Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=3 \cos \left(\frac{1}{2} x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(4\pi\), Phase Shift: \(\frac{\pi}{2}\) units right.

Step 1: Identify the amplitude

The general form of a cosine function is \( y = a \cos(bx - c) + d \). The amplitude, \( a \), is the coefficient of the cosine function. In the given function \( y = 3 \cos\left(\frac{1}{2}x - \frac{\pi}{4}\right) \), the amplitude \( a \) is 3.

Step 2: Determine the period

The period of a cosine function is determined by the formula \( \frac{2\pi}{b} \). In the equation \( y = 3 \cos\left(\frac{1}{2}x - \frac{\pi}{4}\right) \), \( b = \frac{1}{2} \). Thus, the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

Step 3: Calculate the phase shift

The phase shift is given by \( \frac{c}{b} \). In our function, \( c = \frac{\pi}{4} \) and \( b = \frac{1}{2} \), so the phase shift is \( \frac{\frac{\pi}{4}}{\frac{1}{2}} = \frac{\pi}{2} \) units to the right.

Step 4: Sketch the cosine graph

Start by plotting the basic shape of a cosine curve. Mark key points based on the amplitude 3 (peaks at 3 and troughs at -3), the period \( 4\pi \), and the phase shift of \( \frac{\pi}{2} \) units to the right. Adjust the x-values of standard cosine points by \( \frac{\pi}{2} \) to the right. Draw the wave beginning from the new start point, incorporating the identified amplitude and period. Mark the x-axis from \( x = 0 \) to \( x = 4\pi \).

Key Concepts

Amplitude

The amplitude of a cosine function is one of its key features and tells us about the height of the wave's peaks and troughs. Essentially, amplitude refers to how far the function oscillates above and below its central axis. For a general cosine function

• The equation can be written as: \[ y = a \cos(bx - c) + d \] Here, "a" is the amplitude.
• In this formula, the amplitude "a" is the coefficient in front of the cosine term.
• In the exercise's equation, \[ y = 3 \cos\left( \frac{1}{2}x - \frac{\pi}{4} \right) \] the amplitude, "3", tells us that the function will reach up to 3 units above and 3 units below its middle line.
• This means the graph will oscillate from +3 to -3, centered around 0.

Understanding the amplitude will help you grasp the overall vertical stretch of the graph and how intense or soft the oscillations are.

Period

Period determines how long it takes the cosine wave to complete one full cycle before it starts repeating. Think of it as the horizontal length of one complete oscillation.

• In the equation for the cosine function \[ y = a \cos(bx - c) + d \] "b" influences the period.
• The formula to calculate the period is \[ \frac{2\pi}{b} \].
• For the given equation \[ y = 3 \cos \left( \frac{1}{2}x - \frac{\pi}{4} \right) \], "b" is \( \frac{1}{2} \).
• So, the period is \[ \frac{2\pi}{\frac{1}{2}} = 4\pi \].
• This value tells us that every 4π units along the x-axis, the wave will repeat its pattern.

Recognizing the period is crucial for sketching the graph correctly as it provides the spacing between recurring patterns of the waveform especially when transformations involve stretch and compression.

Phase Shift

Phase shift indicates how the entire graph of the cosine function is horizontally shifted along the x-axis. It reveals whether the wave starts to the left or right relative to its standard position. To ascertain phase shift:

• Use the general form of the cosine function \[ y = a \cos(bx - c) + d \].
• Phase shift is calculated as \[ \frac{c}{b} \].
• In the equation \[ y = 3 \cos \left( \frac{1}{2} x - \frac{\pi}{4} \right) \], "c" equals \( \frac{\pi}{4} \) and "b" is \( \frac{1}{2} \).
• The resulting phase shift is \[ \frac{\frac{\pi}{4}}{\frac{1}{2}} = \frac{\pi}{2} \], meaning the entire graph is shifted \( \frac{\pi}{2} \) units to the right.

Understanding this can help in determining where to start plotting the critical points of the graph, ensuring the cosine wave aligns with its transformations effectively.