Graphs of Sine and Cosine Functions Section 4.5

Objectives Sketch the graphs of sine and cosine functions and their variations. Use amplitude and period to help sketch the graphs of sine and cosine functions. Sketch translations of graphs of sine and cosine.

𝒚= 𝐬𝐢𝐧 𝒙

315° 225° 270° 0° 180° 90° 45° 30° 330° 300° 60° 240° 135° 210° 150° 120° y = sin x 1 90° 180° 270° 360° –1

315° 225° 270° 0° 180° 90° 45° 30° 330° 300° 60° 240° 135° 210° 150° 120° y = sin x 90° 180° 270° 360°

Values of (x, y) on the graph of 𝒚= 𝐬𝐢𝐧 𝒙 𝐱 𝟎 𝛑 𝟔 𝛑 𝟑 𝛑 𝟐 𝟐𝛑 𝟑 𝟓𝛑 𝟔 𝛑 𝟕𝛑 𝟔 𝟒𝛑 𝟑 𝟑𝛑 𝟐 𝟓𝛑 𝟑 𝟏𝟏𝛑 𝟔 𝟐𝛑 𝒚= 𝐬𝐢𝐧 𝒙 𝟏 𝟐 𝟑 𝟐 𝟏 − 𝟏 𝟐 − 𝟑 𝟐 −𝟏 𝑦= sin 𝑥, 0≤𝑥≤2𝜋 𝑃𝑒𝑟𝑖𝑜𝑑=2𝜋 Sine starts at the sea shore ∎

Graph of the Sine Function To sketch the graph of y = sin x we don’t need many points, just the key points. These are the maximum points, the minimum points, and the intercepts. -1 1 sin x x Divide the cycle of 2𝜋 into four equal parts. Then, connect the five critical points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x     

Amplitude The amplitude is the maximum height above or below the center line. The standard sine curve y = sin x has an amplitude = 1 y = The equation 𝐴 sin 𝑥 has an amplitude = |A| 𝑦=2 sin 𝑥 has an amplitude of 2. 𝑦=−2 sin 𝑥 has an amplitude of 2. Amplitude

Period The period is the distance between two peaks or valleys. y = sin x has period 2π y = sin(Bx) has period 2𝜋 𝐵

𝑦=𝐴 sin (𝐵𝑥−𝐶) (the cycle) Amplitude (the height) Period: 2𝜋 𝐵 Phase Shift: 𝐶 𝐵 The phase shift is a right or left shift of the graph.

Graphing Variations of 𝒚= 𝐬𝐢𝐧 𝒙 Identify the amplitude and the period. Find the values of x for the five key parts---the x-intercepts (3 for sine), the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right as desired. (phase shift) 𝑷𝒆𝒓𝒊𝒐𝒅 𝟒 y x y = 2 sin x y = sin x y = sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x

Free Graph Paper for Graphing Trig Functions http://mathbits.com/MathBits/StudentResources/GraphPaper/GraphPaper.htm

Online Trig Graphing Calculator https://www.desmos.com/calculator

Graphing Variations of 𝒚= 𝐬𝐢𝐧 𝒙 Graph 𝒚=𝟑 𝐬𝐢𝐧 𝒙 for 𝟎≤𝒙≤𝟐𝝅 Identify the amplitude and the period. Find the values of x for the five key parts---the x-intercepts (3 for sine), the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right as desired.

Graph 𝒚=𝟑 𝐬𝐢𝐧 𝒙 for 𝟎≤𝒙≤𝟐𝝅 Period 2𝜋 𝐵 = 2𝜋 1 =2𝜋 The amplitude = 𝐴 =3

Graph 𝒚= 𝐬𝐢𝐧 𝟑𝒙 for 𝟎≤𝒙≤𝟐𝝅 Period 2𝜋 𝐵 = 2𝜋 3 The amplitude = 𝐴 =1

Use the identity sin (– x) = – sin x: Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: 2 3 = amplitude: |a| = |–2| = 2 Calculate the five key points. 2 –2 y = –2 sin 3x x y x ( , 2) (0, 0) ( , 0) ( , 0) ( , -2)

The Graph of 𝒚=𝑨 𝐬𝐢𝐧 𝑩𝒙−𝑪 The graph of 𝑦=𝐴 sin 𝐵𝑥−𝐶 is obtained by horizontally shifting the graph of 𝑦=𝐴 sin 𝐵𝑥 so that the starting point of the cycle is shifted from 𝑥=0 to 𝑥= 𝐶 𝐵 . If 𝐶 𝐵 >0, the shift is to the right. If 𝐶 𝐵 <0, the shift is to the left. The number 𝐶 𝐵 is called the phase shift. 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. amplitude = |𝐴| period = 2𝜋 𝐵 y y = A sin Bx Amplitude: | A| phase shift = 𝐶 𝐵 x Starting point: x = C/B Period: 2/B

Determine the amplitude, period, and phase shift of 𝑦=2 sin (4𝑥−𝜋) 𝐴=2 𝐵=4 𝐶=𝜋 Amplitude = |𝐴| 2 Maximum height 2𝜋 4 = 𝜋 2 Period = 2𝜋 𝐵 One cycle Phase shift = 𝐶 𝐵 𝜋 4 Starting point to the right

Graph 𝒚=𝟐 𝒔𝒊𝒏 (𝟒𝒙−𝝅) Amp = 2 Period = 𝝅 𝟐 Phase shift = 𝝅 𝟒

𝑦=𝐴 sin 𝐵𝑥−𝐶 +𝐷 Amplitude (height) Period: 2𝜋 𝐵 Phase Shift: 𝐶 𝐵 (cycle) Phase Shift: 𝐶 𝐵 (Shift right or left) Vertical Shift (Up or down)

y = a sin(bx – c) + d Vertical shift is the amount that the red curve is moved up (or down) compared to the reference (black) curve. A positive shift means an upward displacement while a negative shift means a downward displacement. vertical shift This black curve is for reference.

