Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=\frac{2}{3} \cos \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Amplitude: \(\frac{2}{3}\); Period: \(2\pi\); Phase shift: Left \(\frac{\pi}{2}\); Vertical translation: None; Range: \([-\frac{2}{3}, \frac{2}{3}]\).

Step 1: Identify the Amplitude

The amplitude of a cosine function of the form \(y = a \cos(bx + c) + d\) is the absolute value of the coefficient \(a\). In this function, \(a = \frac{2}{3}\), so the amplitude is \(\frac{2}{3}\).

Step 2: Determine the Period

The period of a cosine function is given by \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the cosine function. Here, \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Calculate the Phase Shift

The phase shift is determined by solving \(-\frac{c}{b}\) from the equation \(b(x + \frac{c}{b})\). In this function, \(c = \frac{\pi}{2}\) and \(b = 1\), so the phase shift is \(-\frac{\pi}{2}\). This means the graph shifts \(\frac{\pi}{2}\) units to the left.

Step 4: Identify the Vertical Translation

The vertical translation is determined by \(d\) in the general form \(y = a \cos(bx + c) + d\). In this function, \(d = 0\), meaning there is no vertical translation.

Step 5: Calculate the Range

The range of the cosine function \(y = a \cos(bx + c) + d\) is \([d - |a|, d + |a|]\). Here, \(d = 0\) and \(a = \frac{2}{3}\), so the range is \([-\frac{2}{3}, \frac{2}{3}]\).

Step 6: Graph the Function Over One Period

Graph the function by plotting key points for one period \([0, 2\pi]\). Start with points \((\frac{\pi}{2}, 0)\), \((0, -\frac{2}{3})\), \((\frac{3\pi}{2}, \frac{2}{3})\), \((\pi, 0)\), and \((\frac{5\pi}{2}, -\frac{2}{3})\), accounting for amplitude, period, and phase shift.

Key Concepts

Amplitude

Understanding amplitude is crucial when studying trigonometric functions like the cosine function. The amplitude represents the maximum distance between the highest point of the wave and the middle of the wave. It effectively determines how "tall" or "short" the wave is. In the function given, the amplitude is discovered by looking at the coefficient in front of the cosine. If we have a cosine function of the form \(y = a \cos(bx + c) + d\), the amplitude is \(|a|\). For the expression \(y = \frac{2}{3} \cos\left(x + \frac{\pi}{2}\right)\), the amplitude is \(\frac{2}{3}\), indicating that the peaks of the cosine wave are \(\frac{2}{3}\) units above and below the centerline.

Period of Function

The period of a trigonometric function is the length of the interval needed for the function to complete one full cycle. Think of it as the space it takes for the wave to complete its shape before repeating again. For the cosine function \(y = a\cos(bx + c) + d\), we find the period using the formula \(\frac{2\pi}{b}\). Here, \(b\) is the number that multiplies \(x\). In our example \(y = \frac{2}{3} \cos\left(x + \frac{\pi}{2}\right)\), \(b = 1\). Thus, the period is \(\frac{2\pi}{1} = 2\pi\). This means every \(2\pi\) units along the x-axis, the wave starts to repeat.

Phase Shift

Phase shift essentially describes the horizontal movement of the wave along the x-axis. It tells us how much the whole wave is shifted left or right from its usual starting position. To calculate the phase shift in the cosine function \(y = a\cos(bx + c) + d\), apply the formula \(-\frac{c}{b}\). In the given function \(y = \frac{2}{3} \cos\left(x + \frac{\pi}{2}\right)\), we find \(c = \frac{\pi}{2}\) and \(b = 1\), giving us a phase shift of \(-\frac{\pi}{2}\). This means that the entire graph of the cosine function is moved to the left by \(\frac{\pi}{2}\) units.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It helps model periodic phenomena like sound waves, light waves, and tides. In general, a basic cosine function is represented as \(y = \cos(x)\), showing how the function oscillates as \(x\) changes. When parameters \(a\), \(b\), \(c\), and \(d\) are introduced, the function transforms to \(y = a\cos(bx + c) + d\).

• \(a\) affects the amplitude (how tall the wave peaks are).
• \(b\) modifies the period (how quickly the wave completes its cycle).
• \(c\) influences the phase shift (horizontal movement).
• \(d\) moves the graph vertically up or down.

For the specific problem \(y = \frac{2}{3}\cos\left(x + \frac{\pi}{2}\right)\), the function maintains the familiar wave pattern, but with the adjustments in amplitude, period, and phase shift we computed earlier.