Pre-Calculus



5.5 Graphing the Other 4 Trig Functions

Cosecant, Secant, Tangent, Cotangent

This is the basic “parent function” of: [pic].

Graphing the COSECANT (CSC) function:

o The cosecant function has a relative minimum at each point the corresponding sine function has a relative maximum. It has a relative maximum at each point the corresponding sine function has a relative minimum.

o The cosecant function has a vertical asymptote at each x-value where the corresponding sine function has a point of inflection (a “zero”).

The values of a, b, c, and d cause the parent function to be shifted, reflected or stretched / compressed, just like with the sine and cosine functions.

Steps to Graph a CSC Function:

1. Graph the corresponding sine function. O-H-O-L-O

2. Draw vertical asymptotes where “sin x = 0”.

3. Flip the sine curve between each pair of asymptotes.

(Different colors are helpful to draw the asymptotes, the sine, and the cosecant curves.)

Graph the following by hand:

1. Graph: [pic]

Start by sketching the graph

of the corresponding sine function.

Determine the domain and range of the function[pic]:

2. [pic] Remember: Start with [pic]

Amplitude:

Period:

Unit:

Phase Shift:

Vertical Shift:

Use your graph to find the domain and range of [pic]

3. [pic] Remember: Start with [pic]

Amplitude:

Period:

Unit:

Phase Shift:

Vertical Shift:

Use your graph to find the domain and range of [pic]

II. Graphing a Secant Function:

The SECANT function can be graphed the same way the cosecant function is graphed – except the key points will be found from the corresponding COSINE function!

4. [pic] Start with [pic].

Amplitude:

Period:

Unit:

Phase Shift:

Vertical Shift:

Use your graph to find the domain and range of [pic]

5. [pic] Start with [pic]

Amplitude

Period:

Unit:

Phase Shift:

Vertical Shift:

Use your graph to find the domain and range of [pic]

III. Graphing Tangent Functions:

The period of the tangent function is [pic]. Therefore, if we can find the shape of the graph over an interval of [pic] units, then the graph will repeat that pattern.

6. Graph: [pic].

Start with a table of values, using points from the unit circle: x y = tan x

[pic]

[pic]

0

[pic]

[pic]

These points will give one period of the function.

Use your graph to find the domain and range of [pic]

Important Features of the tangent curve:

1. The tangent function increases.

2. Asymptotes occur at odd multiples of [pic].

3. Halfway through the period, the function has a zero.

4. One-fourth through the period, the tan = (1, three-fourths through, tan = 1

Let f(x) = a tan(bx – c) + d

Amplitude = [pic] Period = [pic] Unit = [pic]

Vertical Shift = [pic]

2 Consecutive Asymptotes:

[pic] [pic]

Reflection = over x axis if “a” is negative

7. Graph [pic]

Amplitude:

Period:

Unit:

Phase shift:

Vertical shift:

Use your graph to find the domain and range of [pic]

IV. Graphing Cotangent Functions

8. Graph [pic]

Remember: [pic] therefore cot(x) is undefined when sin x = 0

Cotangent is graphed in the same way as tangent, except

1. vertical asymptotes are at …,[pic],… values

2. the function is decreasing

3. cot = 0 halfway between the asymptotes

4. cot = 1 one-fourth of the distance through the period

5. cot = (1 three-fourths of the distance through the period

This is the basic “parent function” of: [pic]

Notice the function decreases!

Let f(x) = a cot(bx – c) + d

Amplitude = [pic] Period = [pic] Unit = [pic] Vertical Shift = [pic]

2 Consecutive Asymptotes:

[pic] [pic]

Reflection = over x axis if “a” is negative

8. Graph [pic]

Amplitude:

Period:

Unit:

Phase shift:

Vertical shift:

-----------------------

x

y

x

y

x

y

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download