I. Graph of the Tangent Function: y = a tan (bx - c) + d

A. The tangent function is odd. tan (-

*x*) = - tan

*x*

B. Therefore the graph of the y = tan x is symmetric with respect to the origin.

C. Period = π

D. Domain: all the reals except

*x*≠ π /2 + n π

E. Range: all the reals

F. Vertical Asymptotes:

*x*= π /2 + n π

II. To sketch the graph y =

*a*tan (

*bx*-

*c*) +

*d*

A. Locate key points that identify the intercepts and asymptotes.

1. to find the asymptotes:

*bx*-

*c*= -π /2

**and**

*bx*-

*c*= π /2

2. The midpoint between two consecutive asymptotes is an

*x*-intercept of the graph.

3. The period of the tangent function is the distance between two consecutive asymptotes. This can also be found by: period = π /

*b*

4. The amplitude of a tangent function is not defined. But if “

*a*” is a negative number, the graph is a reflection in the

*x*-axis

Example 1:

*y*= tan 2

*x*

*bx*-

*c*= -π /2

**OR**b

*x*- c = π / 2

2

*x*= -π /2

**OR**2

*x*= π /2

*x*= -π /4

**OR**

*x*= π /4

(-π /2 + π /2)/2 = 0 so the midpoint is (0,0)

period = π /

*b*= π /2

So therefore some of the key points are: (-π /2 , 0 ), (0,0) , (π /2 , 0)

and some of the asymptotes are

*x*= -3π /4 ,

*x*= -π /4,

*x*= π /4,

*x*= 3π /4

Example 2:

*y*= -3 tan (

*x*/4)

*bx*-

*c*= -π /2

**OR**

*bx*-

*c*= π /2

*x*/4 = -π /2

**OR**

*x*/4 = π /2

*x*= -2π

**OR**

*x*= 4π

(-2π + 2π )/2 = 0 so the midpoint is (0,0)

period = π /

*b*= π /(1/4) = 4π

**Or**(2π --2π ) = 4π

So therefore some of the key points are: (-π , 3 ), (0,0) , (π , -3) and some of the asymptotes are

*x*= -2π ,

*x*= 2π

III. Graph of the Cotangent function

*y*= cot

*x*= (cos

*x*)/(sin

*x*)

has vertical asymptotes wherever sin

*x*= 0 so...

*x*= n π

A.

**The cotangent function is odd**. cot (-

*x*) = -cot

*x*

B. Therefore the graph of the y = cot

*x*is

**symmetric with respect to the orig**in.

C.

**Period**= π

D.

**Domain**: all the reals except x ≠

*n*π

E.

**Range**: all the reals

F.

**Vertical Asymptotes**:

*x*=

*n*π

Some key points are: (-7π/4 , 1), (-3π/2, 0), (-5π/4, -1), (-3π/4, 1), (-π/2, 0), (-π/4, -1)

IV. Cosecant function is

*y*= csc

*x*= 1/(sin

*x*)

so to sketch the graph of the cosecant function, first make a sketch of the sine curve, then take the reciprocals of the y-coordinates to obtain the points needed.

The vertical asymptotes are where sin

*x*= 0 so therefore at

*x*=

*n*π

A.

**The cosecant function is odd**. csc (-

*x*) = -csc

*x*

B. Therefore the graph of the y = csc x is

**symmetric with respect to the origin**.

C.

**Period**= 2π

D.

**Domain**: all the reals except

*x*≠

*n*π

E.

**Range**: all the reals except

*y*≠ [-1,1]

F.

**Vertical Asymptotes**:

*x*=

*n*π

V. Secant function is

*y*= sec

*x*= 1/(cos

*x*)

so to sketch the graph of the secant function, first make a sketch of the cosine curve, then take the reciprocals of the

*y*-coordinates to obtain the points needed.

The

**vertical asy**mptotes are where cos

*x*= 0 so therefore at

*x*= π/2 +

*n*π

A.

**The secant function is even**. sec (-

*x*) = sec

*x*

B. Therefore the graph of the y = sec x is

**symmetric with respect to the y - axis**.

C.

**Period**= 2π

D.

**Domain**: all the reals except

*x*≠ π/2 +

*n*π

E.

**Range**: all the reals except

*y*≠ [-1,1]

F.

**Vertical Asymptotes**:

*x*= π/2 +

*n*π

VI. Damped Trigonometric Graphs:

A product of 2 functions can be graphed using properties of the individual functions.

**Example 1**: f(

*x*) = e

^{-x}cos (

*x*)

key points are (-3 π /2 , 0), (-3.93, -35.89), (-π/2 , 0), (0,1), (π/2 , 0)

The damping factor is e

^{-x}because:

-e

^{-x}≤ e

^{-x}cos

*x*≤ e

^{-x}

As the

*y*-values approach zero, the

*x*-values approach infinity.

**Example**: h(x) = 2

^{(-x/4)}sin

*x*

graph this and see what happens as

*x*approaches zero and as

*y*approaches zero.

**Homework**: #33; pg. 341; #1 - 17 odd, 41, 43, 47, 49, 51-55, 57, 63 - 67 odd, 75,

pg. 330; #12, 15, 21, 29, 45, 99, 101