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?

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 = (sin 3 x)(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 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) = a 1 sin(bx+c 1 ) and g(x) = a 2 cos(bx+c 2 ) both with the same b value then the sum (f+g)(x) = a 1 sin(bx+c 1 ) + a 2 cos(bx+c 2 ) is a sinusoid with period 2π/b

Examples: Determine whether each of the following functions is or is not a sinusoid.

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