Unit 5 Day 9

Graphs of Trig Functions

Warm-upWarm-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)a = 9.2, B = 71, C = 49

x = 90°

Vertex: (0,0) AOS: x = 0

Vertex: (0,5) AOS: x = 0

Vertex: (4,-7) AOS: x = 4

Homework AnswersPart 1:1. A2. B3. C4. E5. F6. D

Part 2:Xmin: 0Xmax : 360Ymin : -5Ymax : 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: 180Midline: y = 1

Homework Answer ContinuedGraph B:Max: NoneMin: NoneIncreasing (0,90)U(90, 180)U(180, 270)U(270, 360)Decreasing: NeverPositive: (45,90)U(135,180)U(225,270)U(315,360)Negative: (0,45)U(90,135)U(180, 225)U(270, 315)Period: 90Midline: 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: 360Midline: 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 continuedMidline: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

Midline Continued

y = sin(x)

y = sin(x) + 1

Midline moved up 1

**Notice: The amplitude did not

change.

Given the standardequation y = a sin (bx),

*How does “a” affect the graph?

*How does “b” affect the graph?

The “a” affects the height of the graphA negative “a” reflects the graph over the x-axis

The “b” affects the period of the graphRemember, 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.

360| |B

2| |

PerB

( ) sin( )f x A Bx

Think of the pi as 180 degrees!!!

360| |B

360 3601

360 360

1

360 720(1/ 2)

360 720(1/ 2)

360 1802

360 1802

( ) sin( )f x A Bx 2(180)| |

PerB

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, 7For each problem, graph one period in the

positive direction and one period in the negative direction.

Remember to label the axes!

Homework