Graph each function defined as follows over a two-period interval. y=-3 tanx

Name: Problem 73, Chapter 13: Calculus with Applications
Uploaded: 2020-08-10T22:04:21-08:00
Duration: 29 s
Channel: Carson Merrill
Description: Graph each function defined as follows over a two-period interval. y=-3 tanx

step 1 So here's the graph of our function. And since it's just being multiplied, the tangent x is bein

Graph each function defined as follows over a two-period...

Key Concepts

Transformations of Trigonometric Functions

Transforming a trigonometric function involves altering its graph through operations such as scaling, reflecting, or translating. In the function y = -3 tan(x), the factor of -3 results in a vertical stretch by a factor of 3 and a reflection over the x-axis. These modifications change the steepness and orientation of the original tangent graph while preserving its periodicity.

Periodicity and Vertical Asymptotes

Periodicity refers to the repeating nature of a function's graph over a fixed interval, with the basic tangent function repeating every ? units. Vertical asymptotes occur at values of x where the function is undefined, typically where the denominator in its sine-over-cosine representation is zero. When transformations are applied, these features are shifted or scaled accordingly, guiding the overall graph structure.

Tangent Function

The tangent function, often defined as sin(x)/cos(x), is a basic trigonometric function characterized by its periodic behavior and vertical asymptotes. Its fundamental period for the basic function is ?, and it does not have a maximum or minimum value because it extends to infinity in both directions near its asymptotes.

Graphing Techniques for Trigonometric Functions

Graphing a transformed trigonometric function involves identifying its period, calculating the new positions of key features such as asymptotes and intercepts, and plotting sufficient points within one period to capture the behavior of the curve. This systematic approach helps in accurately sketching the graph over any specified interval, such as a multiple period interval in this case.

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