Problem 7 - Calculus Concepts

Question

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(g(x)=3.62 \sin (0.22 x+4.81)+7.32\)

Step-by-Step Solution

Answer

Amplitude: 3.62, Period: 28.55, Average value: 7.32, Horizontal shift: -21.86 units right.

Step 1: Identify the Amplitude

For a sine function in the form \(y = A \sin(Bx + C) + D\), the amplitude is represented by the absolute value of \(A\). In the function \(g(x) = 3.62 \sin(0.22x + 4.81) + 7.32\), \(A = 3.62\). Thus, the amplitude is \(\left| 3.62 \right| = 3.62\).

Step 2: Calculate the Period

The period of a sine function is given by the formula \(\frac{2\pi}{B}\). In the function \(g(x) = 3.62 \sin(0.22x + 4.81) + 7.32\), \(B = 0.22\). Therefore, the period is \(\frac{2\pi}{0.22}\), which simplifies to approximately \(28.55\).

Step 3: Determine the Average Value

The average value of a function in the form \(A \sin(Bx+C) + D\) is \(D\). For \(g(x) = 3.62 \sin(0.22x + 4.81) + 7.32\), \(D = 7.32\). So, the average value is \(7.32\).

Step 4: Find the Horizontal Shift

The horizontal shift of a sine function is determined by solving \(Bx + C = 0\) for \(x\). In our function, replace \(B\) and \(C\) with their values: \(0.22x + 4.81 = 0\). Solving for \(x\), we get \(x = -\frac{4.81}{0.22}\), which equals approximately \(-21.86\). This indicates a horizontal shift of \(-21.86\) units to the right.

Key Concepts

Amplitude

When talking about trigonometric functions like sine and cosine, the amplitude is an important concept. It represents the peak value of the waves created by these functions. You can think of it as the height of the wave from its middle (or baseline) to its top (or crest).
In mathematical terms, for a sine function defined as \( y = A \sin(Bx + C) + D \), the amplitude is determined by the absolute value of \( A \). For example, in our specific function \( g(x) = 3.62 \sin(0.22x + 4.81) + 7.32 \), the amplitude is \( |3.62| = 3.62 \).

• This tells us that the wave reaches 3.62 units above and below its average position.
• The amplitude does not affect the width or position of the wave, but purely its vertical stretch.

Understanding amplitude helps in visualizing the size of the oscillations in the wave, which is crucial in many applications like sound and light waves.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. This means how far along the x-axis you go before the wave pattern repeats itself.
For sine and cosine functions, the period can be calculated as \( \frac{2\pi}{B} \) where \( B \) is the coefficient of \( x \) in the function \( y = A \sin(Bx + C) + D \).
In the function \( g(x) = 3.62 \sin(0.22x + 4.81) + 7.32 \), \( B = 0.22 \). Hence, the period is \( \frac{2\pi}{0.22} \), which is approximately \( 28.55 \).

• A larger period means a slower wave, while a smaller period means a faster one.
• This function will repeat itself approximately every 28.55 units along the x-axis.

Knowing the period is essential for predicting the behavior of waves over time.

Average Value

The average value in trigonometric functions typically refers to the baseline or equilibrium position around which the wave oscillates. In functions of the form \( y = A \sin(Bx + C) + D \), this value is represented by \( D \).
For \( g(x) = 3.62 \sin(0.22x + 4.81) + 7.32 \), the average value is \( 7.32 \). This tells us where the central line of the wave is situated on the y-axis.

• Understanding the average value helps to determine the displacement of the wave from the origin.
• This value is crucial in applications like alternating current (AC) voltage plots where the wave oscillates around a central value.

Recognizing the average value gives insight into the overall position of the harmonic motion.

Horizontal Shift

The horizontal shift of a trigonometric function shows how the entire wave is moved left or right along the x-axis. This shift is determined by the phase term \( C \) in the function \( y = A \sin(Bx + C) + D \).
To find the horizontal shift, solve \( Bx + C = 0 \) for \( x \). In the function \( g(x) = 3.62 \sin(0.22x + 4.81) + 7.32 \), we have \( 0.22x + 4.81 = 0 \). Solving this, \( x \approx -21.86 \).

• This means the function has moved approximately 21.86 units to the left.
• Negative values indicate a shift to the right, positive values mean a shift to the left.

Knowing the horizontal shift is important for synchronizing signals or aligning patterns in various fields.