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# Graphs of Trigonometric Functions - PowerPoint PPT Presentation

Graphs of Trigonometric Functions. x. 0. cos x. 1. 0. -1. 0. 1. y = cos x. y. x. Graph of the Cosine Function. 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.

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### Graphs of Trigonometric Functions

0

cos x

1

0

-1

0

1

y = cos x

y

x

Graph of the Cosine Function

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.

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.

0

sin x

0

1

0

-1

0

y = sin x

y

x

Graph of the Sine Function

Sine Function

To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

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.

2. The range is the set of y values such that .

5. Each function cycles through all the values of the range over an x-interval of .

Properties of Sine and Cosine Functions

Properties of Sine and Cosine Functions

The graphs of y = sin x and y = cos x have similar properties:

1. The domain is the set of real numbers.

3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.

6. The cycle repeats itself indefinitely in both directions of thex-axis.

• Identify the key points of your basic graph

• Find the new period (2π/b)

• Find the new beginning (bx - c = 0)

• Find the new end (bx - c = 2π)

• Divide the new period into 4 equal parts to create new interval for the x values in key points

• Adjust the y values of the key points by applying a and the vertical shift (d)

The graph y = sec x, use the identity .

y

Properties of y = sec x

1. domain : all real x

x

4. vertical asymptotes:

Graph of the Secant Function

Secant Function

At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

2. range: (–,–1]  [1, +)

3. period: 2

To graph y = csc x, use the identity .

y

Properties of y = csc x

1. domain : all real x

x

4. vertical asymptotes:

Graph of the Cosecant Function

Cosecant Function

At values of x for which sin x = 0, the cosecant functionis undefined and its graph has vertical asymptotes.

2. range: (–,–1]  [1, +)

3. period: 2

where sine is zero.

• Identify the key points of your reciprocal graph (sine/cosine), note the original zeros, maximums and minimums

• Find the new period (2π/b)

• Find the new beginning (bx - c = 0)

• Find the new end (bx - c = 2π)

• Find the new interval (new period / 4) to divide the new reference period into 4 equal parts to create new x values for the key points

• Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d)

• Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums…

• Graph key points and connect the dots based upon known shape

To graph y = tan x, use the identity .

y

Properties of y = tan x

1. domain : all real x

x

4. vertical asymptotes:

period:

Graph of the Tangent Function

At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.

2. range: (–, +)

3. period: 

To graph y = cot x, use the identity .

Properties of y = cot x

x

1. domain : all real x

4. vertical asymptotes:

vertical asymptotes

Graph of the Cotangent Function

Cotangent Function

At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

2. range: (–, +)

3. period: 

Identify the key points of your basic graph

• Find the new period (π/b)

• Find the new beginning (bx - c = 0)

• Find the new end (bx - c = π)

• Find the new interval (new period / 2) to divide the new reference period into 2 equal parts to create new x values for the key points

• Adjust the y values of the key points by applying the amplitude (a) and the vertical shift (d)

• Graph key points and connect the dots

x

1.Chose the correct equation of the graph: a) y = sin x b) y = cos x c) y = –sin x d) y = –cos x

x

2.Chose the correct equation of the graph: a) y = sin x b) y = cos x c) y = –sin xd) y = –cos x

x

3. Chose the correct equation of the graph: a) y = sin ½ x b) y = 2 sin x c) y = sin 2x d) y = sin x

x

4. Chose the correct equation of the graph: a) y = sin x + 1 b) y = cos x + 1 c) y = 2sin x d) y = 2cos x

x

5. Chose the correct equation of the graph: a) y = sin x + 1 b) y = sin (x + π/2)c) y = sin (x – π/2)d) y = sin x

x

6. Chose the correct equation of the graph: a) y = sin 4x b) y = 4 sin x c) y = sin x + 4 d) y = sin (x + 4)

x

7. Chose the correct equation of the graph: a) y = sin ½ x b) y = 2 sin x c) y = sin 2x d) y = sin x

x

8. Chose the correct equation of the graph: a) y = 4sin x b) y = 3 sin x + 1 c) y = sin 3x + 1 d) y = sin x + 4

x

Secant Function

9. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x

x

Secant Function

10. Chose the correct equation of the graph: a) y = sec x

b) y = 2sec x c) y = sec 2x

d) y = sec x + 2

x

Secant Function

11. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x

x

12. Chose the correct equation of the graph: a) y = sec x

b) y = csc x c) y = -sec x

d) y = -csc x

.

Cosecant Function

x

13. Chose the correct equation of the graph: a) y = csc x

b) y = csc 2x c) y = 2csc x

d) y = csc x – 2

.

Cosecant Function

x

14. Chose the correct equation of the graph: a) y = tan x

b) y = cot x c) y = -tan x

d) y = -cot x

.

Cotangent Function

x

period:

15. Chose the correct equation of the graph: a) y = tan x

b) y = cot x c) y = -tan x

d) y = -cot x

.

period:

x

16. Chose the correct equation of the graph: a) y = 2tan x

b) y = tan 2x c) y = -2cot x

d) y = -cot 2x

.