Problem 66 - Thomas Calculus

Question

For $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$ identify \(A, B, C,\) and \(D\) for the sine functions in Exercises \(65-68\) and sketch their graphs. $$y=\frac{1}{2} \sin (\pi x-\pi)+\frac{1}{2}$$

Step-by-Step Solution

Answer

A = \(\frac{1}{2}\), B = 2, C = 1, D = \(\frac{1}{2}\).

Step 1: Identify Amplitude (A)

The general form of the sine function is given by \(f(x) = A \sin\left(\frac{2\pi}{B}(x-C)\right) + D\). In the given function, \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), the coefficient \(A\) in front of the sine function is \(\frac{1}{2}\). Therefore, \(A = \frac{1}{2}\).

Step 2: Identify Period Modifier (B)

To find \(B\), compare the given sine function's argument, \(\pi x - \pi\), with \(\frac{2\pi}{B}(x - C)\). Here, we see that \(\pi\) is the coefficient of \(x\), which can be further expressed as \(\frac{2\pi}{B} = \pi\). Solving for \(B\), we get \(B = 2\).

Step 3: Identify Horizontal Shift (C)

In the original function form, the horizontal shift \(C\) appears as \(x - C\). From \(\frac{2\pi}{B}(x - C) = \pi x - \pi\), we compare the constant term inside the sine argument. Solving \( \pi(x - 1) = \pi x - \pi\), we find \(C = 1\).

Step 4: Identify Vertical Shift (D)

The vertical shift \(D\) is the constant term added to the sine function. In \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), \(D = \frac{1}{2}\).

Step 5: Sketch the Graph

Plot the function \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\) using the identified values. The function has an amplitude of \(\frac{1}{2}\), period length of \(2\), is shifted to the right by 1 unit (\(C = 1\)), and is vertically shifted upwards by \(\frac{1}{2}\). The sine curve will start its cycle from \(x = 1\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is crucial in many areas of mathematics and science. It is periodic, meaning it repeats its values in regular intervals or periods. In its basic form, the sine function is written as \(y = \sin x\). In graph terms, it starts at zero, rises to a maximum, decreases to zero, continues to a minimum, and then returns to zero. This sequence within one complete cycle characterizes sinusoidal behavior.
For more complex sinusoidal equations like \(f(x)=A \sin(\frac{2\pi}{B}(x-C))+D\), additional parameters modify the basic sine wave, making it more flexible and applicable to a variety of contexts. Each parameter affects the graph shape in unique ways, which we will explore in the subsequent sections.

Amplitude

Amplitude in a sine function is a measure of how "tall" or "short" the waves are. It is defined as the absolute value of \(A\) in the equation \(y = A \sin(\ldots)\). In simpler terms, amplitude refers to the maximum distance the wave reaches from its central axis.
For the specific function \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), the amplitude is \(\frac{1}{2}\).

• This means that starting from the middle value, the wave peaks \(\frac{1}{2}\) units above and dips \(\frac{1}{2}\) units below this central value.
• In graph terms, this determines the height of each wave cycle in the sine function.

Understanding amplitude is key to adjusting the wave's height in applied scenarios.

Period

The period of a sine function is the length of one complete cycle, which can be found by observing the coefficient \(B\) in the function form \(y = A \sin(\frac{2\pi}{B}(x-C)) + D\). The period \(T\) is given by \(T = \frac{2\pi}{B}\).
In the function \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), we determine that \(B = 2\), so the period is \(T = \frac{2\pi}{2} = 2\).

• This means the function completes a full wave cycle every 2 units on the x-axis.

Period is essential for determining how fast or slow a sine wave cycles as it progresses, which is crucial for applications like signal processing.

Horizontal Shift

Also known as "phase shift," the horizontal shift \(C\) moves the sine wave left or right on the x-axis. It is determined from the expression \((x - C)\) in the sine function's formula.
In the function \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), substituting the values, we get \(C = 1\). This means the entire sine wave is shifted 1 unit to the right.

• The horizontal shift is essential for aligning sine waves to start at a different point along the x-axis.
• Understanding this shift is critical in scenarios where timing or phase alignment is important.

Grasping how horizontal shifts work can be particularly useful when synchronizing periodic events.

Vertical Shift

The vertical shift \(D\) translates the entire graph up or down by some constant value. It appears as a constant added to the standard sine function.
For \(y = \frac{1}{2} \sin(\pi x - \pi) + \frac{1}{2}\), the vertical shift is \(D = \frac{1}{2}\). This indicates that the whole graph is moved up by \(\frac{1}{2}\) units.

• Vertical shifts are important in adjusting the average or median level of the wave's oscillation.
• By lifting or lowering the entire wave, it changes the reference level without altering the wave's internal dynamics.

In applications, vertical shifts allow fine-tuning of wave alignments to fit specific requirements.

Graph Sketching

Graph sketching for a modified sine function involves plotting its behavior over a chosen interval. With the identified parameters of the function, you can construct the graph by systematically analysing each characteristic:

• **Amplitude:** The function \(y = \frac{1}{2} \sin(\ldots) + \frac{1}{2}\) rises and falls \(\frac{1}{2}\) units.
• **Period:** The cycle repeats every 2 units along the x-axis.
• **Horizontal Shift:** The graph starts shifting from 1 on the x-axis.
• **Vertical Shift:** The baseline is moved up by \(\frac{1}{2}\).

Start sketching from \(x = 1\). Draw a smooth curve showing these translations and standstills until reaching the cycle's end. Repeat patterns across the x-axis ensuring identical repetition every 2 units. Graph sketching provides a visual understanding, crucial for analysis and predictions in real-world scenarios.