Graphs of Trig Functions Unit 5 Day 9 Graphs of Trig Functions

Warm-up Warm-up: Graph the following and state the vertex and axis of symmetry: y = 3x2 y = x2 +5 y = 3(x-4)2 -7   2. Solve the triangle if Angle A = 60, c = 8, b = 10 3. Solve the trigonometric equation: 2tan(x)sin(x) = 2tan(x) Vertex: (0,0) AOS: x = 0 Vertex: (0,5) AOS: x = 0 Vertex: (4,-7) AOS: x = 4 a = 9.2, B = 71, C = 49 x = 90°

Homework Answers Part 1: Graph A: A Max (45, 4) B Min (135, -2) C F 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) Period: 90 Midline: y = 0 Graph C: Max: (0, 1) and (360,1) 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 I think they can fill out the table and graph what they get on their own. Then we can review features of the graph after.

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

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: When there is no vertical shift, the midline is always the x-axis (y = 0). (Ex: y = sin(x), y = 2sin(x), y = sin(3x) all have a midline of y = 0 ) Midline is y = (Max + Min) OR y = Min + Amp 2

**Notice: The amplitude did not change. 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? *How does “b” affect the graph? The “a” affects the height of the graph A negative “a” reflects the graph over the x-axis Here are just a few more examples of sine functions and their graphs so the kids can the affect the a and the b have on the picture. 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