5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

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1 Chapter 5: Trigonometric Functions and Graphs 1

2 Chapter Graphing Sine and Cosine Functions Pages Complete the following table using your calculator. Round answers to the nearest tenth. 2

3 y 1 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o 270 o 300 o 330 o 360 o 1 3

4 y 1 30 o 60 o 90 o 120 o 150 o 180 o 210 o 240 o 270 o 300 o 330 o 360 o 1 4

5 Periodic Functions a function that repeats itself over regular intervals(cycles) of its domain. Period the horizontal length of the interval of the domain over which a graph repeats itself. Examples: 5

6 Amplitude the maximum vertical distance of a periodic (sinusoidal) function varies above and below the horizontal central axis of the curve. Formula: Examples: 6

7 Function Domain Range Period Amplitude x intercepts y intercepts Standard Tables: y OR y y OR y 7

8 8

9 Example 2 Page 226 Determine the Amplitude of a Sine Function Using your knowledge of transformations and the standard tables: a) On the same set of axes, graph y = 3 sin x, y = 0.5 sin x, and y = 2 sin x for 0 x 2π. b) State the amplitude for each function. c) Compare each graph to the graph of y = sin x. Consider the period, amplitude, domain, and range. 9

10 Example 2: Your Turn Page 227 a) On the same set of axes, graph y = 6 cos x and y = 4 cos x for 0 x 2π. b) State the amplitude for each graph. c) Compare your graphs to the graph of y = cos x. Consider the period, amplitude, domain, and range. d) What is the amplitude of the function y = 1.5 cos x? Answer c), d) Answer a), b) 10

11 For the function Amplitude = 11

Example 3 Page 228 Determine the Period of a Sine Function Using your knowledge of transformations and the standard tables: a) Sketch the graph of the function y = sin 4x for 0 x 360.

12 Example 3 Page 228 Determine the Period of a Sine Function Using your knowledge of transformations and the standard tables: a) Sketch the graph of the function y = sin 4x for 0 x 360. State the period of the function and compare the graph to the graph of y = sin x. b) Sketch the graph of the function y = sin for 0 x 4π. State the period of the function and compare the graph to the graph of y = sin x. a) 12

13 b) 13

Example 3: Your Turn Page 229 a) Sketch the graph of the function y = cos 3x for 0 x 360. State the period of the function and compare the graph to the graph of y = cos x.

14 Example 3: Your Turn Page 229 a) Sketch the graph of the function y = cos 3x for 0 x 360. State the period of the function and compare the graph to the graph of y = cos x. b) Sketch the graph of the function y = cos x for 0 x 6π. State the period of the function and compare the graph to the graph of y = cos x. c) What is the period of the graph of y = cos ( 3x)? Answer b), c) Answer a) 14

15 To determine the period from an equation: To determine the b value to write an equation, just rearrange the formula: 15

16 Example 4 Page 229 Sketch the Graph of y = a cos bx a) Sketch the graph of y = 3 cos 2x for at least one cycle in radians. b) Determine the amplitude the period the maximum and minimum values the x intercepts and the y intercept the domain and range 16

17 Example 4: Your Turn Page 232 a) Graph y = 3 sin 4x, showing at least two cycles in radians. b) Determine the amplitude the period the maximum and minimum values the x intercepts and the y intercept the domain and range Answer 17

18 More Examples: 1. State the amplitude and period, in degrees and radians, of each of the following sinusoidal functions. a) b) c) 18

19 2. Write an equation of the cosine function with the given characteristics: a) amplitude of 3, period b) amplitude of 7, period 150 o 3. Write an equation of the sine function that has: a) amplitude of 0.5, period 720 o b) amplitude of, reflection over the x axis, period 19

20 4. Write an equation for each of the following graphs. a) cosine graph 20

21 b) sine graph 21

Section 5.2 Graphs of the Sine and Cosine Functions

A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in