Graphs of Composite Trig
Functions
Objective:
Be able to combine trigonometric and algebraic functions together.
TS: Demonstrating understanding of concepts.
Warm-Up: Graph each of the below on your calculator. Which
seem to be periodic?
a ) y  sin x  x 2
b) y  x 2 sin x
c) y  (sin x) 2
d ) y  sin( x 2 )
How can we verify if something is
periodic?
If we believe some function f(x) has a
period of a, then to verify we need to show
f(x+ a) = f(x).
Example: Verify y=(sin x)2 is periodic.
You Try:
Is y = (sin3x)(cosx) periodic? Use your
calculator to figure out what the period is.
Graph the following functions one at a time
in the window -2π ≤ x ≤ 2π and -6 ≤ y ≤ 6
a) y  3sin x  2 cos x
b) y  2sin x  3cos x
c) 2sin 3 x  4 cos 2 x
d ) y  2sin(5 x  1)  5cos 5 x
 7x  2 
 7x 
e) y  cos 

sin

  f ) y  3cos 2 x  2sin 7 x
 5 
 5 
Which appear to be sinusoids?
What relationship between the sine and cosine functions
ensures their sum or difference is a sinusoid?
Sums that are Sinusoid Functions
Given the two functions f(x) = a1sin(bx+c1) and
g(x) = a2cos(bx+c2) both with the same b value
then the sum (f+g)(x) = a1sin(bx+c1) + a2cos(bx+c2)
is a sinusoid with period 2π/b
Examples:
Determine whether each of the following
functions is or is not a sinusoid.
a ) f ( x)  5cos x  3sin x
b) f ( x)  cos 5 x  sin 3 x
c) f ( x)  2 cos 3 x  3cos 2 x
 3x 
 3x 
 3x 
d ) f ( x)  a cos    b cos    c sin  
 7 
 7 
 7 
Putting the two together:
Show that g(x) = sin(2x) + cos(3x) is
periodic but not a sinusoid.
What if I just want to graph some crazy trig
functions? (don’t roll your eyes, you know
you want to graph crazy trig functions)
Functions involving the absolute values of Trig
functions:
The key is to remember absolute values
create all positive values.
Examples:
a) f(x) = |tanx|
b) g(x) = |sinx|
Functions involving the absolute values
of Trig functions:
Examples:
b) g(x) = |sinx|
Functions involving a sinusoid and a linear
function
The key is to remember sine and cosine can
be at most 1 and at least -1.
Examples:
a) f(x) = 3x + cosx
b) g(x) = ½x +cosx
Functions involving a sinusoid and a
linear function
Examples:
b) g(x) = ½x +cosx
Dampened Trig Functions (Trig functions
muliplied by a algebraic function)
The key is to remember sine and cosine can
be at most 1 and at least -1.
Example:
f(x) = (2x)cosx