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Magazine Chapter 4: Problem 44
For each equation, identify the amplitude, period, horizontal shift, and phase. Then label the axes accordingly and sketch one complete cycle of the curve. $$ y=\sin \left(\pi x-\frac{\pi}{2}\right) $$
Short Answer
Expert verified
Amplitude: 1, Period: 2, Phase Shift: \(\frac{1}{2}\) right.
Step by step solution
01
Identify the Amplitude
The amplitude of a sine function in the form of \( a \sin(bx - c) + d \) is given by the absolute value of \( a \). In this function, \( y = \sin(\pi x - \frac{\pi}{2}) \), the coefficient in front of the sine is 1. Thus, the amplitude is 1.
02
Determine the Period
The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \) where \( b \) is the coefficient of \( x \) inside the sine function. Here, \( b = \pi \). So the period is \( \frac{2\pi}{\pi} = 2 \).
03
Find the Horizontal Shift
The horizontal shift (or phase shift) is determined by solving \( bx - c = 0 \). For the equation \( \pi x - \frac{\pi}{2} = 0 \), solve for \( x \): \( \pi x = \frac{\pi}{2} \) leads to \( x = \frac{1}{2} \). This means there is a phase shift of \( \frac{1}{2} \) units to the right.
04
Sketching the Graph
With an amplitude of 1, a period of 2, and a phase shift to the right of \( \frac{1}{2} \) units, start plotting key points of the sine wave. The standard sine curve begins at \( (0,0) \), peaks at \( (\frac{1}{2},1) \), crosses zero at \( (1,0) \), reaches minimum at \( (\frac{3}{2},-1) \), and returns to zero at \( (2,0) \). Adjust this start point right by \( \frac{1}{2} \) units to follow the shift. Label the x-axis from \( 0 \) to \( 2 \) and the y-axis from \( -1 \) to \( 1 \).
05
Label the Axes and Complete the Cycle
Label the x-axis with key points shifted by \( \frac{1}{2} \): they are \( \frac{1}{2}, 1, 1.5, \) and \( 2 \). Mark the corresponding y-values of 1, 0, -1, and 0. Draw the sine curve passing through these points. This completes one cycle of the sine curve with the given parameters.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Amplitude
When graphing trigonometric functions, amplitude is a critical concept. It represents the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In simpler terms, it shows how "tall" or "short" the waves are on the sine graph.
To find the amplitude of a sine function in the form of \( a \sin(bx - c) + d \), simply take the absolute value of \( a \).
Always remember, the amplitude affects the "height" of the graph, but it does not influence where the graph starts or the length of each cycle.
To find the amplitude of a sine function in the form of \( a \sin(bx - c) + d \), simply take the absolute value of \( a \).
- If \( a = 2 \), the amplitude is \( 2 \).
- If \( a = -3 \), the amplitude is \( 3 \).
Always remember, the amplitude affects the "height" of the graph, but it does not influence where the graph starts or the length of each cycle.
Period
The period of a sine function determines how long it takes for the function to complete one full cycle. It tells you how "wide" each wave is set on the x-axis.
To find the period, you use the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient that sits before \( x \) inside the trigonometric function.
For example:
This indicates that the wave pattern will repeat itself every two units along the x-axis.
To find the period, you use the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient that sits before \( x \) inside the trigonometric function.
For example:
- If \( b = \pi \), then the period is \( \frac{2\pi}{\pi} = 2 \).
- If \( b = 2 \), the period would be \( \frac{2\pi}{2} = \pi \).
This indicates that the wave pattern will repeat itself every two units along the x-axis.
Phase Shift
The phase shift is a horizontal movement of the graph along the x-axis, making the wave start at a different point. You can think of it as shifting where the wave begins its pattern.
To determine the phase shift, solve for \( x \) in the equation inside the sine function, \( bx - c = 0 \).
For example:
To determine the phase shift, solve for \( x \) in the equation inside the sine function, \( bx - c = 0 \).
For example:
- In \( \pi x - \frac{\pi}{2} = 0 \), rearrange to find \( x = \frac{1}{2} \).
- This step reveals a shift to the right by \( \frac{1}{2} \) units.
Sine Function
The sine function, one of the basic trigonometric functions, models wave-like patterns which are very useful in depicting cyclic behaviors like sound waves or tides.
In a standard sine equation \( y = \sin(x) \), this function describes a wave that starts at zero, reaches a peak of \( 1 \), falls to zero, drops to a minimum of \(-1\), and returns to zero in one complete cycle.
For our equation, \( y = \sin(\pi x - \frac{\pi}{2}) \), the basic form has been adjusted by:
In a standard sine equation \( y = \sin(x) \), this function describes a wave that starts at zero, reaches a peak of \( 1 \), falls to zero, drops to a minimum of \(-1\), and returns to zero in one complete cycle.
For our equation, \( y = \sin(\pi x - \frac{\pi}{2}) \), the basic form has been adjusted by:
- An amplitude of 1, meaning the peaks and troughs reach \( 1 \) and \(-1\).
- A period of 2, indicating each cycle of the wave takes 2 units.
- A phase shift of \( \frac{1}{2} \) units to the right, changing where the wave begins its cycle.
