Problem 32 - Precalculus Essentials

Question

Determine the amplitude of each function. Then graph the function and \(y=\cos x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=3 \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = 3 \cos x\) is 3. Plotting it alongside the function \(y = \cos x\) helps to clearly see the difference in amplitudes between the two functions, with \(y = 3 \cos x\) reaching an amplitude of 3 as opposed to \(y = \cos x\)'s amplitude of 1.

Step 1: Identify the Amplitude

The amplitude of a cosine function \(y = a \cos x\) is given by the absolute value of the coefficient 'a'. In the given function, \(y = 3 \cos x\), the amplitude is the absolute value of 3, which is 3.

Step 2: Sketch the Basic Shape of the Function

The fundamental period of the cosine function extends from 0 to \(2 \pi\). A complete cycle includes a maximum point at \((0,3)\), zero-crossings at \((\frac{\pi}{2},0)\) and \((\frac{3\pi}{2},0)\), and a minimum point at \((\pi,-3)\). Similarly, for \(y = \cos x\), a complete cycle includes a maximum point at \((0,1)\), zero-crossings at \((\frac{\pi}{2},0)\) and \((\frac{3\pi}{2},0)\), and a minimum point at \((\pi,-1)\).

Step 3: Graph the Functions

Draw an x-y plane and label the x-axis from 0 to \(2 \pi\) and the y-axis from -3 to 3. Since \(y = 3 \cos x\) has the maximum point of 3, plot this as the first point. Follow the cycle described in the previous step to plot the entire graph. Do the same for \(y = \cos x\), but its maximum point is 1.

Key Concepts

Amplitude of a Function

When graphing trigonometric functions like the cosine function, the amplitude is one of the key characteristics to understand. The amplitude represents half the distance between the maximum and minimum values of the function. In simpler terms, it indicates how high or low the wave of the cosine function will go.

For the function \(y = a \cos x\), the amplitude is the absolute value of the coefficient \(a\). In our example, the function \(y = 3 \cos x\) has an amplitude of 3 because the coefficient of \(\cos x\) is 3. This means that the graph of the function will reach 3 units above the horizontal axis and 3 units below it at its peak and trough, respectively.

Cosine Function Properties

Cosine functions, noted as \(y = \cos x\), have several distinctive properties that can be easily identified. They are periodic functions, meaning they repeat their values in regular intervals known as the period. The standard cosine function has a period of \(2\pi\), which means it completes one full cycle every \(2\pi\) units along the horizontal axis.

Additionally, cosine functions are even functions, signifying that they are symmetrical with respect to the vertical axis. This means that \(\cos(-x) = \cos(x)\). The cosine function starts at its maximum value when \(x=0\) and oscillates between its maximum and minimum values as \(x\) increases, creating a wave-like graph with crests and troughs.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane consisting of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is defined by an ordered pair of numbers, \( (x, y) \), representing its coordinates.

This system is essential for graphing functions, such as cosine functions, as it provides a reference framework to plot each point determined by the function's rule. In our exercise's context, the cosine functions will be graphed on a horizontal axis running from 0 to \(2\pi\), with a vertical axis scaled appropriately to accommodate the amplitude of the functions.

Period of Trigonometric Function

The period of a trigonometric function, such as a cosine or sine function, is the length of the smallest interval over which the function's values repeat themselves. For the basic cosine function \(y = \cos x\), the period is \(2\pi\), meaning that the function completes one entire cycle every \(2\pi\) units along the x-axis.

When the cosine function is modified with a coefficient, as in \(y = a \cos x\), though the amplitude changes, the period remains the same at \(2\pi\), unless there is a horizontal scaling factor. Understanding the period is crucial for constructing the graph, as it ensures that the wave pattern is appropriately repeated across the chosen interval for \(x\).