Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. Give the period and amplinde. $$y=\cos \frac{3}{4} x$$

Step-by-Step Solution

Answer

Period is \(\frac{8\pi}{3}\), amplitude is 1.

Step 1: Understanding the Basic Form of the Cosine Function

The standard form of the cosine function is \( y = \cos(bx) \). This is a transformation of the basic cosine curve, affecting both the period and the frequency.

Step 2: Identifying the Frequency and Period

For the function \( y = \cos\left(\frac{3}{4}x\right) \), the frequency \( b \) is \( \frac{3}{4} \). The period of a cosine function is calculated as \( \frac{2\pi}{b} \). Substituting the value of \( b \), we get the period as \( \frac{2\pi}{\frac{3}{4}} = \frac{8\pi}{3} \).

Step 3: Calculating the Two-Period Interval

Since each period of the given cosine function is \( \frac{8\pi}{3} \), a two-period interval would be \( 2 \times \frac{8\pi}{3} = \frac{16\pi}{3} \). Hence, we should graph the function from \( x = 0 \) to \( x = \frac{16\pi}{3} \).

Step 4: Determining the Amplitude

The amplitude of a cosine function \( y = a\cos(bx) \) is given by the absolute value of \( a \). For \( y = \cos\left(\frac{3}{4}x\right) \), \( a \) is 1, thus the amplitude is \(|1| = 1\).

Step 5: Graphing the Function

On a graph, plot 5 key points for each period: the maximum point at the starting point of period, the zero point of a quarter through the period, the minimum point halfway through the period, the next zero point three-quarters through the period, and the ending maximum point. Repeat this pattern across two periods from \( x = 0 \) to \( x = \frac{16\pi}{3} \).

Key Concepts

Cosine Function

The cosine function is one of the basic trigonometric functions, often denoted as \(y = \cos(x)\). It describes the relationship between a right-angled triangle's angle and the lengths of its sides. The function is periodic, which means it repeats its values in regular intervals, tracing a wave-like pattern. In its standard form, \(y = a\cos(bx + c) + d\), the parameters \(a\), \(b\), \(c\), and \(d\) can affect the graph's shape and position.

• Basic form: The simplest form of the cosine function is \(y = \cos(x)\).
• Stretching and shrinking: The coefficients \(a\) and \(b\) determine the amplitude and frequency, affecting the function's vertical and horizontal stretching, respectively.

In the exercise, the function is \(y=\cos\left(\frac{3}{4}x\right)\). Here, \(b\) is \(\frac{3}{4}\), indicating a frequency change, but the amplitude remains as 1.

Period of a Function

The period of a periodic function like the cosine function is the length of the smallest interval over which the function repeats its values. For the cosine function \(y = \cos(bx)\), the standard period is \(2\pi\), but it changes based on the coefficient \(b\).

• Formula for period: The period \(T\) of \(y = \cos(bx)\) is computed using \(T = \frac{2\pi}{|b|}\).
• Interval of repetition: The function will complete one full cycle every \(T\) units along the x-axis.

In our example, the function \(y = \cos\left(\frac{3}{4}x\right)\) has a period calculated as \(\frac{2\pi}{\frac{3}{4}} = \frac{8\pi}{3}\), meaning it repeats every \(\frac{8\pi}{3}\) along the x-axis. This impacts how the function is graphed over a specified interval.

Amplitude

Amplitude refers to the maximum vertical distance from the central axis of a trigonometric function to its peak (or trough). It provides an understanding of how "tall" or "short" the oscillations of the function are. For the cosine function \(y = a\cos(bx)\), the amplitude is determined by the absolute value of \(a\).

• The fixed height: The amplitude remains constant regardless of the function's horizontal transformations.
• Graphical representation: It represents half the distance between the highest and lowest points on the graph.

In the given exercise \(y = \cos\left(\frac{3}{4}x\right)\), the amplitude is 1, as \(a = 1\), indicating that the graph oscillates between 1 and -1.