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