Graphing a Function in the Form 𝒚=𝑨 𝐬𝐢𝐧 𝑩𝒙−𝑪 +𝑫 Identify the amplitude and the period, the phase shift, and the vertical shift. Find the values of x for the five key parts---the x-intercepts (3 for sine), the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Find and plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right and up or down as desired Determine the amplitude, period, and phase shift and vertical shift of 𝑦=4 sin 2𝑥− 2𝜋 3 +1. Then graph one period of the function. Graph of Sine Applet

Graph 𝑦=𝐴 sin 𝐵𝑥−𝐶 +𝐷 Example Graph y = 2 sin(4x + ) - 1. Solution Express y in the form A sin(Bx – C) + D. Amplitude = 2 Period = 2𝜋 4 = 𝜋 2 Phase shift = − 𝜋 4 Vertical shift is down 1 unit.

Example 1: Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x y = sin (–x) Use the identity sin (–x) = – sin x y = sin x

Values of (x, y) on the graph of 𝒚= 𝐜𝐨𝐬 𝒙 Cosine starts on the cliff 𝐱 𝟎 𝛑 𝟔 𝛑 𝟑 𝛑 𝟐 𝟐𝛑 𝟑 𝟓𝛑 𝟔 𝛑 𝟕𝛑 𝟔 𝟒𝛑 𝟑 𝟑𝛑 𝟐 𝟓𝛑 𝟑 𝟏𝟏𝛑 𝟔 𝟐𝛑 𝒚= 𝐜𝐨𝐬 𝒙 𝟏 𝟑 𝟐 𝟏 𝟐 − 𝟏 𝟐 − 𝟑 𝟐 −𝟏 Cosine starts on the cliff 𝑦= cos 𝑥, −2𝜋≤𝑥≤2𝜋 𝑃𝑒𝑟𝑖𝑜𝑑=2𝜋

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1 cos x x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = cos x y x

Graphing Variations of 𝒚= 𝐜𝐨𝐬 𝒙 Identify the amplitude and the period. Find the intercepts (one y-intercept and two x-intercepts, the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right as desired.

Graph 𝑦= cos 3𝑥 Amplitude 1 2𝜋 𝐵 = 2𝜋 3 Period

Example: Sketch the graph of y = 3 cos x Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , ,3) ( , 0) ( , 0) ( , –3)

𝑦=𝐴 cos (𝐵𝑥−𝐶) (the cycle) Amplitude (the height) Period: 2𝜋 𝐵 Phase Shift: 𝐶 𝐵 The phase shift is a right or left shift of the graph.

The Graph of 𝒚=𝑨 𝒄𝒐𝒔 𝑩𝒙−𝑪 The graph of 𝑦=𝐴 cos 𝐵𝑥−𝐶 is obtained by horizontally shifting the graph of 𝑦=𝐴 cos 𝐵𝑥 so that the starting point of the cycle is shifted from 𝑥=0 to 𝑥= 𝐶 𝐵 . If 𝐶 𝐵 ≥0, the shift is to the right. If 𝐶 𝐵 <0, the shift is to the left. The number 𝐶 𝐵 is called the phase shift. amplitude = | 𝐴 | period = 2𝜋 𝐵

http://www.youtube.com/watch?v=ijTIr-aykUk

Graphing a Function in the Form 𝒚=𝑨 𝐜𝐨𝐬⁡(𝑩𝒙−𝑪) Determine the amplitude and period of 𝑦=3 cos 𝜋 2 𝑥 Identify the amplitude and the period. Find the intercepts (one y-intercept and two x-intercepts, the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right as desired. Graph of Cosine Applet

http://www.analyzemath.com/cosine/cosine_applet.html

Graphing a Function in the Form 𝒚=𝑨 𝐜𝐨𝐬⁡(𝑩𝒙−𝑪) Determine the amplitude and period of 𝑦= 1 2 cos 4𝑥+𝜋 . Graph one period of the function. Identify the amplitude, the period, and the phase shift. Find the intercepts (one y-intercept and two x-intercepts, the maximum point, and the minimum point. Divide the period into 4 quarter-periods. Plot the five key points. Connect the key points with a smooth graph and graph one complete cycle. Extend the graph in step 4 to the left or right as desired. Graph of Cosine Applet

Amplitude Period: 2π/b Phase Shift: c/b Vertical Shift

Vertical Translation - Example Vertical Translation = 3 units upward, since d > 0 - 2 2 60˚ 30˚ 90˚ 120˚

Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. Use the identity cos (–x) = cos x y x Cosine is an even function. y = cos (–x) y = cos (–x)