Graphs of Tangent and Cotangent Functions

2

Plan for the Day

• Review Homework

• Graphing Tangent and Cotangent

• Homework

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Key Steps in Graphing Sine and Cosine

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

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

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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 amplitude (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

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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

6

7

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.

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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

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10

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.

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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

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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