Periodic and Symmetric Functions

The unit circle has a circumference of C = 2π r = 2π(1) = 2π. Therefore, if a point P travels around the unit circle for a distance of 2π, it ends up where it started. In other words, for any given value q, if 2π is added or subtracted, the coordinates of point P remain unchanged (Figure 1 ).

Figure 1 Periodic coterminal angles.

It follows that

If k is an integer,

Functions that have this property are called periodic functions. A function f is periodic if there is a positive real number q such that f( x + q) = f( x) for all x in the domain of f. The smallest possible value for q for which this is true is called the period of f.

Example 1: If sin y = y = (3/5)/10, then what is the value of each of the following: sin( y + 8π), sin( y + 6π), ( y + 210π)?

All three have the same value of because the sine function is periodic and has a period of 2π.

The study of the periodic properties of circular functions leads to solutions of many real-world problems. These problems include planetary motion, sound waves, electric current generation, earthquake waves, and tide movements.

Example 2: The graph in Figure 2 represents a function f that has a period of 4. What would the graph look like for the interval −10 ⩽ x ⩽ 10?

Figure 2 Drawing for Example 2.

This graph covers an interval of 4 units. Because the period is given as 4, this graph represents one complete cycle of the function. Therefore, simply replicate the graph segment to the left and to the right (Figure 3 ).

Figure 3 Drawing for Example 2.

The appearance of the graph of a function and the properties of that function are very closely related. It can be seen from Figure 4 that

Figure 4 Even and odd trig functions.

The cosine is known as an even function, and the sine is known as an odd function. Generally speaking,

for every value of x in the domain of g. Some functions are odd, some are even, and some are neither odd nor even.

If a function is even, then the graph of the function will be symmetric with the y-axis. Alternatively, for every point on the graph, the point (− x, − y) will also be on the graph.

If a function is odd, then the graph of the function will be symmetric with the origin. Alternatively, for every point ( x, y) on the graph, the point (− x, − y) will also be on the graph.

Example 3: Graph several functions and give their periods (Figure 5 ).

Figure 5 Drawings for Example 3.

Example 4: Graph several odd functions and give their periods (Figure 6 ).

Figure 6 Drawings for Example 4.

Example 5: Is the function f(x) = 2 x³ + x even, odd, or neither?

Because f(−x) = − f(x), the function is odd.

Example 6: Is the function f(x) = sin x – cos x even, odd, or neither?

the function is neither even nor odd. Note: The sum of an odd function and an even function is neither even nor odd.

Example 7: Is the function f( x) = x sin x cos x even, odd, or neither?

Because f(− x) = f( x), the function is even.

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