Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=\frac{3}{4} \cos x$$

Step-by-Step Solution

Answer

The amplitude is \(\frac{3}{4}\), and the graph is a cosine wave from \([-2\pi, 2\pi]\).

Step 1: Identify the Function Type

The function given is in the form of a cosine function. This type of function is a periodic function that oscillates between maximum and minimum values.

Step 2: Determine the Amplitude

The amplitude of a cosine function in the form \(y = a \cos x\) is the absolute value of \(a\). In our function, \(a = \frac{3}{4}\), so the amplitude is \(\left| \frac{3}{4} \right| = \frac{3}{4}\).

Step 3: Set the Interval for Graphing

The problem specifies the interval \([-2 \pi, 2 \pi]\). This means we will sketch our graph along the x-axis from \(-2 \pi\) to \(2 \pi\).

Step 4: Identify Key Points for Graphing

Start by identifying key points of one complete period of the function \(y = \cos x\): \((0, 1), (\pi/2, 0), (\pi, -1), (3\pi/2, 0), (2\pi, 1)\). Multiply the cosine values by \(\frac{3}{4}\) to fit \(y = \frac{3}{4}\cos x\).

Step 5: Extend Key Points Across the Interval

Using the periodic nature of the cosine function, replicate these points to the left and right to cover the interval \([-2 \pi, 2 \pi]\). Remember, the period is \(2\pi\), so repeat these points backward and forward within the interval.

Step 6: Draw the Graph

Plot these adjusted key points on a graph with the x-axis from \(-2\pi\) to \(2\pi\) and the y-axis from \(-1\) to \(1\). Then, sketch the smooth wave-like curve that passes through these key points, ensuring it peaks and troughs at the amplitude of \(\frac{3}{4}\).

Key Concepts

Amplitude

Amplitude is an important concept in the study of trigonometric functions, particularly those like the cosine function. It measures the extent of the function’s variation from its midline, essentially telling us how high or low the function stretches. For cosine functions written as \(y = a\cos x\), the coefficient \(a\) determines the amplitude. It is always the absolute value of \(a\), ensuring the amplitude is non-negative.

• In our example: \(y = \frac{3}{4} \cos x\)
• The amplitude is \(\left| \frac{3}{4} \right| = \frac{3}{4}\)

This means that the graph of the function will reach a maximum of \(\frac{3}{4}\) units from the midline (which is 0 in the case of regular cosine and sine functions) and a minimum of \(-\frac{3}{4}\) units. Understanding amplitude helps predict how far the peaks and troughs of the function will extend vertically.

Cosine Function

The cosine function, denoted as \(\cos x\), is a fundamental trigonometric function that depicts a wave that starts at its maximum value at \(x=0\). The standard form of a cosine function is \(y = a\cos(bx + c) + d\), but typically focuses on \(y = a\cos x\) for simplicity.

• It oscillates between -1 and 1.
• The function has a period of \(2\pi\).
• Each complete cycle of the wave is the period, spanning from crest to crest or trough to trough.

In the exercise \(y = \frac{3}{4} \cos x\), the wave is shifted vertically based on the amplitude but retains the cosine wave's inherent behavior, which includes smooth, continuous oscillation. Recognizing these characteristics makes graphing cosine functions more intuitive and straightforward.

Periodic Function

A periodic function is a function that repeats its values in regular intervals or periods. Trigonometric functions like sine and cosine are classic examples of periodic functions. These functions show specific behaviors where every cycle returns to the initial pattern.

• The cosine function, \(y = \cos x\), has a period of \(2\pi\).
• This tells us that over an interval of \(2\pi\), the function produces one complete wave.

For our function \(y = \frac{3}{4} \cos x\), the periodic nature means that the wave will begin repeating after every \(2\pi\) along the x-axis. Graphing over the interval \([-2\pi, 2\pi]\) involves plotting two complete cycles of the wave:

• One cycle from \(-2\pi\) to 0
• Another cycle from 0 to \(2\pi\)

This regular repetition helps understand how the function behaves over larger intervals and is crucial for graph analysis. Understanding periodicity in functions is essential, as it helps predict behavior beyond individual cycles.