Algebra III
Lesson 43
Periodic Functions – Graphs of Sin θ and
Cos θ
Periodic Functions
One way to think of them is any function whose graph does the
same thing over and over again without end.
Not periodic
Periodic
1
270°
0°
90° 180°
-1
360°
The fist periodic function we’ll look at is:
y = sin θ
1
270°
0°
90° 180°
360°
-1
This graph is described as sinusoidal.
Any graph that resembles this is called a sinusoid.
To find the value of a function, pick and angle
and read the y axis.
f(θ) = sin θ
f(90°) = 1
Next, let’s look at cos θ.
Get some points from the unit circle.
1
270°
0°
90° 180°
360°
-1
Notice, this looks very similar to sin θ.
1
270°
0°
90° 180°
-1
360°
They should look alike.
Recall the unit circle.
What is the max value for sin? Cos?
1
1
What is the min value for sin? Cos?
-1
-1
These two functions cover the same range of numbers
in the same pattern just offset.
What would happen to the graph of f(θ) = sin θ in this
situation, g(θ) = 3 sin θ?
It would be stretched vertically. Peaking at 3 and -3.
3
270°
0°
90° 180°
360°
-3
Stretched and flipped over.
How about h(θ) = -5 cos θ?
5
270°
0°
90° 180°
-5
360°
Example 43.1
Write the equation for this sinusoid.
6
270°
0°
90° 180°
360°
θ
-6
The graph starts at 0 making it a sin graph.
Goes to 6 and -6.
f(θ) = 6 sin θ
f(θ) = sin θ
Example 43.2
Write the equation for this sinusoid.
3
3π/2
0
π/2 π
2π
x
-3
The graph doesn’t start at 0, that makes it a cos graph.
Peaks at 3 and -3.
f(x) = 3 cos θ
Starts low instead of high.
f(x) = -3 cos θ
f(x) = cos θ
Practice
a) Write the equation of the sinusoid.
4
270°
0°
90° 180°
360°
-4
Doesn’t start at 0.
f(θ) = cos θ
Goes to 4 and -4.
f(θ) = 4 cos θ
Starts low, not high.
f(θ) = -4 cos θ
θ
b) Solve for x: log3 6 + log3 3 = log3 (4x + 2)
Multiplication rule.
logb MN = logb M + logb N
log3 6 + log3 3 = log3 6(3)
log3 18 = log3 (4x + 2)
18 = 4x + 2
4x = 16
x=4
c) Simplify: log3 32 + log5 53 + log2 24
Power rule.
log b (M ) = N log b M
N
= 2log3 3 + 3log5 5 + 4log2 2
= 2(1) + 3(1) + 4(1)
=9
d) Find the domain and range of each function whose graph is shown:
domain Æ x
range Æ y
a) domain is {-6 ≤ x ≤ 2}
range is {-2 ≤ y ≤ 6}
b) domain is {-5 ≤ x ≤ 5}
range is {-4 ≤ y ≤ 4}