13-1 Exploring Periodic Data Hubarth Algebra II

A periodic function repeats a pattern of y-values (outputs) at regular intervals. One complete pattern is a cycle. A cycle may begin at any point on the graph of the function. The period of a function is the horizontal length of one cycle. Ex. 1 Identifying Cycles and Periods Analyze this periodic function. Identify one cycle in two different ways. Then determine the period of the function. Begin at any point on the graph. Trace one complete pattern. The beginning and ending x-values of each cycle determine the period of the function. Each cycle is 7 units long. The period of the function is 7.

Ex. 2 Identifying Periodic Functions Determine whether each function is or is not periodic. If it is, find the period. a. b. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. The pattern of y-values in one section repeats exactly in other sections. The function is periodic. Find points at the beginning and end of one cycle. Find points at the beginning and end of one cycle. Subtract the x-values of the points: 2 – 0 = 2. Subtract the x-values of the points: 3 – 0 = 3. The pattern of the graph repeats every 2 units, so the period is 2. The pattern of the graph repeats every 3 units, so the period is 3.

The amplitude of a periodic function measures the amount of variation in the function values Maximum Minimum

Ex 3. Finding Amplitude of a periodic Function Find the amplitudes of the two functions in example 2. a. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [2 – (–2)] Substitute. 1 2 = (4) = 2 Subtract within parentheses and simplify. 1 2 The amplitude of the function is 2. b. amplitude = (maximum value – minimum value)   Use definition of amplitude. 1 2 = [6 – 0] Substitute. 1 2 = (6) = 3 Subtract within parentheses and simplify. 1 2 The amplitude of the function is 3.

Ex. 4 Real-World Connection The oscilloscope screen below shows the graph of the alternating current electricity s, in volts, supplied to homes in the United States. Find the period and amplitude. 1 unit on the t-axis = s 1 360 1 60 One cycle of the electric current occurs from 0 s to s. The maximum value of the function is 120, and the minimum is –120. period = – 0   Use the definitions. = Simplify. 1 60 amplitude = [120 – (–120)] = (240) = 120 1 2 The period of the electric current is s. 1 60 The amplitude is 120 volts.