Textbook Question
Graph y = 1/2 sin x + 2cos x, 0 ≤ x ≤ 2π.
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4. Graphing Trigonometric Functions
Graphs of the Sine and Cosine Functions
7:23 minutesProblem 15
Textbook Question
In Exercises 7–16, determine the amplitude and period of each function. Then graph one period of the function. y = -sin 2/3 x
Verified step by step guidance1
Identify the general form of the sine function: \(y = A \sin(Bx)\), where \(A\) is the amplitude and \(B\) affects the period.
From the given function \(y = -\sin \frac{2}{3} x\), recognize that the amplitude \(A\) is the absolute value of the coefficient in front of the sine, which is \(|-1| = 1\).
Determine the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the sine function. Here, \(B = \frac{2}{3}\).
Calculate the period as \(\frac{2\pi}{\frac{2}{3}} = 2\pi \times \frac{3}{2} = 3\pi\) (do not compute the numerical value, just set up the expression).
To graph one period, plot the function from \(x = 0\) to \(x = 3\pi\), noting that the amplitude is 1 and the sine wave is reflected over the x-axis due to the negative sign.
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Video duration:
7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Sine Function
The amplitude of a sine function is the absolute value of the coefficient in front of the sine term. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak or trough of the wave. For y = -sin(2/3 x), the amplitude is |-1| = 1.
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Period of a Sine Function
The period of a sine function is the length of one complete cycle along the x-axis. It is calculated by dividing 2π by the absolute value of the coefficient of x inside the sine function. For y = -sin(2/3 x), the period is 2π ÷ (2/3) = 3π.
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Graphing One Period of a Sine Function
Graphing one period involves plotting the sine curve from the start to the end of one full cycle, based on the period. Key points include the midline, maximum, minimum, and zeros, spaced evenly according to the period. The negative sign reflects the graph across the x-axis.
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