Recognizing Trigonometric Graphs

Given the graph of a trigonometric function, how do we find the formula for the corresponding trigonometric function? In this summary, we explore properties of trigonometric graphs that will help us in recognizing and finding these corresponding formulas.

As we have seen, trigonometric functions satisfy the important property that they are periodic. It turns out that all periodic behavior can be characterized by combinations of trigonometric functions. This makes trigonometric functions very useful in describing many natural phenomena, such as the motion of waves in the ocean, the behavior of a spring in motion, and sound waves in the air. In order to characterize trigonometric graphs, it is useful to begin by asking the following questions:

• Is the graph periodic? If so, what is the period?

• Does the graph have a maximum and a minimum? If so, what is the amplitude?

• Does the graph have any vertical asymptotes?

• If the graph is unbounded, does the range of the graph include all real numbers? For example, the tangent and cotangent graphs have range all real numbers, while the secant and cosecant graphs have range \( (-\infty, -1] \cup [1, \infty) \).

If the graph is bounded, then it may be a transformation of the \(\sin(x)\) or \(\cos(x)\) functions. By finding the amplitude, period, horizontal shift, and vertical shift of the graph from \(\sin(x)\) or \(\cos(x)\), we can find the formulas using the methods from graphical transformations of trigonometric graphs.

If the graph is not bounded and has vertical asymptotes, and the range of the graph includes all real numbers, then it may be a transformation of the \(\tan(x)\) or \(\cot(x)\) functions. By finding the period, horizontal, and vertical shift of the graph, we can characterize the transformations necessary from the \(\tan(x)\) or \(\cot(x)\) functions to achieve the graph.

If the graph is not bounded and has vertical asymptotes, and the range of the graph does not include all real numbers, then it may be a transformation of the \(\sec(x)\) or \(\csc(x)\) functions. By finding the period, horizontal, and vertical shift of the graph, we can characterize the transformations necessary from the \(\sec(x)\) or \(\csc(x)\) functions to achieve the graph.