key term - Period

Definition

The period of a function is the distance or interval along the independent variable axis over which the function's shape or pattern repeats itself. It is a fundamental concept in the study of periodic functions, such as trigonometric functions, and is essential for understanding their properties and graphs.

5 Must Know Facts For Your Next Test

1. The period of a function determines the interval over which the function's values repeat, and it is often denoted by the symbol $T$.
2. For trigonometric functions, the period is directly related to the frequency, with the period being the reciprocal of the frequency.
3. Periodic functions, such as the sine and cosine functions, are essential in modeling and analyzing various phenomena in science, engineering, and physics.
4. The period of a function is crucial in understanding the behavior and characteristics of its graph, including the shape, amplitude, and phase shift.
5. Identifying the period of a function is important when solving trigonometric equations and analyzing the properties of periodic functions.

Review Questions

• Explain how the period of a function is related to the frequency of the function.
  • The period of a function is inversely related to its frequency. The period, denoted as $T$, represents the distance or interval along the independent variable axis over which the function's shape or pattern repeats itself. The frequency, denoted as $f$, represents the number of complete cycles that occur within a given interval of the independent variable. The relationship between period and frequency is given by the formula $T = \frac{1}{f}$. This means that as the frequency of a function increases, the period decreases, and vice versa. Understanding this relationship is crucial in the study of periodic functions, such as trigonometric functions.
• Describe how the period of a function affects the graph of the function.
  • The period of a function has a significant impact on the appearance and characteristics of its graph. A smaller period results in a more compressed or 'tighter' graph, with the function's shape repeating more frequently along the independent variable axis. Conversely, a larger period leads to a more stretched-out or 'wider' graph, with the function's shape repeating less frequently. The period also affects the amplitude, or the maximum and minimum values, of the function's graph. Changing the period while keeping the amplitude constant can alter the overall shape and appearance of the graph, which is crucial in understanding the properties of periodic functions, such as sine and cosine.
• Explain the importance of the period in solving trigonometric equations.
  • The period of a trigonometric function, such as sine or cosine, is essential in solving trigonometric equations. When solving these equations, the period helps determine the possible solutions and the intervals over which the solutions repeat. Knowing the period allows you to identify the critical points, or the points where the function's value is equal to a given value, and then use this information to find the solutions to the equation. Additionally, the period is necessary for understanding the behavior of the function and its graph, which is crucial in applying trigonometric concepts to real-world problems in fields like physics, engineering, and mathematics.

Related terms

Amplitude:

The amplitude of a periodic function is the distance from the midline to the maximum or minimum value of the function.

Frequency:

The frequency of a periodic function is the number of complete cycles that occur within a given interval of the independent variable.

Phase Shift:

The phase shift of a periodic function is the horizontal displacement of the function's graph relative to a reference position.