Look over Unit Circle! Quiz in 5 minutes!! Warm Up Look over Unit Circle! Quiz in 5 minutes!!

Sine and COsine Graphs

Sine Graph of y = sin(x) 1 -4π -3π -2π -π π 2π 3π 4π -1

Cosine Graph of y = cos(x) 1 -4π -3π -2π -π π 2π 3π 4π -1

Period and Amplitude

Period of a Function When the values of a function regularly repeat themselves, we say that the function is periodic. 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 because that is the length of the piece that repeats.

Amplitude Amplitude is the distance from the midpoint to the highest and lowest point of the function. (Half the distance from the max to the min.) 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

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

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

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

y = asin(tx) y = acos(tx) Amplitude of the graph = a Period of the graph = t is not the period! We evaluate this expression with different values of t to find the period of a function.

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

Example 2: Determine the period and amplitude of the function Example 2: Determine the period and amplitude of the function. Then, graph it! y = 5sin(2x)

Example 3: Determine the period and amplitude of the function Example 3: Determine the period and amplitude of the function. Then, graph it! y = ⅔cos(4x)

Warm Up Determine the period and amplitude of the function Warm Up Determine the period and amplitude of the function. Then, graph it! y = -16sin(x)

Trig graphs: Phase Shifts

y = asin(tx - h) y = acos(tx - h) Phase Shift (h) y = asin(tx - h) y = acos(tx - h) When a graph shifts left or right, it is called a phase shift. h tells us the direction and how far to shift. Note that h is subtracted, so go the opposite direction that you would assume. negative  moves right positive  moves left

Example 1: 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(tx - h) + k y = acos(tx - h) + k Vertical Shift (k) y = asin(tx - h) + k y = acos(tx - h) + k When a graph moves up or down, it is called a vertical shift. c tells us which direction and how far to shift. Negative  moves down Positive  moves up

Example 2: 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(tx - h) + k y = acos(tx - h) + k Amplitude of the graph = a Period of the graph = 2π/t Horizontal translation = h (note: opposite direction) Vertical translation = k