Graph each function over a one-period interval. See Examples 1–3. y = tan 4x
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Period of Trigonometric Functions
The period of a trigonometric function is the length of one complete cycle of the function. For the tangent function, the standard period is π. However, when the function is modified, such as in y = tan(4x), the period is adjusted by the coefficient of x. In this case, the period becomes π/4, meaning the function will complete one full cycle over this interval.
The tangent function, y = tan(x), has specific characteristics, including vertical asymptotes and points of intersection with the x-axis. The graph of y = tan(4x) will have vertical asymptotes where the function is undefined, specifically at x = (π/8) + (nπ/4) for integers n. Understanding these features is crucial for accurately sketching the graph.
Transformations involve changes to the basic form of a function, affecting its position, shape, or size. In the case of y = tan(4x), the '4' indicates a horizontal compression, which alters the spacing of the function's features, such as its period and asymptotes. Recognizing how transformations impact the graph is essential for accurate representation.