Graph each function over a one-period interval. See Examples 1–3. y = tan 4x
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Step 1: Identify the basic properties of the tangent function. The tangent function, , has a period of and vertical asymptotes where the function is undefined.
Step 2: Determine the period of . The period of is given by . For , the period is .
Step 3: Identify the vertical asymptotes. For , the vertical asymptotes occur at , where is an integer.
Step 4: Determine the x-intercepts. The x-intercepts of occur at , where is an integer.
Step 5: Sketch the graph over one period. Plot the x-intercepts and vertical asymptotes, and draw the curve of the tangent function, which increases from negative to positive infinity between each pair of asymptotes.
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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.