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:
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