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Graphing Tangent and Cotangent Functions

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1. Finding the asymptotes Recall that asymptotes occur when the function is undefined. Therefore, we need to look at the denominator of the function. Rewrite the function exactly, except replace the tan or cot with cos or sin, respectively and sketch that sinusoid. For example, to graph y = 2 + 3tan(2(x - π)), we would graph: y = 2 + 3 cos (2(x - π)) because cos is in the denominator of tan. Now, Graph the Cos graph on your graph paper.

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Here is a sketch of the sinusoid. Keep in mind that you should always label the “special points” on the x-axis (usually in terms of π).

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*** The asymptotes of the tan or cot graph will occur where the sinusoid crosses the MIDDLE (which you find the D-value). Here they are in our example.

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2. Finding one point per cycle Once the asymptotes are sketched in, you can easily find one point per cycle because the tan or cot graph will always cross the middle line of the sinusoid exactly half way between the asymptotes.

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Here’s our graph with the points filled in.

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3. Sketching the general shape. Now you can sketch in the general shape. There will be one cycle between each pair of asymptotes. +tan and -cot are increasing. -tan and +cot are decreasing.

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Our example was a +tan, so it is increasing and is shown below.

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Notes: You should be able to sketch these graphs without your calculator The period of tan & cot graphs is always half the period of its corresponding sinusoid. Therefore for tan and cot, period = π/B

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