☰ Contents · Mathematics

Periodic processes, sine and cosine

Lessons 28–29 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
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Periodic processes and observing them

Textbook, Part 2: pp. 23–25
GoalUnderstand periodic processes and periodic functions; find the period, the midline and the amplitude from a graph or a table.
New words
period · davrperiodic function · davriy funksiyamidline · bosh o‘qamplitude · amplituda
Explanation

Many processes repeat after a fixed time: seasons, average monthly temperature, rising and falling water levels, oscillations. If for a function f defined on X there is a number T > 0 with f(x + T) = f(x) for every x in X, f is called periodic; the least such T is its period. If T is a period then so are 2T, 3T, …, which is why we take the least one. If the greatest value is max and the least is min, the midline is y = (max + min)/2 and the amplitude is (max − min)/2. The graph is made by repeating one period-length piece, so knowing one period lets us draw the whole graph. Many processes are not periodic: for instance, a quickly growing population does not repeat.

Worked examples
A city’s monthly mean temperature has a maximum of 30 °C and a minimum of 6 °C and repeats every 12 months. The period is 12 months; the midline is y = (30 + 6)/2 = 18; the amplitude is (30 − 6)/2 = 12.
A table: for x = 0, 1, …, 8, y = 1, 3, 1, −1, 1, 3, 1, −1, 1. The values repeat every 4 steps: f(x + 4) = f(x), so the period is 4. max = 3, min = −1, midline y = 1, amplitude 2.
Class activity

Observe the clock hand or a small hanging pendulum (observe only, do not pull the string) and time one full swing with a stopwatch 5 times; take the average as the period.

Practice
1
What is the period of a periodic function?
2
A function has greatest value 15 and least −5. The midline is y = ?
3
If f(x + 6) = f(x) and f(2) = 3, find f(20).
4
Why do we take the least T as the period?