MCR3U1
U5L1
PERIODIC FUNCTIONS
(and their PROPERTIES)
PART A ~ INTRODUCTION
A functions that produces a graph that has a regular repeating
pattern over a constant interval is called a periodic function.
It describes something that happens in a cycle, repeating in the
same way over and over.
PART B ~ DEFINITIONS
periodic function:
a function whose graph repeats at regular intervals;
the y-values show a repetitive pattern when the x-values
change by the same increment
cycle:
one complete pattern that may begin at any point on the graph
period:
the change in the independent variable (typically x)
corresponding to one cycle (the “horizontal length” of a cycle)
equation of axis:
the equation of the horizontal line halfway between the
maximum and the minimum values
y=
amplitude:
max 𝑣𝑎𝑙𝑢𝑒 + min 𝑣𝑎𝑙𝑢𝑒
2
half the difference between the maximum and minimum values;
it is also the vertical distance from the function’s axis to the
maximum or minimum value (it is always positive)
amplitude =
max 𝑣𝑎𝑙𝑢𝑒 − min 𝑣𝑎𝑙𝑢𝑒
2
trough:
the minimum point on a graph
peak:
the maximum point on a graph
MCR3U1
U5L1
PART C ~ IDENTIFYING A PERIODIC FUNCTION FROM ITS GRAPH
Determine whether the term periodic can be used to describe the graph for each
example below. If so, state the period, the e
y
a)
Periodic?
x
period:
equation of axis:
amplitude:
y
b)
Periodic?
x
period:
c)
equation of axis:
amplitude:
y
Periodic?
x
MCR3U1
U5L1
PART D ~ PERIODIC GRAPHS IN REAL–LIFE SITUATIONS
The given graph shows Max’s height, in metres, above the ground in terms of time,
in seconds, while riding a ferris wheel at the Mathville Amusement Park.
Height on a Ferris Wheel
h
20
Height
(m)
16
12
8
4
0
4
8
12
16
20
24
28
32
36
40
44
t
Time (s)
a)
Explain why riding a ferris wheel models a periodic function.
b)
State the period of this function. What does this represent?
c)
State the amplitude of this function. What does this represent?
d)
State the equation of the axis. What does this represent?
e)
State at what times Max is at the top of the wheel.
f)
State at what height Max boards the ferris wheel.
HOMEWORK: p.352–355 #1–5, 8