Problem 168 - Precalculus

Question

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes. $$ f(x)=-3 \cos x+3 $$

Step-by-Step Solution

Answer

Amplitude is 3, period is \(2\pi\), midline is \(y=3\), no asymptotes.

Step 1: Identify the Function Type

The given function is \( f(x) = -3 \cos x + 3 \), which is a cosine function. Recall that a cosine function is generally of the form \( f(x) = a \cos(bx+c)+d \). Here, \( a = -3 \), \( b = 1 \), \( c = 0 \), and \( d = 3 \).

Step 2: Determine the Amplitude

The amplitude of a cosine function \( f(x) = a \cos(bx+c)+d \) is given by \( |a| \). In this case, \( a = -3 \), so the amplitude is \( |-3| = 3 \).

Step 3: Determine the Period

The period of a cosine function \( f(x) = a \cos(bx+c)+d \) is given by \( \frac{2\pi}{b} \). Here, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Determine the Midline

The midline of the function is the horizontal line \( y = d \). For this function, \( d = 3 \), thus the midline is \( y = 3 \).

Step 5: Determine the Asymptotes

Cosine functions do not have vertical asymptotes, so for \( f(x) = -3 \cos x + 3 \), there are no asymptotes.

Step 6: Graph the Function

To graph \( f(x) = -3 \cos x + 3 \), plot the midline at \( y = 3 \). Plot the maximum and minimum using the amplitude, setting points at \( y = 6 \) (3+3) and \( y = 0 \) (3-3). Since the period is \( 2\pi \), repeat the pattern over two periods, from \( x = 0 \) to \( x = 4\pi \). Ensure the highest points reach \( y = 0 \), and the lowest points reach \( y = 6 \), starting at \( y = 0 \) for \( x = 0 \) due to the negative amplitude which inverts the typical cosine.

Key Concepts

Cosine Function

Cosine functions are a fundamental type of trigonometric function. They often appear in the form \( f(x) = a \cos(bx + c) + d \), where:

• \(a\) is the amplitude, or vertical stretch factor.
• \(b\) determines the period of the function.
• \(c\) results in horizontal shifts.
• \(d\) moves the function vertically.

The basic shape of the cosine function resembles a wave, starting from its maximum value, decreasing to the minimum, and then returning to the maximum. When graphed, cosine functions exhibit a symmetrical pattern about their midline. They are periodic, which means they repeat their values at regular intervals. In the given problem, the cosine function is \( f(x) = -3 \cos x + 3 \), which has a negative amplitude. This indicates a reflection over the midline, flipping the wave pattern compared to a standard cosine function.

Amplitude

The amplitude of a cosine function reflects the height of its peaks and the depth of its troughs. Mathematically, it is calculated as the absolute value of \( a \) in the function \( f(x) = a \cos(bx + c) + d \). So, amplitude = \(|a|\). For the function \( f(x) = -3 \cos x + 3 \), \(a = -3\), resulting in an amplitude of 3.
Amplitude determines how "tall" the graph appears. For example, an amplitude of 3 means that the distance from the midline to either the peak or the trough is three units. Importantly, amplitude affects only the vertical stretch of the graph but does not move the midline. This value remains consistent, whether the function is shifted up, down, or reflected across the midline.

Period

A function's period is the distance (along the x-axis) required for the function to complete one full cycle of its wave pattern. For cosine functions of the form \( f(x) = a \cos(bx + c) + d \), the period is calculated as \( \frac{2\pi}{b} \).
In the example \( f(x) = -3 \cos x + 3 \), since \(b = 1\), the period is \( \frac{2\pi}{1} = 2\pi \).
This means that between any two points on the x-axis separated by \(2\pi\), the cosine function will display one complete pattern of peaks and troughs. Periodicity is a fundamental property, making sine and cosine functions powerful in modeling repetitive phenomena like sound waves and tides.

Midline

The midline of a trigonometric function like cosine sets the average value or "baseline" of the wave's oscillation. It is calculated as \( y = d \) in the function \( f(x) = a \cos(bx + c) + d \). For the function \( f(x) = -3 \cos x + 3 \), the midline is \( y = 3 \).
This midline indicates the horizontal axis around which the graph of the function oscillates. The peaks of the function rise above the midline by the amplitude amount, whereas the troughs dive below it by the same amount. Midlines are especially useful for graphing since they help establish the central reference point for other key features of the cosine wave.

Asymptotes

In trigonometry, asymptotes are lines that a graph of a function approaches but never actually reaches. Unlike some other trigonometric functions, the basic cosine function does not have vertical asymptotes. Asymptotes are more typical in functions like tangents or secants.

• For the function \( f(x) = -3 \cos x + 3 \), there are no vertical asymptotes.

Cosine functions are defined for all real numbers, contributing to their continuous wave-like pattern. Therefore, the graph of \( f(x) = -3 \cos x + 3 \) smoothly oscillates without approaching any singular line indefinitely. This predictable behavior is part of what makes the cosine function an invaluable tool in various applications spanning from physics to engineering.