Problem 29 - Algebra and Trigonometry

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=-2 \sin \left(x-\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{6}\) right.

Step 1: Identify the Amplitude

The amplitude of the function is the absolute value of the coefficient in front of the sine function. In the given function, the coefficient is \-2. So, the amplitude is \(|-2| = 2\).

Step 2: Calculate the Period

The period of a sine function is calculated using the formula \((2\pi)/b\), where \(b\) is the coefficient of \(x\) inside the function. In this case, the coefficient of \(x\) is \1. Therefore, the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Determine the Phase Shift

The phase shift of a sine function is calculated by shifting the graph horizontally by \(c/b\), where \(c\) is the constant inside the parenthesis. Here, \(c = \frac{\pi}{6}\) and \(b = 1\), so the phase shift is \(\frac{\pi}{6}\). The graph is shifted to the right by \(\frac{\pi}{6}\).

Step 4: Graph the Function

To graph the function, start from the phase shift, \(x = \frac{\pi}{6}\), and plot the points from one complete period with the range from \(-2\) to \(2\). The function starts from \(y = 0\) at \(x = \frac{\pi}{6}\), goes to the minimum of \(-2\) at \(x = \frac{\pi}{2}\), returns to \(y = 0\) at \(x = \frac{5\pi}{6}\), goes to the maximum of \2\ at \(x = \frac{3\pi}{2}\), and returns to \(y = 0\) at \(x = \frac{7\pi}{6}\).

Key Concepts

Amplitude

When examining trigonometric functions like sine and cosine, understanding the amplitude is crucial. The amplitude reveals how "tall" or "short" the waves of the function are which directly impacts the overall graph's appearance. The amplitude is assessed by taking the absolute value of the number in front of the sine function, also known as the coefficient. It directly affects both the peaks and troughs of the wave.
In the function provided, the expression is \( y = -2 \sin \left( x - \frac{\pi}{6} \right) \). The coefficient here is -2. To find the amplitude, take the absolute value: \( |-2| = 2 \). Hence, the amplitude is 2. This means that the wave's highest point is 2 and its lowest point is -2. The amplitude determines the vertical stretch or compression of the wave.

Period

The period of a trigonometric function tells us the length of one complete cycle of the wave. For sine functions, the period is always influenced by the coefficient of \(x\) found inside the equation. The general formula used to calculate the period is \( \frac{2\pi}{b} \), where \(b\) is the coefficient of \(x\).
In our example, \( y = -2 \sin \left( x - \frac{\pi}{6} \right) \), the coefficient \(b\) is 1. Plugging it into the formula gives us \( \frac{2\pi}{1} = 2\pi \). This tells us that the wave completes a full cycle every \(2\pi\) units. The period defines how the wave stretches horizontally and whether the wave is widened or compressed.

Phase Shift

Phase shift is the horizontal movement of a trigonometric wave. It indicates how far the wave has moved from its standard position left or right. To find the phase shift, we use the amount inside the parenthesis following the function, typically in the form \( \sin(x - c) \). The formula \( \frac{c}{b} \) helps determine the exact shift.
In our function, \( y = -2 \sin \left( x - \frac{\pi}{6} \right) \), \(c\) is \(\frac{\pi}{6}\), and \(b\) is 1. Hence, the phase shift is \( \frac{\pi}{6} \). A positive value suggests a shift to the right, while a negative value indicates a shift to the left. Therefore, this wave shifts to the right by \( \frac{\pi}{6} \) units. Phase shift fundamentally influences the starting point of the wave along the horizontal axis, allowing for precise control over the wave's orientation.