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Accelerated Precalculus Name: _____________________________ Graphing Other Trig Functions

Graphing Secant and Cosecant Functions

We will use the GC to explore the relationship of reciprocal functions.

Using 2 different colors, graph both functions on the same grid over 2 periods.

1. ( ) sinf x x= , ( ) cscg x x=

2. ( ) ( )cos , secg x x g x x= =

2

3. ( ) ( )sin 2 1, csc2 1f x x g x x= − − = − −

4. ( ) ( )2cos , 2sec2 2

f x x g x x

= + = +

What is the relationship of the functions?

What do you notice about their graphs?

How to graph secant and cosecant functions:

1. Graph the corresponding cosine (for secant) or sine (for cosecant) wave.

2. Identify the midline of the graph.

3. Draw asymptotes through the points that cross the midline (vertical through zeros

of the sine or cosine function).

4. Draw U’s between the asymptotes.

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Graph 2 periods of the given function. *VS means vertical stretch.

1.

VS_______period_______phase shift_______vertical shift_______reflection_______midline_______

x 2− 3

2

− −

2

− 0

2

2

3 2

( ) cos=f x x 1 0 - 1 0 1 0 - 1 0 1

2. VS_______period_______phase shift_______vertical shift_______reflection_______midline_______

x 2− 3

2

− −

2

− 0

2

2

3 2

( ) sinf x x= 0 1 0 - 1 0 1 0 - 1 0

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3. VS _______period_______phase shift_______vertical shift_______reflection_______midline_______

4. ( )

= +

csc 22

f x x

VS_______period_______phase shift_______vertical shift_______reflection_______midline_______

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Graphing Tangent and Cotangent Functions Complete the tables below then graph the points and connect them with a smooth curve:

x - 2 - 7

4

-2

3 - 5

4

− - 3

4

-2

-

4

0 4

2

3

4

5

4

2

3

7

4

2

Tanx

x - 2 - 7

4

-2

3 - 5

4

− - 3

4

-2

-

4

0 4

2

3

4

5

4

2

3

7

4

2

Cotx

6

( ) tanf x x=

Remember: sin

c stan

o

xx

x=

( ) cotf x x=

Since Tangent and Cotangent are reciprocals…the VA become the x-intercepts and the x-intercepts become the

VA; the entire graph is reflected vertically.

You can also think of it using the quotient identity cos

sicot

n=

xx

x and find the vertical asymptotes and zeros from

there

Tangent has vertical asymptotes

whenever cosx=0

Tangent has zeros

(x-intercepts) whenever sinx=0

2

2

−

3

2

−

3

2

2− −

2

2

If you plot enough

points between the

asymptotes and zeros

then you will find

that the tangent

graph looks like this

2

3

2

2 2

− − 3

2

−

2−

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Properties of Tangent and Cotangent Functions and their graphs:

F(x)=Tan(x)

▪ Period:

▪ Odd : f(-x)=-f(x)

▪ Symmetric: origin

▪ Domain: x Odd Multiples of 2

▪ Range: ( )− ,

▪ Zeros: Multiples of

F(x)=Cot(x)

▪ Period:

▪ Odd : f(-x)=-f(x)

▪ Symmetric: origin

▪ Domain: x Multiples of

▪ Range: ( )− ,

▪ Zeros: Odd Multiples of 2

General Forms:

tan ( )

cot ( )

= − +

= − +

y a b x c d

y a b x c d

If a >1, the graph is vertically stretched. If 0< a <1, the graph is vertically compressed.

*For tangent and cotangent, this just speeds up the graph or slows it down.

b

=period (how long does it take for one complete graph to form)

*Notice this is different from the period of sine and cosine functions.

a < 0 reflects vertically b < 0 reflects horizontally

c = phase shift (right or left) d = vertical shift (up or down)

In order to graph tangent and cotangent functions using transformations, begin with the parent

table of values.

The new x-values are found using +1

x cb

, and

the new y-values are found using ay d+ .

Next page: Complete the table and graph 2 periods.

ACC Pre-Calculus Name:_________________________

Graphing Tangent and Cotangent Functions Date___________________Blk_____

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1) 2 tan 3y x= VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

2) 1

cot 12

y x= − + VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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3) 3cot2

y x

= −

VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

4) tan(3 )y x = − VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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5) ( )cot 5 2y x= − + VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

6) 5tan 3 22

y x

= + +

VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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Practice Graphing (all 6) Trig Functions

Graph each function over 2 periods using transformations and identify characteristics. Be sure to label axes.

1. 1)4sin(2 +−= xy Amp: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

2. 28cos3 +−= xy Amp: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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3. 82tan += xy VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

4. )6

(3cot5

1 −= xy VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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5. )5sec(4 += xy VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

6. 2csc4 3y x= − − VS: _____ Period:_____ Phase Shift:_____ Vertical Shift:_____ Reflections:___

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7. Using the graphs below, write an equation of the function two different ways, once as a sine function and

once as a cosine function.

a. Sine: ____________________________ a. Sine: ____________________________

b. Cosine: __________________________ b. Cosine: __________________________

X

Y

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

0

X

Y

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

0

8. Using the graphs below, write an equation of the tangent (or cotangent) function.

a. _________________________________ b. ____________________________________

X

Y

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

0

X

Y

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

0

8

2

6

4