Graphing Trig Functions

Sine Graph of y = sin(x) Starts at the origin. 1 -4π -3π -2π -π π 2π π 2π 3π 4π Starts at the origin. -1

Cosine Graph of y = cos(x) Starts at the max. 1 -4π -3π -2π -π π 2π 3π π 2π 3π 4π -1

The period of y = sin(x) is 2π! The period of a function is the length of the piece of the graph that repeats itself. π 2π 3π 4π 5π 6π The period of y = sin(x) is 2π!

Find the period of each graph: π 2π 3π 4π 5π 6π π 2π 3π 4π 5π 6π π 2π 3π 4π 5π 6π

Amplitude Amplitude is the distance from the midpoint to the highest and lowest point of the function. Always measure the amplitude from “sea level.” Sea level changes as the center of the graph moves up and down. 1 -4π -3π -2π -π π 2π 3π 4π -1

Find the amplitude of each graph: 1 -1 π 2π 3π -2 5 -3 π 2π 3π π 2π 3π -16

Sine vs. Cosine Graphs Both graphs have a period of 2π. 1 -4π -3π -2π -π π 2π 3π 4π -1 y = sin(x) starts at 0. Both graphs have a period of 2π. Both graphs have an amplitude of 1. 1 y = cos(x) starts at 1. -4π -3π -2π -π π 2π 3π 4π -1

y = asin(bx) y = acos(bx) Amplitude of the graph = a Period of the graph = b is the frequency. It tells us the number of full cycles within 2π.

Example 1: Determine the period and amplitude of each trig function. y = 7cos(2x) y = -8sin(3x) y = ¼cos(6x) y = -sin(πx)

y = asin(b(x – h)) y = acos(b(x – h)) Phase Shift (h) y = asin(b(x – h)) y = acos(b(x – h)) When a graph shifts left or right, it is called a phase shift. GO THE OPPOSITE DIRECTION!

Example 2: State the direction and distance that each function has shifted. 1. y = sin(x + π) 2. y = cos(x – ) 3. y = cos(x + ) 4. y = sin(x – )

y = asin(b(x – h)) + k y = acos(b(x – h)) + k Vertical Shift (k) y = asin(b(x – h)) + k y = acos(b(x – h)) + k When a graph moves up or down, it is called a vertical shift. k tells us which direction and how far to shift.

Example 3: State the direction and distance that each function has shifted. y = cos(x) – 3 y = sin(x) – 2 y = sin(x) + 4 y = cos(x) + ½

Putting it all together! y = asin(b(x – h)) + k y = acos(b(x – h)) + k Amplitude of the graph = a Period of the graph = Horizontal translation = h (note: opposite direction) Vertical translation = k