Warm-Up 4/8 Q&A on assignment. Give equation for each graph. y = 2sin x y = sin x y = 1 2 sin x Q&A on assignment.

1, 3, 7, 5 1, 3, 7, 5 2, 4, 6 2, 4, 7, 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. MA.912. A.2.11

Trig 6 Graphing Cosine and Tangent Functions

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

Even function: cos (–x) = cos x x-intercepts: y- intercept: Continuity: Extrema: End Behavior: 𝜋 2 +n, n ϵ Z Domain: Range: Amplitude (a): period (b): Phase shift (c): Midline (d): Oscillation: Symmetry: (0, 1) continuous on 1 Maximum of 1 at x =2n, n ϵ Z Minimum of –1 at x = +2n, n ϵ Z y = 0 between – 1 and 1 Even function: cos (–x) = cos x

f(x) = a cos(bx + c) + d – a: reflects in the x-axis PERIOD: 𝟐𝝅 𝒃 Frequency: 𝒃 𝟐𝝅 PHASE SHIFT: −𝒄 𝒃 VERTICAL SHIFT: d MIDLINE: 𝒚=𝒅 AMPLITUDE: 𝒂 – a: reflects in the x-axis 𝟎< 𝒂 <𝟏: Vertically Compressed 𝒂 >𝟏: Vertically Expanded 𝟎< 𝒃 <𝟏: Horizontally Expanded 𝒃 >𝟏: Horizontally Compressed

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 𝑜𝑟 3. x f(x) g(x) 1 –3 𝜋 2  – 1 3 3𝜋 2 2 5𝜋 2 3 7𝜋 2 4

Example 2: Describe how the graph of f(x) = cos x and g(x) = cos 𝑥 3 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 2𝜋 1 3 𝑜𝑟 6. x f(x) 1 𝜋 2  – 1 3𝜋 2 2 5𝜋 2 3 7𝜋 2 4 x g(x) 1 3𝜋 2 3 – 1 9𝜋 2 6

Fill in Chart. x tan x − − 3𝜋 4 − 𝜋 2 − 𝜋 4 𝜋 4 𝜋 2 3𝜋 4  1 und – 1 (0, 1) Fill in Chart. x tan x − − 3𝜋 4 − 𝜋 2 − 𝜋 4 𝜋 4 𝜋 2 3𝜋 4  (– 1, 0) (1, 0) 1 und – 1 (0, – 1) Vertical Asymptote Vertical Asymptote

tan 𝑥= sin 𝑥 cos 𝑥

Period: 𝝅 𝒃 Amplitude = 1 2 [𝑚𝑎𝑥−𝑚𝑖𝑛] Amplitude does not exist for the tangent function.

Vertical Asymptotes: 𝒃𝒙−𝒄=− 𝝅 𝟐 𝐚𝐧𝐝 𝒃𝒙−𝒄= 𝝅 𝟐

f(x) = a tan(bx + c) + d – a: reflects in the x-axis PERIOD: 𝝅 𝒃 PHASE SHIFT: −𝒄 𝒃 VERTICAL SHIFT: d 𝒂 – a: reflects in the x-axis 𝟎< 𝒂 <𝟏: Vertically Compressed 𝒂 >𝟏: Vertically Expanded 𝟎< 𝒃 <𝟏: Horizontally Expanded 𝒃 >𝟏: Horizontally Compressed

Domain: Range: period (b): Phase shift (c): x-intercepts:  y- intercept: Oscillation: Symmetry: Asymptotes: Continuity: End Behavior:  𝜋n, n ϵ Z (0, 0) between – ∞ and ∞ Origin (odd function) discontinuous at

Example 3: Locate the vertical asymptotes, and sketch the graph of y = tan 2𝑥. tan x − 𝜋 2 V.A. − 𝜋 4 – 1 𝜋 4 1 𝜋 2 x y − 𝜋 4 V.A. − 𝜋 8 – 1 𝜋 8 1 𝜋 4 Vertical Asymptotes 𝒃𝒙−𝒄=− 𝝅 𝟐 𝐚𝐧𝐝 𝒃𝒙−𝒄= 𝝅 𝟐 𝟐𝒙=− 𝝅 𝟐 𝐚𝐧𝐝 𝟐𝒙= 𝝅 𝟐 𝒙=− 𝝅 𝟒 𝐚𝐧𝐝 𝒙= 𝝅 𝟒

