Problem 32 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=3 \cos \left(x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(2\pi\), Phase shift: \(\frac{-\pi}{4}\) (left).

Step 1: Identify the Basic Form of the Function

The given function is \[ y = 3 \cos \left( x + \frac{\pi}{4} \right) \]This is a cosine function of the form \[ y = a \cos(bx + c) \] where \(a\), \(b\), and \(c\) are constants.

Step 2: Determine the Amplitude

The amplitude of a cosine function in the form \( y = a \cos(bx + c) \) is given by the absolute value of \(a\). In this case, \(a = 3\), so the amplitude is \[ |3| = 3 \] Thus, the amplitude is 3.

Step 3: Calculate the Period

The period of a cosine function \( y = a \cos(bx + c) \) is given by the formula \[ \text{Period} = \frac{2\pi}{b} \] Here, \(b = 1\), so the period of this function is \[ \frac{2\pi}{1} = 2\pi \] Therefore, the period is \(2\pi\).

Step 4: Determine the Phase Shift

The phase shift of a cosine function \( y = a \cos(bx + c) \) can be calculated using the formula \[ \text{Phase Shift} = -\frac{c}{b} \] With \(c = \frac{\pi}{4}\) and \(b = 1\), the phase shift is \[ -\frac{\pi}{4} \] Thus, the function is shifted \(\frac{\pi}{4}\) units to the left.

Step 5: Graph the Function

To graph one complete period of the function, note that:- Start at \(-\frac{\pi}{4}\) due to the phase shift.- The interval of one period is from \(-\frac{\pi}{4}\) to \(-\frac{\pi}{4} + 2\pi\), which is \(\left(-\frac{\pi}{4}, \frac{7\pi}{4}\right)\).- The amplitude is 3, so the graph oscillates from -3 to 3 on the y-axis.- Sketch the cosine wave beginning at the maximum value (3) at \(-\frac{\pi}{4}\), going down to the minimum value (-3), and returning to 3 by \(\frac{7\pi}{4}\).

Key Concepts

Amplitude

Amplitude is a fundamental concept when dealing with trigonometric functions like sine and cosine. The amplitude determines the height of the peaks and the depth of the troughs of the graph of the function.
In mathematical terms, the amplitude is the absolute value of the coefficient in front of the cosine or sine function. For the function \(y = 3 \cos \left(x + \frac{\pi}{4}\right)\), the amplitude is easily found by taking the absolute value of 3, which gives us:

• Amplitude = \(|3| = 3\)

This means that the graph of this function will oscillate 3 units above and below its central axis, which is typically the x-axis if no vertical shifts are present. This value tells us about the vertical stretch or compression of the wave.
Remember, amplitude is never negative. It's always the distance between the maximum peak and the middle of the wave, or the distance between the trough and the middle.

Period

In trigonometry, the period is the horizontal length of one complete cycle of the wave. It tells us how often the function repeats itself over the x-axis.
The period of a cosine function has the general formula

• Period = \(\frac{2\pi}{b}\)

where \(b\) is the coefficient of \(x\) inside the cosine function. For \(y = 3 \cos \left(x + \frac{\pi}{4}\right)\), \(b\) is 1. So, the calculation of the period is:

• Period = \(\frac{2\pi}{1} = 2\pi\)

What does \(2\pi\) mean? It means that it will take \(2\pi\) units along the x-axis for the cosine wave to complete a full oscillation and begin to repeat its cycle! Knowing the period is crucial for understanding the horizontal spacing in the wave pattern of the graph.

Phase Shift

The last concept is the phase shift, which tells us the horizontal movement or shift of the graph from its usual position. Essentially, phase shift changes where the graph starts its cycle on the x-axis.
To calculate the phase shift, use the formula

• Phase Shift = \(-\frac{c}{b}\)

where \(c\) is the constant added or subtracted inside the parentheses with \(x\), and \(b\) again is the coefficient of \(x\). For the given function \(y = 3 \cos \left(x + \frac{\pi}{4}\right)\), \(c = \frac{\pi}{4}\) and \(b = 1\). Therefore, the phase shift is:

• Phase Shift = \(-\frac{\frac{\pi}{4}}{1} = -\frac{\pi}{4}\)

This tells us that the graph of the function is shifted \(\frac{\pi}{4}\) units to the left compared to a standard cosine graph. By understanding phase shifts, we can manipulate when and where waves start, giving us control over the timing of the wave in practical scenarios.