Problem 57
Question
Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=2 \sin (x-\pi)$$
Step-by-Step Solution
Verified Answer
Amplitude: 2; Period: \(2\pi\); Phase shift: \(\pi\) right; No vertical translation; Range: \([-2, 2]\).
1Step 1: Identify the Amplitude
The amplitude of a sine function in the form \( y = a \sin(bx - c) + d \) is the absolute value of \( a \). Here, \( a = 2 \). Thus, the amplitude is 2.
2Step 2: Determine the Period
The period of a sine function is given by \( \frac{2\pi}{b} \). For the function \( y = 2 \sin(x - \pi) \), \( b = 1 \). Therefore, the period is \( \frac{2\pi}{1} = 2\pi \).
3Step 3: Calculate the Phase Shift
The phase shift is determined by the horizontal translation \( \frac{c}{b} \). Here \( c = \pi \) and \( b = 1 \). Thus, the phase shift is \( \frac{\pi}{1} = \pi \) to the right.
4Step 4: Vertical Translation
The vertical translation is determined by the value \( d \) in \( y = a\sin(bx - c) + d \). Since \( d = 0 \), there is no vertical translation.
5Step 5: Find the Range
The range of a sine function \( y = a\sin(bx - c) + d \) is \([d-a, d+a]\). Here, \( d = 0 \) and \( a = 2 \), so the range is \([-2, 2]\).
Key Concepts
AmplitudePeriodPhase ShiftVertical Translation
Amplitude
When we talk about the amplitude of a trigonometric function, we are referring to the height of the wave, specifically the distance from the middle of the wave (the equilibrium position) to its peak or its trough. In mathematical terms, the amplitude is given by the absolute value of the coefficient in front of the sine or cosine function, denoted as \( a \) in the general form \( y = a \sin(bx - c) + d \).
For the equation \( y = 2 \sin(x - \pi) \), the amplitude is simply the absolute value of \( 2 \), which is \( 2 \).
This means that the wave will reach 2 units above and 2 units below the midline of the graph, which in this situation is the x-axis as there is no vertical translation. Understanding amplitude helps you predict how tall or short the waves in the graph will be.
For the equation \( y = 2 \sin(x - \pi) \), the amplitude is simply the absolute value of \( 2 \), which is \( 2 \).
This means that the wave will reach 2 units above and 2 units below the midline of the graph, which in this situation is the x-axis as there is no vertical translation. Understanding amplitude helps you predict how tall or short the waves in the graph will be.
Period
The period of a trigonometric function indicates the interval after which the wave repeats itself, completing one full cycle. For sine and cosine functions, the period is determined by the coefficient \( b \) as seen in the expression \( y = a \sin(bx - c) + d \). The formula to calculate the period is \( \frac{2\pi}{b} \).
In our example \( y = 2 \sin(x - \pi) \), the value of \( b \) is \( 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).
This means every \( 2\pi \) units along the x-axis, the shape of the sine wave starts to repeat itself.
In our example \( y = 2 \sin(x - \pi) \), the value of \( b \) is \( 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).
This means every \( 2\pi \) units along the x-axis, the shape of the sine wave starts to repeat itself.
- Recognizing the period helps in predicting where the graph will begin another identical cycle.
Phase Shift
Phase shift describes the horizontal movement of the trigonometric function along the x-axis. It shows how much the graph of the function is shifted to the right or left compared to the standard position of the sine or cosine graph.
This shift is calculated using the formula \( \frac{c}{b} \) in the function form \( y = a \sin(bx - c) + d \).
For \( y = 2 \sin(x - \pi) \), \( c \) is \( \pi \) and \( b \) is \( 1 \). Therefore, the phase shift is \( \frac{\pi}{1} = \pi \).
Because \( c \) is positive, the graph shifts to the right by \( \pi \) units. Knowing about phase shifts means you can accurately predict the starting point of the wave form along the x-axis and compare it to its standard position.
This shift is calculated using the formula \( \frac{c}{b} \) in the function form \( y = a \sin(bx - c) + d \).
For \( y = 2 \sin(x - \pi) \), \( c \) is \( \pi \) and \( b \) is \( 1 \). Therefore, the phase shift is \( \frac{\pi}{1} = \pi \).
Because \( c \) is positive, the graph shifts to the right by \( \pi \) units. Knowing about phase shifts means you can accurately predict the starting point of the wave form along the x-axis and compare it to its standard position.
Vertical Translation
Vertical translation involves moving the entire graph of the function up or down along the y-axis. In the function format \( y = a \sin(bx - c) + d \), the value of \( d \) specifies this vertical shift.
If \( d \) is positive, the graph shifts upward, and if \( d \) is negative, the graph shifts downward.
Looking at our function \( y = 2 \sin(x - \pi) \), \( d \) is \( 0 \) meaning there's no vertical shift.
If \( d \) is positive, the graph shifts upward, and if \( d \) is negative, the graph shifts downward.
Looking at our function \( y = 2 \sin(x - \pi) \), \( d \) is \( 0 \) meaning there's no vertical shift.
- This means the midline of the sine wave remains on the x-axis.
Other exercises in this chapter
Problem 56
Find the angle of least positive measure that is co terminal with the given angle. $$-\pi$$
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With your calculator in radian mode, work Exercises in order. Let \(s\) represent the number of letters in your first name. Find an approximation for \(\cos s\)
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Give the reference angle for each angle measure. $$\frac{4 \pi}{3}$$
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Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$30^{\circ}$$
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