(y = 0\).

• This indicates the new midline is at \(y = -1\).

This translation reflects in how the peaks and troughs are centered around \(y = -1\) instead of the horizontal axis.

Graph of Function

The graph of a trigonometric function like cosine is a wave that repeats its pattern based on its amplitude, period, and shifts. For \( y = -1 - 2 \cos(5x) \):

• The amplitude is 2, meaning the peaks and valleys reach 2 units above and below the midline \(y = -1\).
• The period is \( \frac{2\pi}{5} \), dictating how quickly the wave repeats.
• There's no phase shift, so the wave's cycle begins at the origin.
• Finally, the vertical translation downshifts the entire graph by 1 unit.

When you plot this, you draw a wave that dips and rises within the bounds of \(-3\) and \(1\), with its central line at \(y = -1\). Understanding these aspects ensures you accurately depict the behavior and positioning of this trigonometric function on a graph.