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Graphs of Trigonometric FunctionsPowerPoint Presentation

Graphs of Trigonometric Functions

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0

cos x

1

0

-1

0

1

y = cos x

y

x

Graph of the Cosine Function

Cosine FunctionTo 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 FunctionTo 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 FunctionsThe 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.

Key Steps in Graphing Sine and Cosine

- 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 FunctionAt 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 FunctionAt 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.

Key Steps in Graphing Secant and Cosecant

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

2. range: (–, +)

3. period:

Key Steps in Graphing Tan and Cot

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 xx

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 xx

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 xx

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 xx

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 + 4x

Secant Function9. 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 Function10. 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 Function11. 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 Functionx

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 Functionx

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 Functionx

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

.

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