Warm-Up 4/8Give equation for each graph.

y = 2sin x

y = sin xy = sin x

Q&A on assignment.

1, 3, 7, 5 1, 3, 7, 5 2, 4, 7, 62, 4, 6

Rigor:You will learn how to graph

transformations of the cosine and tangent functions.

Relevance:You will be able to use sinusoidal

functions to solve real world problems.

Trig 6 Graphing Cosine and

Tangent Functions

f(x) = a cos(bx + c)+ d

Domain: Range:Amplitude (a):period (b):Phase shift (c): Midline (d): Oscillation:Symmetry:

,

1,1

2

x-intercepts:y- intercept:Continuity:Extrema:

End Behavior:

1

0

y = 0

Even function: cos (–x) = cos x

n, n ϵ Z

(0, 1)continuous on ,

Maximum of 1 at x =2n, n ϵ Z

between – 1 and 1

Minimum of –1 at x = +2n, n ϵ Z

AMPLITUDE:PERIOD:

Frequency:

PHASE SHIFT: VERTICAL SHIFT: d

MIDLINE:

f(x) = a cos(bx + c) + d

: Vertically Compressed

: Vertically Expanded

: Horizontally Expanded

: Horizontally Compressed

– a: reflects in the x-axis

Example 1: Describe how the graph of f(x) = cos x and g(x) = – 3cos x are related. Then find the amplitude of g(x), and sketch two periods of both functions on the same coordinate axes. The graph g(x) is the graph of f(x) expanded vertically and the reflected in the x-axis. The Amplitude of g(x) is 3.

x f(x) g(x)

0 1 –3

0 0

– 1 3

0 0

2 1 –3

0 0

3 – 1 3

0 0

4 1 –3

Example 2: Describe how the graph of f(x) = cos x and g(x) = cos are related. Then find the period of g(x), and sketch at least one period of both functions on the same coordinate axes.The graph g(x) is the graph of f(x) expanded horizontally. The Period of g(x) is 6.

x f(x)

0 1

0

– 1

0

2 1

0

3 – 1

0

4 1

x g(x)

0 1

0

3 – 1

0

6 1

Fill in Chart.x tan x

0

0

1

und

– 1

0

1

und

– 1

0

Vertical Asymptote

Vertical Asymptote

x

y

(1, 0)

(0, 1)

(– 1, 0)

(0, – 1)

tan𝑥=¿sin𝑥cos𝑥

¿

Period:

Amplitude =

Amplitude does not exist for the tangent function.

Vertical Asymptotes:

f(x) = a tan(bx + c) + d

|𝒂|PERIOD: PHASE SHIFT: VERTICAL SHIFT: d

: Vertically Compressed

: Vertically Expanded

: Horizontally Expanded

: Horizontally Compressed

– a: reflects in the x-axis

Domain:

Range:

period (b):

Phase shift (c):

x-intercepts:

y- intercept:

Oscillation:

Symmetry:

Asymptotes:

Continuity:

End Behavior:

,

0

Origin (odd function)

n, n ϵ Z

(0, 0)

discontinuous at

sintan

cos

xy x

x

between – and

Example 3: Locate the vertical asymptotes, and sketch the graph of y = . x tan x

V.A.– 1

0 01

V.A.

x y

V.A.– 1

0 01

V.A.

Vertical Asymptotes

Example 4a: Locate the vertical asymptotes, and sketch the graph of y = . x -tan x

V.A.1

0 0–1

V.A.

x y

V.A.1

0 0–1

V.A.

Vertical Asymptotes

Example 5b: Locate the vertical asymptotes, and sketch the graph of y = .

x tan x

V.A.–1

0 01

V.A.

x y

V.A.– 1

01

V.A.

Vertical Asymptotes

Phase Shift:

1. Find the amplitude and period of .

2. Find the frequency and phase shift of .

3. Find the phase shift and vertical shift of .

4. Find the vertical asymptotes of .

amplitude = 4 period =

frequency = phase shift =

phase shift = vertical shift

Checkpoints:

4 𝑥=−𝜋2

14

∙ 4 𝑥=14

∙−𝜋2

𝑥=−𝜋8

4 𝑥=𝜋2

14

∙4 𝑥=14

∙𝜋2

𝑥=𝜋8

Assignment:Trig 6 WS, 1-6 all

Unit Circle & Trig Test Wednesday 4/9

7th Warm-Up 4/81. Find the amplitude and period of .

2. Find the frequency and phase shift of .

3. Find the phase shift and vertical shift of .

4. Find the vertical asymptotes of .

amplitude = 4 period =

frequency = phase shift =

phase shift = vertical shift

4 𝑥=−𝜋2

14

∙ 4 𝑥=14

∙−𝜋2

𝑥=−𝜋8

4 𝑥=𝜋2

14

∙4 𝑥=14

∙𝜋2

𝑥=𝜋8

Assignment:Trig 6 WS, 1-6 all

Unit Circle & Trig Test Wednesday 4/9