Example 4a: Locate the vertical asymptotes, and sketch the graph of y = −tan 𝑥 2 . -tan x − 𝜋 2 V.A. − 𝜋 4 1 𝜋 4 –1 𝜋 2 x y −𝜋 V.A. − 𝜋 2 1 𝜋 2 –1 𝜋 Vertical Asymptotes 𝒃𝒙−𝒄=− 𝝅 𝟐 𝐚𝐧𝐝 𝒃𝒙−𝒄= 𝝅 𝟐 𝒙 𝟐 =− 𝝅 𝟐 𝐚𝐧𝐝 𝒙 𝟐 = 𝝅 𝟐 𝒙=−𝝅 𝐚𝐧𝐝 𝒙=𝝅

Example 5b: Locate the vertical asymptotes, and sketch the graph of y = tan 𝑥− 3𝜋 2 . Phase Shift: 3𝜋 2 Vertical Asymptotes 𝒃𝒙−𝒄=− 𝝅 𝟐 𝐚𝐧𝐝 𝒃𝒙−𝒄= 𝝅 𝟐 x tan x − 𝜋 2 V.A. − 𝜋 4 –1 𝜋 4 1 𝜋 2 x y 𝜋 V.A. 5𝜋 4 – 1 3𝜋 2 7𝜋 4 1 2𝜋 𝒙− 𝟑𝝅 𝟐 =− 𝝅 𝟐 𝐚𝐧𝐝 𝒙− 𝟑𝝅 𝟐 = 𝝅 𝟐 𝒙=𝝅 𝐚𝐧𝐝 𝒙=𝟐𝝅

Checkpoints: 1. Find the amplitude and period of 𝑓 𝑥 = 4cos 2𝑥 . 2. Find the frequency and phase shift of 𝑓 𝑥 = cos (2𝑥− 𝜋 4 ) . phase shift = 𝜋 4 2 = 𝜋 8 frequency = 1 𝜋 3. Find the phase shift and vertical shift of 𝑓 𝑥 = 1 2 cos 𝑥 6 − 𝜋 2 −5 . phase shift = 𝜋 2 1 6 =3𝜋 vertical shift =−5 4. Find the vertical asymptotes of 𝑓 𝑥 = 1 2 tan 4𝑥 . 4𝑥=− 𝜋 2 4𝑥= 𝜋 2 1 4 ∙4𝑥= 1 4 ∙ −𝜋 2 1 4 ∙4𝑥= 1 4 ∙ 𝜋 2 𝑥=− 𝜋 8 𝑥= 𝜋 8

Assignment: Trig 6 WS, 1-6 all Unit Circle & Trig Test Wednesday 4/9

7th Warm-Up 4/8 1. Find the amplitude and period of 𝑓 𝑥 = 4cos 2𝑥 . 2. Find the frequency and phase shift of 𝑓 𝑥 = cos (2𝑥− 𝜋 4 ) . phase shift = 𝜋 4 2 = 𝜋 8 frequency = 1 𝜋 3. Find the phase shift and vertical shift of 𝑓 𝑥 = 1 2 cos 𝑥 6 − 𝜋 2 −5 . phase shift = 𝜋 2 1 6 =3𝜋 vertical shift =−5 4. Find the vertical asymptotes of 𝑓 𝑥 = 1 2 tan 4𝑥 . 4𝑥=− 𝜋 2 4𝑥= 𝜋 2 1 4 ∙4𝑥= 1 4 ∙ −𝜋 2 1 4 ∙4𝑥= 1 4 ∙ 𝜋 2 𝑥=− 𝜋 8 𝑥= 𝜋 8

Assignment: Trig 6 WS, 1-6 all Unit Circle & Trig Test Wednesday 4/9