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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?
2
2
2
2
) sin
) sin
) (sin )
) sin( )
a y x x
b y x x
c y x
d y x
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
) 3sin 2cos ) 2sin 3cos
) 2sin 3 4cos 2 ) 2sin(5 1) 5cos5
7 2 7) cos sin ) 3cos 2 2sin 7
5 5
a y x x b y x x
c x x d y x x
x xe y f y x x
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.) ( ) 5cos 3sin
) ( ) cos5 sin 3
) ( ) 2cos3 3cos 2
3 3 3) ( ) cos cos sin
7 7 7
a f x x x
b f x x x
c f x x x
x x xd f x a b c
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