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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 39
Capitolo 2, Problema 39

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = −1/2 sec πx

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Identify the given function: \(y = -\frac{1}{2} \sec(\pi x)\). This is a secant function with amplitude scaling and reflection.
Recall that the secant function is the reciprocal of the cosine function, so \(\sec(\theta) = \frac{1}{\cos(\theta)}\). The graph of \(y = \sec(\theta)\) has vertical asymptotes where \(\cos(\theta) = 0\).
Determine the period of the function. The standard period of \(\sec(x)\) is \(2\pi\). For \(\sec(bx)\), the period is \(\frac{2\pi}{b}\). Here, \(b = \pi\), so the period is \(\frac{2\pi}{\pi} = 2\).
Since the problem asks for two periods, the interval to graph is from \(x = 0\) to \(x = 4\) (or any interval of length 4). Identify vertical asymptotes by solving \(\cos(\pi x) = 0\), which occurs at \(\pi x = \frac{\pi}{2} + k\pi\), so \(x = \frac{1}{2} + k\) for integers \(k\).
Plot the key points of \(y = -\frac{1}{2} \sec(\pi x)\) by first plotting \(y = \cos(\pi x)\), then taking the reciprocal to get \(\sec(\pi x)\), applying the vertical stretch by \(\frac{1}{2}\) and reflection (negative sign). Mark vertical asymptotes at \(x = \frac{1}{2} + k\) and sketch the graph between these asymptotes over two periods.

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Secant Function and Its Properties

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It has vertical asymptotes where cos(x) = 0, and its graph consists of branches that extend to infinity near these asymptotes. Understanding its periodicity and behavior is essential for graphing.
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Graphs of Secant and Cosecant Functions

Amplitude and Vertical Stretch/Compression

The coefficient in front of the secant function, here −1/2, affects the vertical stretch and reflection of the graph. A factor of 1/2 compresses the graph vertically, while the negative sign reflects it across the x-axis. This changes the height and orientation of the secant branches.
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Stretches and Shrinks of Functions

Period of the Secant Function with Horizontal Scaling

The period of sec(x) is 2π, but when the function is sec(πx), the period changes to 2 because the input is scaled by π. This horizontal scaling compresses or stretches the graph along the x-axis, affecting where the asymptotes and key points occur. Knowing how to find the period is crucial for graphing two full cycles.
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Graphs of Secant and Cosecant Functions
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