4.6 Graphs of Other Trigonometric Functions

Graph of Tangent Function:

Period:, 
Amplitude:  none, graphs go on forever in vertical directions.
Domain: all 
Range:
Vertical Asymptote:

Important Things to Remember:

• The function   is odd, so its graph will be symmetricwith respect to the origin.
• Because  anytime  there will be an x intercept.
• And anytime  there will be a vertical asymptote.

Sketching the Graph:

1. Find the vertical asymptotes so you can find the domain.
2. Determine values for the range.
3. Calculate the graph's x-intercepts.
4. Figure out what's happening to the graph between the intercepts and the asymptotes.

Graph of Cotangent Function:

Period:

Amplitude:  none, graphs go on forever in vertical directions.

Domain:

Range:

Vertical Asymptotes:

Sketching the Graph of a Cotangent Function:

1. Find the vertical asymptotes so you can find the domain.
2. Determine values for the range.
3. Calculate the graph's x-intercepts.
4. Figure out what's happening to the graph between the intercepts and the asymptotes.

Graphs of the Reciprocal Functions:

Period:

Domain:

Range: all y not in (-1,1)

Vertical Asymptotes:, at 

Symmerty: Origin

Period:

Domain: all 

Range:all y not in (-1,1)

Vertical Asymptotes:

Symmerty: y-axis

Comparing the graphs of the secant and cosecant functions with those of the sine and cosine functions, note that the "peaks" and "pits" are interchanged.

Sketching Reciprocal Functions:

1. Make a sketch of the reciprocal function
2. Find the x intercepts of the graph and draw vertical asymptotes at such points.
3. Calculate what happens to the graph at the first interval between the asymptotes.
4. Repeat Step 2 for the second interval
5. Repeat Step 2 for the last interval
6. Find the domain and range of the graph.

Damped Trigonometric Graphs

A product of two functions can be graphed using properties of the individual functions. Such as...

as the product of the functions  and . Using properties of absolute value and the fact that   , you have . Consequently...

which means that the graph of  lines between the lines  and .

In the equation of  the factor x is called the damping factor.