In Exercises 45–52, graph two periods of each function. y = sec(2x + π/2) − 1

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Identify the given function: \(y = \sec(2x + \frac{\pi}{2}) - 1\). This is a secant function with a horizontal transformation inside the argument and a vertical shift downward by 1.

Determine the period of the function. The general period of \(\sec(bx)\) is \(\frac{2\pi}{|b|}\). Here, \(b = 2\), so the period is \(\frac{2\pi}{2} = \pi\).

Since the problem asks for two periods, calculate the interval length to graph: \(2 \times \pi = 2\pi\). So, you will graph the function over an interval of length \(2\pi\).

Find the phase shift caused by the \(+ \frac{\pi}{2}\) inside the argument. Solve \(2x + \frac{\pi}{2} = 0\) for \(x\) to find the horizontal shift: \(x = -\frac{\pi}{4}\). This means the graph is shifted left by \(\frac{\pi}{4}\).

Identify key points and vertical asymptotes of the secant function by setting the inside of the secant equal to values where cosine is zero (since \(\sec \theta = \frac{1}{\cos \theta}\)). Solve \(2x + \frac{\pi}{2} = \frac{\pi}{2} + k\pi\) for \(x\) to find vertical asymptotes, then plot the graph accordingly over two periods, applying the vertical shift of \(-1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Period of a Trigonometric Function

The period of a trigonometric function is the length of the interval over which the function completes one full cycle. For secant functions, the period is derived from the coefficient of x inside the function. Specifically, the period of y = sec(bx) is 2π/|b|, which determines how the graph repeats.

Phase Shift

Phase shift refers to the horizontal translation of the graph caused by adding or subtracting a constant inside the function's argument. For y = sec(2x + π/2), the phase shift is found by solving 2x + π/2 = 0, which shifts the graph left or right, affecting where key features like asymptotes and peaks occur.

Graphing the Secant Function

The secant function is the reciprocal of cosine and has vertical asymptotes where cosine equals zero. When graphing y = sec(2x + π/2) − 1, identify asymptotes from the cosine zeros, plot key points from the cosine graph, and apply vertical shifts. Understanding these features helps accurately sketch two periods of the function.