Problem 33 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=-4 \sin 2\left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Identify Amplitude

The amplitude of a sine function of the form \( y = a \sin(bx + c) \) is the absolute value of \( a \). In this function, \( a = -4 \). Therefore, the amplitude is \( \left| -4 \right| = 4 \).

Step 2: Determine the Period

The period of a sine function is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) within the sine function. Here, \( b = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate the Phase Shift

The phase shift of a sine function is calculated as \( -\frac{c}{b} \), where \( c \) is the horizontal translation. In this function, \( c = 2 \times \frac{\pi}{2} = \pi \). Thus, the phase shift is \( -\frac{\pi}{2} = -\frac{\pi}{2} \).

Step 4: Graph One Complete Period

To graph one complete period of the function, we start at the phase shift \(-\frac{\pi}{2}\) and mark the end at \(-\frac{\pi}{2} + \pi = \frac{\pi}{2}\). The key points within this interval are \(-\frac{\pi}{2}, 0, \frac{\pi}{4}, \frac{\pi}{2} \). Plot these points using the sine wave structure, with the sine starting at 0, moving to the maximum amplitude, back to 0, to the minimum amplitude, and returning to 0.

Key Concepts

Amplitude

The amplitude of a trigonometric function refers to the height of the peaks or the depth of the troughs from the center line of the wave. It tells us how far the wave rises and falls from its midline, which is typically the x-axis in a standard sine or cosine function. This is a crucial feature that determines the overall shape and size of the wave.

In general, when dealing with a function of the form \( y = a \sin(bx + c) \), the amplitude is the absolute value of \( a \). The sign of \( a \) indicates whether the wave is flipped upside down or not, but the amplitude itself is always a positive number.

• For instance, in \( y = -4 \sin 2(x + \frac{\pi}{2}) \), \( a = -4 \).
• The amplitude is therefore \( \left| -4 \right| = 4 \).
• This means the wave reaches 4 units up and down from the midline of the graph.

Knowing the amplitude is vital when sketching the graph of a trigonometric function accurately, as it helps you scale the wave correctly.

Period

The period of a trigonometric function is the distance required for the function to complete one full cycle of its pattern. For sine and cosine functions, this cycle is the distance it takes to travel from start to finish before repeating the same sequence again. Understanding the period allows you to know how frequently the waves repeat over a given interval.

The period is calculated using the formula \( \frac{2\pi}{b} \) in functions like \( y = a \sin(bx + c) \). Here, \( b \) is a scaling factor for the x-axis, stretching or compressing the function horizontally.

• In our function, \( y = -4 \sin 2(x + \frac{\pi}{2}) \), \( b = 2 \).
• To find the period, calculate \( \frac{2\pi}{2} = \pi \).
• This reflects the fact that one complete wave cycle occurs over a span of \( \pi \) units along the x-axis.

With the period of \( \pi \), you know how to space your graph, marking key points to show the function’s ups and downs over one complete cycle.

Phase Shift

Phase shift in a trigonometric function describes the horizontal displacement of the wave. It indicates how much the entire function is shifted to the left or right from its original position. This concept is important for accurately positioning the wave on a graph, influencing when key features of the wave like peaks and troughs occur.

Phase shift is determined by the formula \( -\frac{c}{b} \) when you have a function of form \( y = a \sin(bx + c) \). Here, \( c \) represents the horizontal shift due to the inside of the function’s parentheses.

• In \( y = -4 \sin 2(x + \frac{\pi}{2}) \), we have \( c = 2 \times \frac{\pi}{2} = \pi \).
• The phase shift then becomes \( -\frac{\pi}{2} = -\frac{\pi}{2} \).
• This indicates a shift of \( \frac{\pi}{2} \) units to the left on the x-axis.

Understanding phase shift is crucial for accurately plotting the function’s starting point and correctly aligning the cycle on the graph.