EXAMPLE 4 Combine a translation and a reflection Graph y = –2 sin (x – ). 2 3 π 2 SOLUTION STEP 1Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude: a = –2 = 2 Horizontal shift: π 2 h = period : b 2π2π 2π2π 3 22 =3π3π = Vertical shift: k = 0 STEP 2 Draw the midline of the graph. Because k = 0, the midline is the x -axis.

EXAMPLE 4 Combine a translation and a reflection STEP 3Find the five key points of y = –2 sin (x – ). 2 3 π 2 On y = k: (0 +, 0) π 2 = (, 0); π 2 ( +, 0) 3π3π 2 π 2 = (2π, 0) π 2 (3π +, 0) 7π7π 2 = (, 0) Maximum: ( +, 2) 3π3π 4 π 2 5π5π 4 = (, 2) Minimum: ( +, –2) 9π9π 4 π 2 11π 4 (, –2) = STEP 4 Reflect the graph. Because a < 0, the graph is reflected in the midline y = 0.

EXAMPLE 4 Combine a translation and a reflection So, (, 2) becomes (, –2 ) 5π5π 4 5π5π 4 and becomes. 11π 4 (, –2) 11π 4 (, 2) STEP 5 Draw the graph through the key points.

EXAMPLE 5 Combine a translation and a reflection Graph y = –3 tan x + 5. SOLUTION STEP 1Identify the period, horizontal shift, and vertical shift. Period: π Horizontal shift: h = 0 Vertical shift: k = 5 STEP 2 Draw the midline of the graph, y = 5. STEP 3 Find the asymptotes and key points of y = –3 tan x + 5.

EXAMPLE 5 Combine a translation and a reflection Asymptotes: x π 2 1 – = = ; π 2 – x π 2 1 = π 2 = On y = k : (0, 0 + 5) = (0, 5) Halfway points: (–, –3 + 5) π 4 (–, 2); π 4 = (, 3 + 5) π 4 (, 8) π 4 = STEP 4Reflect the graph. Because a < 0, the graph is reflected in the midline y = 5. So, (–, 2) π 4 (–, 8) π 4 becomes and (, 8) π 4 (, 2). π 4 becomes

EXAMPLE 5 Combine a translation and a reflection STEP 5Draw the graph through the key points.

EXAMPLE 6 Model with a tangent function Glass Elevator You are standing 120 feet from the base of a 260 foot building. You watch your friend go down the side of the building in a glass elevator. Write and graph a model that gives your friend’s distance d (in feet) from the top of the building as a function of the angle of elevation .

EXAMPLE 6 Model with a tangent function SOLUTION Use a tangent function to write an equation relating d and . Definition of tangent tan  opp adj = = 260 – d 120 Multiply each side by tan  260 – d = Subtract 260 from each side. 120 tan  – 260 –d–d = Solve for d. –120 tan  d = The graph of d = –120 tan  is shown at the right.

GUIDED PRACTICE for Examples 4, 5, and 6 Graph the function. 4. y = – cos ( x + ) π 2 SOLUTION

GUIDED PRACTICE for Examples 4, 5, and 6 Graph the function. 5. y = –3 sin x SOLUTION

GUIDED PRACTICE for Examples 4, 5, and 6 Graph the function. 6. f(x) = – tan 2 x – 1 SOLUTION

GUIDED PRACTICE for Examples 4, 5, and 6 7. What if ? In example 6, how does the model change if you are standing 150 feet from a building that is 400 feet tall ? ANSWER The graph is shifted 400 units up instead of 260. The new equation would be d = –150 tan 