EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude: a = 2 Horizontal shift: h = 0 Period: 2 b π = 2 4 π = π 2 Vertical shift: k = 3 STEP 2 Draw the midline of the graph, y = 3. STEP 3 Find the five key points.

EXAMPLE 1 Graph a vertical translation On y = k: (0, 0 + 3)= (0, 3); π 4 (, 0 + 3) = (, 3); π 4 (, 0 + 3) π 2 = (, 3) π 2 Maximum: (, 2 + 3) π 8 = (, 5) π 8 Minimum: (, –2 + 3) 3π3π 8 = (, 1) 3π3π 8 STEP 4 Draw the graph through the key points.

EXAMPLE 2 Graph a horizontal translation Graph y = 5 cos 2(x – 3π ). SOLUTION STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude: a = 5 Horizontal shift: h = 3π Period: 2 b π 2 2 π = π = Vertical shift: k = 0 STEP 2 Draw the midline of the graph. Because k = 0, the midline is the x- axis. STEP 3 Find the five key points.

EXAMPLE 2 Graph a horizontal translation On y = k: ( + 3π, 0) π 4 = (, 0); 13π 4 ( + 3π, 0) 3π3π 4 = (, 0) 15π 4 Maximum: (0 + 3π, 5) = (3π, 5) (π + 3π, 5) = (4π, 5) Minimum: ( + 3π, –5) π 2 = (, –5) 7π7π 2 STEP 4 Draw the graph through the key points.

EXAMPLE 3 Graph a model for circular motion Ferris Wheel Suppose you are riding a Ferris wheel that turns for 180 seconds. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the equation π 20 h = 85 sin (t – 10) a. Graph your height above the ground as a function of time. b. What are your maximum and minimum heights?

EXAMPLE 3 Graph a model for circular motion SOLUTION The amplitude is 85 and the period is = 40. The wheel turns = 4.5 times in 180 seconds, so the graph below shows 4.5 cycles. The five key points are (10, 90), (20, 175), (30, 90), (40, 5), and (50, 90). a. π 20 2 π2 π

EXAMPLE 3 Graph a model for circular motion Your maximum height is = 175 feet and your minimum height is 90 – 85 = 5 feet. b.

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the function. y = cos x SOLUTION

GUIDED PRACTICE for Examples 1, 2, and 3 SOLUTION Graph the function. y = 3 sin (x – ) 2. π 2

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the function. f(x) sin (x + π) – 1 3. SOLUTION