Problem 210 - Precalculus

Question

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline. $$ f(x)=5 \cos x $$

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(2\pi\), Midline: \(y=0\).

Step 1: Identify the Amplitude

The amplitude of a trigonometric function in the form \(f(x) = a \cos(bx)\) is the absolute value of \(a\). In this case, \(a = 5\), so the amplitude is \( |5| = 5\).

Step 2: Determine the Period

The period of a cosine function is determined by the formula \(\frac{2\pi}{b}\) where \(b\) is the coefficient of \(x\). Here, \(b = 1\), thus the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Find the Midline

The midline of the function is the horizontal line that the graph oscillates around. For the function \(f(x) = 5 \cos x\), the midline is \(y=0\) since there is no vertical shift.

Step 4: Sketch the Function

To sketch the graph of \(f(x) = 5 \cos x\) for two full periods, plot the points starting at \((0, 5)\), then at \(\left(\frac{\pi}{2}, 0\right)\), \((\pi, -5)\), \(\left(\frac{3\pi}{2}, 0\right)\), and finally \((2\pi, 5)\) to complete one period. Repeat this pattern for a second period up to \(4\pi\). Remember that the graph is symmetric about the midline \(y=0\).

Key Concepts

Cosine Function

The cosine function, which is one of the primary trigonometric functions, is instrumental in many mathematical fields, especially where periodic phenomena are analyzed. In a general form, a cosine function can be expressed as \( f(x) = a \cos(bx + c) + d \). Here:

• \(a\) determines the amplitude of the function.
• \(b\) affects the period or frequency.
• \(c\) controls the horizontal phase shift.
• \(d\) shifts the graph vertically, affecting the midline.

For the given function \( f(x) = 5 \cos x \), we observe that the amplitude is 5, the period is determined using the coefficient of \(x\), which is 1, and is typical for standard cosine functions with no phase shift and a midline at \(y = 0\). Recognizing these components helps in sketching and analyzing the behavior of the graph effectively.

Amplitude

Amplitude measures how far the peaks and troughs of a trigonometric function extend from its midline. For any function \( f(x) = a \cos(bx + c) + d \), the amplitude is the absolute value \(|a|\). This height, measured from the midline to a peak or trough, indicates how much the function values deviate from their average value.In our exercise, \( f(x) = 5 \cos x \), the amplitude is 5. This means:

• The graph reaches a maximum height of 5 units above the midline.
• It also descends to 5 units below the midline, giving a total vertical span of 10 units between peak and trough.

Understanding amplitude is crucial for visualizing the scale of oscillations within the graph and predicting the impact of changes in \(a\) on the function's appearance.

Period

The period of a trigonometric function defines the length over which the function repeats itself. Calculated as \( \frac{2\pi}{b} \) for a cosine function \( f(x) = a \cos(bx + c) + d \), it tells us how quickly the wave cycles through its phases.For \( f(x) = 5 \cos x \), since \(b = 1\), the period is \( \frac{2\pi}{1} = 2\pi \). This means the wave completes one full cycle every \(2\pi\) units along the x-axis. Recognizing the period aids in:

• Predicting where peaks, troughs, and crossings of the midline occur.
• Replicating the pattern across multiple cycles as needed to represent more extended functions.

Understanding periods offers insights into the frequency of the oscillations and how they might adapt under various transformations.

Midline

The midline is the horizontal line that runs through the middle of a wave, equidistant from both its highest and lowest points. It represents the average value of the function. In equations like \( f(x) = a \cos(bx + c) + d \), the midline is represented by the constant \(d\).For our function \( f(x) = 5 \cos x \), the midline is at \(y = 0\), as there is no vertical shift (\(d = 0\)). Key points about the midline:

• It provides a baseline from which we can measure the amplitude above and below.
• The graph oscillates symmetrically about this line.

Understanding the concept of the midline helps in visualizing the central path around which the oscillations occur, crucial for accurately sketching trigonometric graphs.