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Chapter 13: Problem 25

Sketch one cycle of the graph of each sine function. $$ y=2 \sin \pi \theta $$

Short Answer

Expert verified
The graph of the function \(y = 2 \sin \pi \theta\) starts at the origin, reaches its maximum at \((\frac{1}{2}, 2)\), returns to the midline at (1,0), reaches its minimum at \((\frac{3}{2}, -2)\) and finishes the cycle at (2,0).

Step by step solution

01

Identify the amplitude

The amplitude of the function is the absolute value of the numeric coefficient of the sine function. In this case, amplitude is \(A = 2\), because the coefficient attached to the sine function is 2. This tells us the maximum and minimum points the function can reach.
02

Identify the period

The period of the function is obtained from the coefficient attached to the variable within the sine function. Here, the coefficient of \(\theta\) is \(\pi\). So the period of the function is \(\frac{2\pi}{\pi} = 2\), instead of the conventional \(2\pi\) for a simple sine function. This tells us how long it takes for the function to complete one cycle.
03

Identify phase shifts and vertical shifts

Phase shifts and vertical shifts are identified from any added constants within or outside the sine function. In this case, there are no such constants, which means neither a phase shift nor a vertical shift exists. The function starts at the origin and has no vertical displacement.
04

Sketch the graph

With the amplitude, period, phase shifts and vertical shifts known, now it can be plotted. The graph starts at the origin (0,0), reaches its maximum at \((\frac{1}{2}, 2)\), comes back to the midline at (1,0), reaches its minimum at \((\frac{3}{2}, -2)\) and comes back to the midline to finish the cycle at (2,0).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude in Sine Functions
The amplitude of a sine function refers to the height of the wave from its centerline. Imagine the waves of the ocean reaching higher and higher; the amplitude measures how high those waves go from a calm, flat sea. In our function, \(y = 2 \sin \pi \theta\), the amplitude is \(2\). This means from the center of the sine wave, it will rise up 2 units and drop down 2 units.
  • The amplitude is always a positive value.
  • It is found by taking the absolute value of the coefficient of the sine term.
  • Because it is '2', the wave reaches its highest at '2' and lowest at '-2'.
In real-world scenarios, imagine sound waves鈥攖he louder the sound, the greater its amplitude on a graph.
Period of a Sine Function
The period of a sine function tells us how long it takes for the wave to repeat itself. It's like knowing how long a dance circle exists before it starts over with the same moves. Typically, a standard sine function has a period of \(2\pi\). However, coefficients inside the function can change this.

In our example, \(y = 2 \sin \pi \theta\), the coefficient multiplying \(\theta\) is \(\pi\). Therefore, the period is calculated as \(\frac{2\pi}{\pi} = 2\). This tells us the graph completes one cycle every 2 units of \(\theta\).
  • To find the period, use \(\frac{2\pi}{\text{B}}\), where \(\text{B}\) is the coefficient of \(\theta\).
  • The period helps position the peaks and troughs of the wave correctly on the graph.
  • The shorter the period, the faster the function repeats itself.
Phase Shift in Sine Functions
The phase shift in a sine function indicates a horizontal movement of the graph along the \(\theta\)-axis. It tells us how the wave is shifted from its usual starting position. Think of it like the wave moving left or right before it starts dancing across a new cycle.

In the function \(y = 2 \sin \pi \theta\), there is no added constant inside the sine function. This means there is no phase shift, so the wave starts precisely where we anticipate鈥攁t the origin \((0, 0)\).
  • A phase shift is typically indicated by a horizontal translation of the sine graph.
  • If inside the function you'll find something like \(\sin(\theta + C)\), then \(C\) would tell us about the phase shift.
  • In this specific function, the absence of \(C\) suggests no horizontal movement is needed.
The absence of a phase shift means this sine wave behaves in a standard, expected pattern, with no left or right dislocations.

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