1. 10.2 Translate and Reflect Trigonometric Graphs
How do you translate trigonometric graphs?
How do you reflect trigonometric graphs?

2. p. 619

3. 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 = 4
Vertical shift:
k = 3
STEP 2
Draw the midline of the graph, y = 3.
STEP 3
Find the five key points.

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

5. 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 =
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.

6. On y = k:
( π , 0) π 4
= ( , 0);
13π
( π, 0) 3π 4
= ( , 0)
15π
Maximum:
(0 + 3π , 5)
= (3π, 5)
(π + 3π , 5)
= (4π, 5)
Minimum:
( π, –5) π 2
= ( , –5) 7π
STEP 4
Draw the graph through the key points.

7. 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) + 90.
a. Graph your height above the ground as a function
of time.
b. What are your maximum and minimum heights?

8. 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 π 40
180
Your maximum height is = 175 feet and your minimum height is 90 – 85 = 5 feet. b.

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

10. On y = k:
( π , 0 +4) 1 4
= ( , 4); π 2 3 4
( π, 0 + 4)
= ( , 4) 3π 2
Maximum:
(0 , 1 + 4)
= (0,5)
(2π ,1 + 4)
= (2π, 5)
Minimum:
( π, –1 + 4) 1 2
= (π , 3)

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

12. On y = k:
( , 0) π 2
= ( , 0);
= ( , 0) 3π 2
( π , 0) 1 π
= ( , 0) 5π 2
( 2π , 0) π
Maximum:
( π + , 3)
= (π, 3) 1 4 π 2
Minimum:
( π , –3) 3 4
= ( , –3) 2π π 2
( , –3) 3π =

13. Graph the function.
f(x) sin (x + π) – 1 3.
SOLUTION
STEP 1
Identify the amplitude, period, horizontal shift, and vertical shift.
Amplitude: 1
Horizontal shift:
h =
– π
Period: 2 b π 1 = 2π
Vertical shift:
k = –1
STEP 2
Draw the midline of the graph. y = – 1.
STEP 3
Find the five key points.

14. On y = k:
(0 – π , 0 – 1 )
= (– π, – 1);
= (0, –1) 1 2
( π – π , –1)
= (π , – 1)
( 2π – π , 0 – 1)
Maximum: π 2
= (– ,0)
( π – π , 1 + 1) 1 4
Minimum:
( π – π, –1 –1) 3 4
( , – 2) π 2 =

15. How do you translate trigonometric graphs?
The graphs of y = 𝑎 sin⁡𝑏 (𝑥−ℎ)+𝑘 and 𝑦 = 𝑎 cos⁡𝑏(𝑥−ℎ)+𝑘, where a>0 and b>0 are horizontal translations h units and vertical translations k units of the graphs of 𝑦=𝑎 sin⁡𝑏𝑥 and 𝑦=𝑎 cos⁡𝑏𝑥 respectively, and have amplitude a, period 2𝜋 𝑏 , and midline 𝑦=𝑘
How do you reflect trigonometric graphs?
In general, when 𝑎<0, the graphs of 𝑦 =a sin⁡𝑏𝑥 and 𝑦 =𝑎 cos⁡𝑏𝑥 are the reflections of the graphs of 𝑦= 𝑎 sin 𝑏𝑥 𝑎𝑛𝑑 𝑦= 𝑎 cos 𝑏𝑥, respectively, in the midline 𝑦 = 𝑘

16. 10.2 Assignment, day 1
Page 623, odd

17. 10.2 Translate and Reflect Trigonometric Graphs, day 3
How do you translate trigonometric graphs?
How do you reflect trigonometric graphs?

18. p. 621

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

20. STEP 3
Find the five key points of y = –2 sin (x – ). 2 3 π
On y = k:
( , 0) π 2
= ( , 0);
( , 0) 3π
= (2π, 0) π 2
(3π , 0) 7π
= ( , 0)
Maximum:
( , 2) 3π 4 π 2 5π
= ( , 2)
Minimum:
( , –2) 9π 4 π 2
11π
( , –2) =
STEP 4
Reflect the graph. Because a < 0, the graph is reflected in the midline y = 0.
So, ( , 2) becomes ( , –2 ) 5π 4
and becomes
11π 4
( , –2)
( , 2)
STEP 5
Draw the graph through the key points.

21. 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.

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

23. 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 q .

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

25. Graph the function.
y = – cos ( x ) π 2 4.
SOLUTION

26. Graph the function. 5.
y = – 3 sin x + 2 1 2
SOLUTION

27. Graph the function. 6.
f(x) = – tan 2 x – 1
SOLUTION

28. Graphing tangent functions using translations and reflections is similar to graphing sine and cosine functions. When a tangent function has a horizontal shift, the asymptotes also have a horizontal shift. 𝑎<0, ℎ<0, and 𝑘>0

29. 10.2 Assignment, day 3
Page 623, odd, odd