Lesson 4-6Graphs of Secant and

Cosecant

2

Get out your graphing calculator…

Graph the following

y = cos x

y = sec x

What do you see??

3

2

3

y

x

2

2

2 3

2

5

4

4

xy cos

Graph of the Secant Function

2. range: (–,–1] [1, +)

3. period: 24. vertical asymptotes:

kkx 2

1. domain : all real x)(

2 kkx

cos

1sec

xx The graph y = sec x, use the identity .

Properties of y = sec x

xy sec

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

4

First graph:

• y = 2cos (2x – π) + 1

Then try:

• y = 2sec (2x – π) + 1

5

Graph

Graph the following

y = sin x

y = csc x

What do you see??

6

2

3

x

2

2

2

2

5

y

4

4

Graph of the Cosecant Function

2. range: (–,–1] [1, +) 3. period: 2

where sine is zero.

4. vertical asymptotes: kkx

1. domain : all real x kkx

sin

1csc

xx To graph y = csc x, use the identity .

Properties of y = csc x xy csc

xy sin

At values of x for which sin x = 0, the cosecant function

is undefined and its graph has vertical asymptotes.

7

First graph:

• y = -3 sin (½x + π/2) – 1

Then try:

• y = -3 csc (½x + π/2) – 1

8

Key Steps in Graphing Secant and Cosecant

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

2. Find the new period (2π/b)3. Find the new beginning (bx - c = 0)4. Find the new end (bx - c = 2π)5. 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

6. Adjust the y values of the key points by applying the change in height (a) and the vertical shift (d)

7. Using the original zeros, draw asymptotes, maximums become minimums, minimums become maximums…

8. Graph key points and connect the dots based upon known shape

Graphs of Tangent and Cotangent Functions

10

Tangent and Cotangent

Look at:ShapeKey pointsKey featuresTransformations

Graph

Set window

Domain: -2π to 2π

x-intervals: π/2

(leave y range)

Graph

y = tan x

11

12

y

x

2

3

2

32

2

Graph of the Tangent Function

2. range: (–, +)

3. period:

4. vertical asymptotes: nnx

2

1. domain : all real x nnx

2

Properties of y = tan x

period:

To graph y = tan x, use the identity .x

xx

cos

sintan

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

13

Graph

y = tan x and y = 4tan x in the same window

What do you notice?

y = tan x and y = tan 2x

What do you notice?

y = tan x and y = -tan x

What do you notice?

Graph

Set window

Domain: 0 to 2π

x-intervals: π/2

(leave y range)

Graph

y = cot x

14

15

Graph of the Cotangent Function

2. range: (–, +)

3. period: 4. vertical asymptotes:

nnx

1. domain : all real x nnx

Properties of y = cot x

y

x

2

2

2

32

3

2

xy cot

0xvertical asymptotes xx 2x

To graph y = cot x, use the identity .x

xx

sin

coscot

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

16

Graph Cotangent

y = cot x and y = 4cot x in the same window

What do you notice?

y = cot x and y = cot 2x

What do you notice?

y = cot x and y = -cot x

What do you notice?

y= cot x and y = -tan x

17

Key Steps in Graphing Tangent and Cotangent

Identify the key points of your basic graph1. Find the new period (π/b)2. Find the new beginning (bx - c = 0)3. Find the new end (bx - c = π)4. 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

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

6. Graph key points and connect the dots