Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -¼ cos (½ x + π/2)
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4. Graphing Trigonometric Functions
Graphs of the Sine and Cosine Functions
Problem 4.37
Textbook Question
Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = ½ cos π x
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Verified step by step guidance1
Identify the standard form of the cosine function: \( y = a \cos(bx + c) + d \). In this case, \( a = \frac{1}{2} \), \( b = \frac{\pi}{2} \), \( c = 0 \), and \( d = 0 \).
Determine the amplitude of the function, which is the absolute value of \( a \). Here, the amplitude is \( |\frac{1}{2}| = \frac{1}{2} \).
Calculate the period of the function using the formula \( \text{Period} = \frac{2\pi}{|b|} \). Substitute \( b = \frac{\pi}{2} \) to find the period.
Graph the function over a two-period interval. Start by plotting key points for one period, such as the maximum, minimum, and intercepts, then repeat the pattern for the second period.
Label the graph with the amplitude and period, ensuring the x-axis covers two full periods and the y-axis reflects the amplitude.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Period of a Trigonometric Function
The period of a trigonometric function is the length of one complete cycle of the wave. For cosine functions, the standard period is 2π. However, when the function is modified, such as by a coefficient in front of x, the period can change. Specifically, for the function y = ½ cos(π/2 x), the period can be calculated using the formula 2π divided by the coefficient of x, which in this case is π/2, resulting in a period of 4.
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Period of Sine and Cosine Functions
Amplitude of a Trigonometric Function
The amplitude of a trigonometric function refers to the maximum distance from the midline of the graph to its peak or trough. It is determined by the coefficient in front of the cosine function. In the function y = ½ cos(π/2 x), the amplitude is ½, indicating that the graph will oscillate between ½ and -½, which defines the vertical stretch of the wave.
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Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of the function over a specified interval. For the function y = ½ cos(π/2 x), one would typically plot points for x values within a two-period interval, which is from 0 to 8. The graph will exhibit a wave-like pattern, reflecting the calculated period and amplitude, allowing for visual interpretation of the function's behavior.
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