Problem 2 - Calculus Concepts

Question

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(f(x)=0.1 \sin (4 x-2)-0.5\)

Step-by-Step Solution

Answer

Amplitude: 0.1, Period: \(\frac{\pi}{2}\), Average value: -0.5, Horizontal shift: 0.5 units right.

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function in the form \(f(x) = a \sin(bx - c) + d\) is given by the absolute value of \(a\). Here, \(a = 0.1\), so the amplitude is \(0.1\).

Step 2: Determine the Period

The period of a sinusoidal function \(f(x) = a \sin(bx - c) + d\) is calculated by \(\frac{2\pi}{|b|}\). For \(b = 4\), the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 3: Calculate the Average Value

The average value of a sinusoidal function is given by \(d\), which is the vertical shift. For \(f(x) = 0.1 \sin(4x - 2) - 0.5\), the average value is \(-0.5\).

Step 4: Find the Horizontal Shift

The horizontal shift, also known as the phase shift, is calculated by \(\frac{c}{b}\), where \(c = 2\) and \(b = 4\). So the horizontal shift is \(\frac{2}{4} = 0.5\) units to the right.

Key Concepts

Amplitude

In trigonometric functions, the amplitude is a measure of how much the function oscillates above and below its midline. For a sine or cosine function given by the standard form \( f(x) = a \sin(bx - c) + d \), the amplitude is determined by the coefficient \( a \). This value dictates the maximum and minimum values that the function can reach from the midline.
For the function \( f(x) = 0.1 \sin(4x - 2) - 0.5 \), the amplitude is the absolute value of \( a \), which is \( 0.1 \). This means that the peaks and troughs of the sine wave extend only 0.1 units above and below the midline. In simple terms:

• The range of the function spans from \( -0.5 + 0.1 = -0.4 \) to \( -0.5 - 0.1 = -0.6 \).
• This small amplitude indicates that the sine wave is quite flat.

Amplitude affects the steepness or the height of the oscillations, and in this instance, the wave's oscillations are gentle.

Period

The period of a trigonometric function describes how long it takes for the function to repeat its cycle. This is especially important for sine and cosine functions. The period can be determined from the standard equation \( f(x) = a \sin(bx - c) + d \) using the formula \( \frac{2\pi}{|b|} \).
For \( f(x) = 0.1 \sin(4x - 2) - 0.5 \), \( b \) is 4, meaning the function repeats every \( \frac{2\pi}{4} = \frac{\pi}{2} \) units. Here’s a breakdown of why the period is important:

• The period determines the width of one cycle of the sine wave on the graph.
• A period of \( \frac{\pi}{2} \) means the wave completes a full oscillation in half the time it takes a standard sine curve to do so, leading to more frequent oscillations.
• This makes the graph appear more frequent and closely packed.

In essence, a smaller period indicates quicker and more frequent oscillations.

Phase Shift

The phase shift in trigonometric functions, also known as the horizontal shift, indicates how the wave is shifted left or right from its usual position. For the function \( f(x) = a \sin(bx - c) + d \), the phase shift can be calculated as \( \frac{c}{b} \). This helps in repositioning the wave along the x-axis.
For our function \( f(x) = 0.1 \sin(4x - 2) - 0.5 \), with \( c = 2 \) and \( b = 4 \), the phase shift is \( \frac{2}{4} = 0.5 \) units to the right. Here's why phase shift matters:

• It tells how much the starting point of the sine wave has moved horizontally on the graph.
• A phase shift of 0.5 to the right means that every point on the standard sine wave has been moved 0.5 units to the right.
• This can affect the timing of the oscillation cycle.

Understanding phase shift is crucial for graphing and analyzing the timing and alignment of trigonometric functions.