Unit 5 Day 9 Graphs of Trig Functions

Warm-up: Graph the following and state the vertex and axis of symmetry: • y

Homework Answers Part 1: 1. A 2. B 3. C 4. E 5. F

Homework Answer Continued Graph B: Max: None Min: None Increasing (0, 90)U(90, 180)U(180, 270)U(270,

Amplitude What are the similarities and differences between the 3 graphs? ? Given the

Summary: Amplitude: *Amplitude is the height of the graph from the midline* a. A

Summary continued Midline: The midline is the line that “cuts the graph in half.

Midline Continued Midline moved up 1 y = sin(x) + 1 y = sin(x)

Given the standard equation y = a sin (bx), *How does “a” affect the

Period of a Function *Period is the length of 1 cycle. * Y =

Let’s Graph one together! Notes pg. 32 We’ll graph one period in the positive

Practice – you try the others! Notes pg. 33 #5, 6, 7 For each

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Unit 5 Day 9 Graphs of Trig Functions

Warm-up: Graph the following and state the vertex and axis of symmetry: • y = 3 x 2 Vertex: (0, 0) AOS: x = 0 • y = x 2 +5 Vertex: (0, 5) AOS: x = 0 • y = 3(x-4)2 -7 Vertex: (4, -7) AOS: x = 4 2. Solve the triangle if Angle A = 60, c = 8, b = 10 a = 9. 2, B = 71, C = 49 3. Solve the trigonometric equation: 2 tan(x)sin(x) = 2 tan(x) x = 90°

Homework Answers Part 1: 1. A 2. B 3. C 4. E 5. F 6. D Part 2: Xmin: 0 Xmax : 360 Ymin : -5 Ymax : 5 Graph A: Max (45, 4) Min (135, -2) Increasing (0, 45)U(135, 225)U(315, 360) Decreasing (45, 135)U(225, 315) Positive (0, 105) U (165, 285) Negative(105, 165) U (285, 360) Period: 180 Midline: y = 1

Homework Answer Continued Graph B: Max: None Min: None Increasing (0, 90)U(90, 180)U(180, 270)U(270, 360) Decreasing: Never Positive: (45, 90)U(135, 180)U(225, 270)U(315, 360) Negative: (0, 45)U(90, 135)U(180, 225)U(270, 315) Graph C: Period: 90 Max: (0, 1) and (360, 1) Midline: y = 0 Min: (180, -3) Increasing (180, 360) Decreasing: (0, 180) Positive (0, 45)U(315, 360) Negative(45, 315) Period: 360 Midline: y = -1

Amplitudes, Midlines, and Period

Amplitude What are the similarities and differences between the 3 graphs? ? Given the standard equation y=asin(bx), How does “a” affect the graph? The “a” affects the height of the graph.

Summary: Amplitude: *Amplitude is the height of the graph from the midline* a. A graph in the form of: has an amplitude of . b. The amplitude of a standard sine or cosine graph is 1.

amp = | a | = | max – min | 2

Summary continued Midline: The midline is the line that “cuts the graph in half. ” The midline is halfway between the max and the min. The midline can be found by using the following formula: Midline is y = (Max + Min) 2 OR y = Min + Amp When there is no vertical shift, the midline is always the x-axis (y = 0). (Ex: y = sin(x), y = 2 sin(x), y = sin(3 x) all have a midline of y = 0 )

Midline Continued Midline moved up 1 y = sin(x) + 1 y = sin(x) **Notice: The amplitude did not change.

Given the standard equation y = a sin (bx), *How does “a” affect the graph? The “a” affects the height of the graph A negative “a” reflects the graph over the x-axis *How does “b” affect the graph? The “b” affects the period of the graph Remember, period = 360 |b|

Period of a Function *Period is the length of 1 cycle. * Y = sin(x) has a period of 360. y = cos(x) has a period of 360. y = tan(x) has a period of 180. Let’s go back to the graphs and take a look at what this means graphically.

Think of the pi as 180 degrees!!!

Let’s Graph one together! Notes pg. 32 We’ll graph one period in the positive direction and one period in the negative direction. 4. y = 0. 5 sin (x) Amplitude: ______ Midline: ______ Period: _____ Always find the period before graphing! Label both axes!

Practice – you try the others! Notes pg. 33 #5, 6, 7 For each problem, graph one period in the positive direction and one period in the negative direction. Remember to label the axes!

Homework