Domanda del libro di testo
Graph each function over a two-period interval.
y = cos (x - π/2 )
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 40
Graph each function over a one-period interval.
y = -½ cos (πx - π)
Guida verificata passo dopo passo1
Identify the given function: \(y = -\frac{1}{2} \cos(\pi x - \pi)\).
Recall that the general form of a cosine function is \(y = A \cos(Bx - C)\), where the period is given by \(\frac{2\pi}{|B|}\).
Calculate the period of the function: since \(B = \pi\), the period is \(\frac{2\pi}{\pi} = 2\).
Determine the one-period interval for \(x\). Since the period is 2, a natural choice is any interval of length 2, for example, \([0, 2]\) or \([-1, 1]\).
Analyze the transformations: the amplitude is \(\frac{1}{2}\) (due to the coefficient \(-\frac{1}{2}\)), the negative sign reflects the graph about the x-axis, and the phase shift is found by solving \(\pi x - \pi = 0\) which gives \(x = 1\). This means the graph is shifted right by 1 unit.
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Period of a Trigonometric Function
The period is the length of one complete cycle of a trigonometric function. For cosine functions of the form y = cos(bx), the period is calculated as 2π divided by the absolute value of b. Understanding the period helps determine the interval over which to graph the function.
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Percorso guidato
5:33
Period of Sine and Cosine Functions
Phase Shift in Trigonometric Functions
Phase shift refers to the horizontal translation of the graph caused by adding or subtracting a constant inside the function's argument. For y = cos(bx - c), the phase shift is c/b. It affects where the graph starts within the period and is essential for accurate plotting.
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Percorso guidato
6:31Phase Shifts
Amplitude and Reflection
Amplitude is the absolute value of the coefficient multiplying the cosine function, indicating the maximum displacement from the midline. A negative coefficient reflects the graph across the x-axis. In y = -½ cos(πx - π), the amplitude is ½, and the negative sign flips the graph vertically.
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Percorso guidato
5:05Amplitude and Reflection of Sine and Cosine
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