Problem 31 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y=2 \cos x\) is 2.

Step 1 - Determining the Amplitude

Amplitude is the distance from the middle of the oscillation (or wave) to the peak. In the case of the function \(y=2 \cos x\), the amplitude is given by the coefficient of \(\cos x\), which in this case is 2.

Step 2 - Drawing the Graph of \(y=2 \cos x\)

Start by plotting the peak at y=2, the trough at y=-2, and the midline at y=0. The periods here are exactly the same, \(2\pi\). The wave starts at the peak when \(x=0\), goes down to 0 at \(\frac{\pi}{2}\), then to -2 at \(\pi\), back to 0 at \(\frac{3\pi}{2}\), and finally back to peak at \(2\pi\).

Step 3 - Drawing the Graph of \(y=\cos x\)

Now we'll plot the function \(y=\cos x\) on top of the previous graph. This graph starts at a peak when \(x=0\), goes down to 0 at \(\frac{\pi}{2}\), then to -1 at \(\pi\), back to 0 at \(\frac{3\pi}{2}\), and finally back to the peak point at \(2\pi\). The amplitude of this function is 1.

Key Concepts

Amplitude of Trigonometric Functions

Understanding the amplitude of trigonometric functions is pivotal when dealing with waves and oscillations in mathematics. The term 'amplitude' refers to the maximum displacement or distance of a point on the wave from its equilibrium position.

In trigonometric functions like cosine and sine, the amplitude can be identified quite easily: it's the coefficient that multiplies the function. For example, in the function \(y=2 \cos x\), the amplitude is 2. This coefficient dictates how 'tall' or 'short' the wave appears. If the amplitude is greater than 1, the wave stretches vertically, and if it's between 0 and 1, it compresses. An amplitude of 1 means the wave is at its standard height, equivalent to the function \(y=\cos x\).

When graphing trigonometric functions, always note the amplitude first as it sets the upper and lower bounds of the wave, which are essential for accurately drawing the graph.

Trigonometric Graphs

Graphing Steps for \(y=2 \cos x\)

When graphing trigonometric functions such as \(y=2 \cos x\), a systematic approach helps ensure accuracy. Begin by identifying key properties like amplitude, period, phase shift, and vertical shift. The amplitude, previously discussed, sets the peaks and troughs.

Next, plot the central points: start with the peak at \(y=2\), the trough at \(y=-2\), and the baseline, or midline, at \(y=0\). The period of the cosine function by default is \(2\pi\), meaning it completes a full cycle within an interval of \(2\pi\) radians. This function will complete one of its cycles as \(x\) changes from 0 to \(2\pi\).

Graphing \(y=\cos x\) follows a similar pattern. Its amplitude is 1, so the peak is at \(y=1\), and the trough at \(y=-1\). The period remains \(2\pi\). Placing both graphs on the same axes allows comparison of how the differing amplitudes affect the shape and size of the waves.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is the framework within which trigonometric graphs are plotted. It consists of two perpendicular axes: the horizontally running x-axis and the vertically running y-axis.

Each point on this system is determined by an ordered pair of numbers, known as coordinates. These numbers denote the distances from the origin (0,0) along the x-axis (first number) and the y-axis (second number). With trigonometric functions like \(y=2 \cos x\) and \(y=\cos x\), the x-axis typically represents the angle in radians, while the y-axis corresponds to the value of the cosine function at that angle.

By plotting these functions within the rectangular coordinate system, we can visualize the periodic and oscillatory nature of trigonometric functions. This graphical representation serves as a fundamental tool in mathematics, physics, and engineering to understand and predict wave behaviors.