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Magazine Chapter 6: Problem 30
Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-2 \sin \left(\frac{1}{2} x+\frac{\pi}{2}\right) $$
Short Answer
Expert verified
Amplitude: 2, Period: \(4\pi\), Phase Shift: \(-\pi\).
Step by step solution
01
Identifying the amplitude
The amplitude of a sinusoidal function of the form \(y = a \sin(bx + c)\) is the absolute value of \(a\). Here, \(a = -2\). Therefore, the amplitude is \(|-2| = 2\).
02
Finding the period
The period of a sine function is given by \(\frac{2\pi}{|b|}\). In this case, \(b = \frac{1}{2}\), so the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).
03
Determining the phase shift
The phase shift is calculated as \(-\frac{c}{b}\). For the function \(-2 \sin(\frac{1}{2}x + \frac{\pi}{2})\), \(c = \frac{\pi}{2}\) and \(b = \frac{1}{2}\). The phase shift is \(-\frac{\frac{\pi}{2}}{\frac{1}{2}} = -\pi\).
04
Sketching the graph
Given the amplitude (2), period \(4\pi\), and phase shift (-\pi), we can now sketch the graph. Starting from the phase shift at \(-\pi\), it peaks at \((-\pi + 1\cdot\frac{4\pi}{4}), -2)\), dips to \((-\pi + 2\cdot\frac{4\pi}{4}), 2)\), and completes one full cycle by \((-\pi + 4\pi, 0)\). The negative amplitude reflects the sine wave upside down.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude
When discussing trigonometric functions, amplitude is a crucial concept, especially with sine waves. In simple terms, the amplitude of a sine function determines how "tall" or "short" the wave appears. It represents the maximum distance from the wave's midline to its peak or trough.
For a general sine function of the form \( y = a \sin(bx + c) \), the amplitude is the absolute value of \( a \).
For a general sine function of the form \( y = a \sin(bx + c) \), the amplitude is the absolute value of \( a \).
- If \( a \) is negative, as in the function \( y = -2 \sin \left(\frac{1}{2}x + \frac{\pi}{2}\right) \), the amplitude remains a positive number. Hence, the amplitude in this case is \( |-2| = 2 \).
- A higher absolute value of \( a \) implies a taller wave, while a lower value implies a shorter wave.
- The amplitude affects only the vertical stretch or shrink of the graph.
Period
The period of a trigonometric function measures how long it takes for the function to complete one full cycle of its pattern and return to its starting point.
In mathematics, for a function \( y = a \sin(bx + c) \), the period is derived from the value of \( b \) and calculated using the formula \( \frac{2\pi}{|b|} \).
In mathematics, for a function \( y = a \sin(bx + c) \), the period is derived from the value of \( b \) and calculated using the formula \( \frac{2\pi}{|b|} \).
- The function \( y = -2 \sin \left(\frac{1}{2}x + \frac{\pi}{2}\right) \) has \( b = \frac{1}{2} \), thus the period of this sine function becomes \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).
- This means the wave takes a length of \( 4\pi \) along the x-axis to complete one cycle.
- Changes in \( b \) affect the horizontal stretch/shrink of the wave, with larger values causing more cycles within shorter intervals.
Phase Shift
Phase shift describes the horizontal translation of a trigonometric function along the x-axis. It indicates how much the entire graph of the function is shifted right or left from its original position.
In the context of the function \( y = a \sin(bx + c) \), the phase shift is calculated using the formula \( -\frac{c}{b} \).
In the context of the function \( y = a \sin(bx + c) \), the phase shift is calculated using the formula \( -\frac{c}{b} \).
- Considering the function \( y = -2 \sin \left(\frac{1}{2}x + \frac{\pi}{2}\right) \), the values of \( c \) and \( b \) are \( \frac{\pi}{2} \) and \( \frac{1}{2} \), respectively. This results in a phase shift of \( -\frac{\frac{\pi}{2}}{\frac{1}{2}} = -\pi \).
- The negative sign indicates a shift to the left.
- The phase shift helps in identifying where the wave鈥檚 cycle starts.
Graph of Sine Functions
The visual representation of sine functions involves plotting them on a graph, clearly presenting characteristics like amplitude, period, and phase shift.
Sketching the graph of a function like \( y = -2 \sin \left(\frac{1}{2}x + \frac{\pi}{2}\right) \) involves combining all these concepts:
Sketching the graph of a function like \( y = -2 \sin \left(\frac{1}{2}x + \frac{\pi}{2}\right) \) involves combining all these concepts:
- First, account for the phase shift of \(-\pi\) and plot starting from this point.
- The period of \(4\pi\) suggests that by \(-\pi + 4\pi = 3\pi \), the wave would have completed one full cycle.
- Peaks occur at \(-2\) (maximum height), while troughs reach \(2\), demonstrating a reflected amplitude due to the negative sign of \( -2 \).
