EXAMPLE 4 Combine a translation and a reflection

Graph y = –2 sin (x – ).23

π2

SOLUTION

STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift.

Amplitude: a = –2 = 2 Horizontal shift: π2h =

period : b2π 2π

322= 3π= 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 3 Find the five key points of y = –2 sin (x – ).23

π2

On y = k: (0 + , 0)π2

= ( , 0);π2 ( + , 0)

3π2

π2 = (2π, 0)

π2(3π + , 0) 7π

2 = ( , 0)

Maximum: ( + , 2)3π4

π2

5π4

= ( , 2)

Minimum: ( + , –2)9π4

π2

11π4

( , –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 )5π4

5π4

and becomes .11π

4( , –2)

11π4

( , 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 1 Identify 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: xπ

2 1–= = ;π

2– xπ

2 1= π

2=

On y = k: (0, 0 + 5) = (0, 5)

Halfway points: (– , –3 + 5)π4

(– , 2);π4= ( , 3 + 5)π

4 ( , 8)π4=

STEP 4 Reflect the graph. Because a < 0, the graph is reflected in the midline y = 5.

So, (– , 2) π4 (– , 8)π

4becomes

and ( , 8)π4

( , 2) .π4becomes

EXAMPLE 5 Combine a translation and a reflection

STEP 5 Draw 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 tangenttan oppadj= =

260 – d 120

Multiply each side by 120.120 tan 260 – d =

Subtract 260 from each side.120 tan – 260 –d=

Solve for d.–120 tan + 260 d=

The graph of d = –120 tan + 260 is shown at the right.

GUIDED PRACTICE for Examples 4, 5, and 6

Graph the function.

4. y = – cos ( x + )π2

SOLUTION

GUIDED PRACTICE for Examples 4, 5, and 6

Graph the function.

5. y = –3 sin x + 2 12

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 + 400.