EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the...

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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.

Transcript of EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the...

Page 1: 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:

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: 2bπ

=24π

=π2

Vertical shift: k = 3

STEP 2 Draw the midline of the graph, y = 3.

STEP 3 Find the five key points.

Page 2: 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:

EXAMPLE 1 Graph a vertical translation

On y = k: (0, 0 + 3) = (0, 3);π4( , 0 + 3) = ( , 3);

π4 ( , 0 + 3)

π2

= ( , 3)π2

Maximum: ( , 2 + 3)π8 = ( , 5)

π8

Minimum: ( , –2 + 3)3π8 = ( , 1)

3π8

STEP 4 Draw the graph through the key points.

Page 3: 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:

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: 2bπ 2

= 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.

Page 4: 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:

EXAMPLE 2 Graph a horizontal translation

On y = k: ( + 3π , 0)π4 = ( , 0);

13π4

( + 3π, 0)3π4 = ( , 0)

15π4

Maximum: (0 + 3π , 5) = (3π, 5)

(π + 3π , 5) = (4π, 5)

Minimum: ( + 3π, –5)π2 = ( , –5)

7π2

STEP 4 Draw the graph through the key points.

Page 5: 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:

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

20h = 85 sin (t – 10) + 90.

a. Graph your height above the ground as a functionof time.

b. What are your maximum and minimum heights?

Page 6: 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:

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. π20

2 π

40180

Page 7: 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:

EXAMPLE 3 Graph a model for circular motion

Your maximum height is 90 + 85 = 175 feet and your minimum height is 90 – 85 = 5 feet.

b.

Page 8: 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:

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.

Page 9: 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:

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.

Page 10: 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:

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.

Page 11: 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:

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.

Page 12: 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:

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

Page 13: 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:

EXAMPLE 5 Combine a translation and a reflection

STEP 5 Draw the graph through the key points.

Page 14: 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:

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 .

Page 15: 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:

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.