Textbook Question
In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x
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4. Graphing Trigonometric Functions
Graphs of the Sine and Cosine Functions
5:50 minutesProblem 43
Textbook Question
In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = cos(x − π/2)
Verified step by step guidance1
Identify the general form of the cosine function: \(y = A \cos(B(x - C))\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) is the phase shift.
Compare the given function \(y = \cos(x - \frac{\pi}{2})\) to the general form. Here, \(A = 1\) (since there is no coefficient in front of cosine), \(B = 1\) (coefficient of \(x\)), and \(C = \frac{\pi}{2}\).
Calculate the amplitude, which is the absolute value of \(A\): \(\text{Amplitude} = |A| = 1\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Since \(B = 1\), the period is \(2\pi\).
Determine the phase shift, which is \(C\). Since the function is \(\cos(x - \frac{\pi}{2})\), the phase shift is \(\frac{\pi}{2}\) units to the right.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Trigonometric Function
Amplitude is the maximum absolute value of a trigonometric function from its midline. For functions like y = cos(x), the amplitude is the coefficient before the cosine term, indicating the height of peaks and depth of troughs. In y = cos(x − π/2), the amplitude is 1, as there is no coefficient other than 1.
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Period of a Trigonometric Function
The period is the length of one complete cycle of the function, typically 2π for sine and cosine functions. It can be adjusted by a coefficient inside the function's argument. For y = cos(x − π/2), since the coefficient of x is 1, the period remains 2π.
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Phase Shift of a Trigonometric Function
Phase shift refers to the horizontal translation of the graph, determined by the value added or subtracted inside the function's argument. For y = cos(x − π/2), the graph shifts π/2 units to the right, meaning the entire cosine curve moves rightward by π/2 along the x-axis.
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