Problem 38 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=\frac{1}{2} \cos \left(x-\frac{\pi}{4}\right)\)

Step-by-Step Solution

Answer

Sketch one cycle of the transformed cosine function, starting at \(x = \frac{\pi}{4}\) and ending at \(x = 2\pi + \frac{\pi}{4}\), with amplitude \(\frac{1}{2}\) and shifted right by \(\frac{\pi}{4}\).

Step 1: Identify the basic properties

The given function is a cosine function, which can generally be written as \(y = a \cos(bx - c)\). In this function, \(a=\frac{1}{2}\), \(b=1\), and \(c=\frac{\pi}{4}\). This indicates an amplitude of \(\frac{1}{2}\), a horizontal shift of \(\frac{\pi}{4}\) to the right, and a period of \(2\pi\) since the period is \(\frac{2\pi}{b}\).

Step 2: Calculate the x-values (critical points)

To sketch one cycle of the cosine function, calculate the key points for one period, starting from the shifted point. The start of one period is when \(bx - c = 0\). Solving \(x - \frac{\pi}{4} = 0\) gives \(x = \frac{\pi}{4}\). The end of one period is when \(bx - c = 2\pi\). Solving \(x - \frac{\pi}{4} = 2\pi\) gives \(x = 2\pi + \frac{\pi}{4}\).

Step 3: Determine the key y-values

The cosine function normally reaches its maximum at \(y = a\), its minimum at \(y = -a\), and it crosses the x-axis at \(y = 0\). For this function: Maximum at \(\frac{1}{2}\), Minimum at \(-\frac{1}{2}\), and crosses the x-axis at critical points between maximum and minimum. Use these key values to guide your sketch.

Step 4: Sketch the function

On the graph, mark the start at \(\frac{\pi}{4}\) and end at \(2\pi + \frac{\pi}{4}\). Divide this interval into four equal sections to locate critical points of the cycle: these will be at \(\frac{\pi}{4}\), \(\pi + \frac{\pi}{4}\), \(2\pi + \frac{\pi}{4}\), and in the middle \(\frac{3\pi}{4}\), \(\frac{5\pi}{4}\), etc. Use this to plot the points starting at the maximum value, then crossing to zero, reaching the minimum, crossing back to zero, and back to the maximum. Connect the points smoothly to form a typical "wave" shape of the cosine function.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \(\cos(x)\). A standard cosine wave is characterized by its smooth, repeating oscillations that extend both horizontally and vertically. These waves repeat over a regular interval, known as a cycle, which makes them predictable and easy to graph once you understand their properties.
The general form of a cosine function is given by:

• \(y = a \cos(bx - c) + d\)

Here:

• \(a\) is the amplitude, affecting the vertical stretch or compression.
• \(b\) influences the period, or how long it takes for the wave to repeat.
• \(c\) represents the horizontal shift, moving the graph right or left.
• \(d\) gives the vertical shift, moving the graph up or down.

Cosine waves start at their maximum value, which distinguishes them from sine waves that begin by crossing the horizontal axis. This results in a distinct wave pattern that peaks and dips evenly around the horizontal axis, defined by its specific values of amplitude, period, and horizontal shift.

Amplitude

Amplitude in trigonometric functions, such as the cosine function, refers to the height from the middle of the wave to its peak. In the function \(y = a \cos(bx - c)\), the amplitude is the absolute value of \(a\).
For example, in the equation \(y = \frac{1}{2} \cos(x - \frac{\pi}{4})\), the amplitude is \(\frac{1}{2}\). This means the wave reaches a maximum of \(\frac{1}{2}\) above and a minimum of \(\frac{1}{2}\) below its central axis, effectively compressing the wave vertically compared to the standard \(\cos(x)\) wave, which has an amplitude of 1.
The amplitude affects how "tall" or "short" the waves appear:

• Higher amplitude increases the wave's height, making it steeper.
• Lower amplitude results in a flatter wave.

Amplitude does not affect the horizontal stretching of the graph, but rather only the vertical 'extent' from the baseline to the peak of the cosine function.

Period

The period of a trigonometric function like the cosine wave is crucial as it determines how frequently the waves complete a full cycle. The period can be calculated with the formula \(\frac{2\pi}{b}\) when the standard form \(y = a \cos(bx - c)\) is used.
In our example, \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\). This means the wave completes one full cycle over the interval of \(2\pi\), highlighting a repeating pattern every \(2\pi\) units along the x-axis.
A shorthand for remembering periods:

• When \(b\) is greater than 1, the period shortens, making the waves more frequent.
• When \(b\) is less than 1, the period lengthens, making the waves less frequent.

This provides a straightforward way to gauge how stretched or compressed your wave graph will appear horizontally. Period is a fundamental component in understanding the rhythm and predictability of trigonometric functions.

Horizontal Shift

The horizontal shift in a cosine function refers to the sideways movement along the x-axis. It is also termed as "phase shift." In the equation \(y = a \cos(bx - c)\), the quantity \(\frac{c}{b}\) gives the horizontal shift.
For the example given, \(y = \frac{1}{2} \cos(x - \frac{\pi}{4})\), the horizontal shift is \(+\frac{\pi}{4}\) units to the right. This indicates that every point on the cosine wave, including maximums, minimums, and axis crossings, moves right by \(\frac{\pi}{4}\).
In simple terms:

• A positive value (\(c>0\)) shifts the graph to the right.
• A negative value (\(c<0\)) shifts the graph to the left.

Horizontal shifts are essential in tailoring the precise starting point of wave cycles in various trigonometric calculations. They help in determining the exact phase at which the wave begins its oscillations with respect to a given reference point on the x-axis.