Problem 1 - Calculus Concepts

Question

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(f(x)=5 \sin (2 x-1)+3\)

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(\pi\), Average Value: 3, Horizontal Shift: \frac{1}{2} to the right.

Step 1: Identify the Amplitude

In the sine function given by \(f(x) = a \sin(bx - c) + d\),\ the amplitude is the absolute value of \(a\). In this function, \(a = 5\). Thus, the amplitude is \(5\).

Step 2: Determine the Period

The period of a sine function \(\sin(bx - c)\) is calculated by the formula \(\frac{2\pi}{b}\). Here, \(b = 2\), so the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Calculate the Average Value

The average value, often referred to as the vertical shift or midline, is given by \(d\) in the equation. In this function, \(d = 3\), so the average value is \(3\).

Step 4: Find the Horizontal Shift

The horizontal shift, also known as the phase shift, is given by \(\frac{c}{b}\), where \(c = 1\) and \(b = 2\) in the equation \(\sin(bx - c)\). The horizontal shift is \(\frac{1}{2}\) to the right.

Key Concepts

Amplitude in Sine Functions

Amplitude in trigonometric functions, particularly in sine functions, represents how much the function oscillates above and below its midline or average value. It's a measure of the function's maximum displacement from its average. For a standard sine function of the form \(f(x) = a \sin(bx - c) + d\), the amplitude is represented by the absolute value of \(a\). Calculating the amplitude is straightforward:

• Identify the coefficient \(a\) in the sine function.
• Take the absolute value of \(a\), as amplitude is always a positive quantity.

In the given function \(f(x) = 5 \sin(2x - 1) + 3\), the amplitude is \(|5| = 5\). This tells us that the function's peaks reach 5 units above and 5 units below the midline.

Period of Trigonometric Functions

The period of a trigonometric function is the length over which the function completes one full cycle of its wave pattern. For sine functions of the form \(f(x) = a \sin(bx - c) + d\), the period is calculated using the formula \(\frac{2\pi}{b}\). Understanding and finding the period is crucial when analyzing the frequency of the wave:

• Identify the coefficient \(b\) from the sine function term \(\sin(bx - c)\).
• Plug \(b\) into the formula \(\frac{2\pi}{b}\) to find the period.

In our example \(f(x) = 5 \sin(2x - 1) + 3\), we have \(b = 2\). Thus, the period is \(\frac{2\pi}{2} = \pi\). This indicates that the sine wave completes one cycle every \(\pi\) units along the x-axis.

Phase Shift in Functions

Phase shift, or horizontal shift, is a measure of the horizontal displacement of a trigonometric function along the x-axis. In the function \(f(x) = a \sin(bx - c) + d\), the horizontal shift can be calculated using \(\frac{c}{b}\):

• Determine the coefficient \(c\) inside the sine function, after factoring out \(b\).
• Compute the phase shift via the formula \(\frac{c}{b}\).
• The direction of the shift depends on the sign of \(c\): positive for a shift to the right, negative for a shift to the left.

For \(f(x) = 5 \sin(2x - 1) + 3\), the values are \(c = 1\) and \(b = 2\), resulting in a phase shift of \(\frac{1}{2}\). This is a shift to the right by \(\frac{1}{2}\) units.

Calculating Average Value in Transformations

The average value of a sine function, also known as the vertical shift or midline, provides the baseline around which the function oscillates. It is represented by \(d\) in the equation \(f(x) = a \sin(bx - c) + d\). Calculating the average value involves:

• Identifying the constant \(d\) in the equation.

This constant dictates where the center of the oscillation lies vertically on the graph. For example, in the function \(f(x) = 5 \sin(2x - 1) + 3\), \(d = 3\), which means the wave oscillates around the line \(y = 3\). This shift upward corresponds to the fact that the function’s midline is 3 units above the x-axis, ensuring the waveform hovers evenly above and below this line.