Problem 4 - Calculus Concepts

Question

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(j(u)=7 \sin (2 u+\pi)-6\)

Step-by-Step Solution

Answer

Amplitude: 7, Period: \(\pi\), Average value: -6, Horizontal shift: \(-\frac{\pi}{2}\) units right.

Step 1: Identify the Amplitude

The amplitude of a sine function, given by the form \(a \sin(bx + c) + d\), is \(|a|\). In the function \(j(u) = 7 \sin(2u + \pi) - 6\), the amplitude \(|a|\) is \(|7| = 7\).

Step 2: Determine the Period

The period of a sine function is calculated using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of the angle variable inside the sine function. Here, \(b = 2\), so the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Find the Average Value

The average value of the sine function \(a \sin(bx + c) + d\) is the value of \(d\). In this function, the average value is \(-6\).

Step 4: Calculate the Horizontal Shift

The horizontal shift is determined from the phase shift \(\frac{-c}{b}\), where \(c\) is the constant added to \(bx\) in the sine function. Here, \(c = \pi\), so the horizontal shift is \(\frac{-\pi}{2}\). This means the function is shifted \( \frac{-\pi}{2} \) units to the right.

Key Concepts

Amplitude

Amplitude is a crucial concept when dealing with trigonometric functions like sine and cosine. It measures the distance from the middle of the wave to its peak. In mathematical terms, for sinusoidal functions such as \( a \sin(bx + c) + d \), amplitude is denoted by \(|a|\).

• Amplitude represents the height of the wave above or below its midline.
• In the given function \( j(u) = 7 \sin(2u + \pi) - 6 \), the amplitude is \( |7| = 7 \).
• The amplitude informs us how "tall" or "short" the wave is, contributing to the wave's energy.

Understanding amplitude helps in predicting the maximum displacement from the equilibrium. It is fundamentally significant in various real-world applications, including sound waves, electricity, and even tides.

Period

The period of a trigonometric function determines how long it takes to repeat a full cycle. It essentially tells us the amount of input needed for the function to start over. For sine and cosine functions, calculating the period involves the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\).

• In the function \( j(u) = 7 \sin(2u + \pi) - 6 \), the coefficient \(b\) is \(2\).
• This leads to a period calculation of \(\frac{2\pi}{2} = \pi\).
• A period of \(\pi\) implies the function rewinds after every \(\pi\) units along the horizontal axis.

The concept of period is vital in various fields, from analyzing repetitive behaviors in nature to designing circuits that oscillate at specific frequencies.

Phase Shift

Phase shift indicates the horizontal displacement of a trigonometric function. It defines how far the function has shifted to the right or left compared to the usual position. For the function \( a \sin(bx + c) + d \), the phase shift is calculated as \( \frac{-c}{b} \).

• In the function \( j(u) = 7 \sin(2u + \pi) - 6 \), "\(c\)" is \(\pi\).
• The phase shift, therefore, is \( \frac{-\pi}{2} \).
• This indicates that the sine wave is shifted \( \frac{-\pi}{2} \) units to the right.

Phase shift plays a substantial role in modeling scenarios where timing is crucial, such as in engineering and physics. It helps synchronize waves or signals to create desired outcomes.