section 4.1 basic graphs. periodic function a function that repeats its values in regular intervals
TRANSCRIPT
Chapter 4:Graphing &
Inverse Functions
Section 4.1 Basic Graphs
periodic functiona function that repeats its values in regular intervals
is periodic if , .f x f x p f x x
x x p
f x f x p
Examine each of the following graphs.
Does the graph represent
a periodic function?
Yes
Yes
No
Yes
No
No
Yes
No
amplitudemagnitude of change in the oscillating variable
If the greatest value of f(x) is M and the least value of f(x) is m, then the amplitude of the graph of f(x) is
1
2A M m
M
m
1
2A M m
M
m
11 1
2A
12
2A
1A
For trig functions,
amplitude is distance
to max or min value
from the midline.
What is the amplitude of the periodic function?
1.75
What is the amplitude of the periodic function?
2
What is the amplitude of the periodic function?
1.5
What is the amplitude of the periodic function?
undefined
none
periodsmallest positive distance at which a function repeats
For any function f(x), the smallest positive number p for which
is called the period of f(x).
,f x p f x x
2p 0flet 0x
2f 2then x p
f x f x p
0 0
0 2f f
,f x p f x x
2p
2f
2let x
3
2f
2hen
3t x p
f x f x p
1 1
3
2 2f f
,f x p f x x
2p f x f x p ,f x p f x x
3et
2l x
2then x p
3
2f
2
f
3
2 2f f
1 1
What is the period of the periodic function?
6.5
What is the period of the periodic function?
3p
What is the period of the periodic function?
2
What is the period of the periodic function?
1
Look for repeatingy-values to determine
period…butthe period is
the length along the x-
axis.
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0
0,1
0, 1
4
2
3
4
5
4
3
2
7
4
6
3
2
3
5
6
7
6
4
3
5
3
11
6
1,00 2or
Sine Graph: f(q)= sin q
qsine
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
Sine Graph: f(q)= sin q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
Sine Graph: f(q)= sin q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
Sine Graph: f(q)= sin q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
Sine Graph: f(q)= sin q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
3 1,
2 2
6
2 2,
2 2
4
1 3,
2 2
3
Sine Graph: f(q)= sin q
Amplitude:
Period:
Sine Graph: y = sin q
Domain:
Range:
y-int:
Graph of one cycle
all
1 1y
1
2 0,0
1,1
,
Cosine Graph: f(q)= cos q
q
cosine
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
cos ,sin
q
cosine
Cosine Graph: f(q)= cos q
Cosine Graph: f(q)= cos q
Graph of one cycle
Cosine Graph: y = cos q
Domain:
Range:
y-int:
all
1 1y
1
2 0,1
Amplitude:
Period:
Graph of one cycle
q
tangent
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
sintan
cos
0
tan1
tan 0
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
1tan
0
sintan
cos
tan undefined
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
0tan
1
sin
tancos
tan 0
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
1tan
0
sintan
cos
tan undefined
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
tan 0
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
sintan
cos
22 22tan
22 22
tan 1
22tan2
2
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
22 22tan
22 22
sintan
cos
22tan
22
tan 1
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
tan 1
22 22tan
22 22
sintan
cos
22tan2
2
Tangent Graph: f(q)= tan q
2 2,
2 2
2 2,
2 2
2 2,
2 2
2 2,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
3 1,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1 3,
2 2
1,0 1,0
0,1
4
2
3
4
5
4
7
4
0 2or
6
3
2
3
5
6
7
6
4
3
5
3
11
6
0, 13
2
sintan
cos
tan 1
22 22tan
22 22
sintan
cos
22tan2
2
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Graph of one cycle
Graph of one
cycle is
usually to
2
2
Amplitude:
Period:
Asymptotes:
Tangent Graph: y = tan q
Domain:
Range:
y-int:
all ,2
x k
all undefined
0,0
2x k
Graph of one cycle
k
Cosecant Graph: y = csc q
To graph the cosecant, start
with its reciprocal: sine.
y = sin x
Cosecant Graph: y = csc q
Where sin Ѳ = 0, csc Ѳ is undefined.These are asymptotes.
Cosecant Graph: y = csc q
Where sin Ѳ = 1, csc Ѳ = 1.Where sin Ѳ = -1, csc Ѳ = -1.
Cosecant Graph: y = csc q
Where sin Ѳ = ½ (/6, 5/6), csc Ѳ = 2.Where sin Ѳ = -½ (7/6, 11/6), csc Ѳ = -2.
Cosecant Graph: y = csc q
Repeat for reciprocals of other y-values.
Amplitude:
Period:
Asymptotes:
Cosecant Graph: y = csc q
Domain:
Range:
y-int:
all , x k
1 or 1y y
undefined
2none x k
Graph of one cycle
, 1 1,
Secant Graph: y = sec q
To graph the secant, start with
its reciprocal: cosine.
y = cos x
Secant Graph: y = sec q
Where cos Ѳ = 0, sec Ѳ is undefined.These are asymptotes.
Secant Graph: y = sec q
Add points where y = 1 and y = -1.Sketch rest of graph using asymptotes.
Amplitude:
Period:
Asymptotes:
Secant Graph: y = sec q
Domain:
Range:
y-int:
all ,2
x k
1 or 1y y undefined
2 0,1
2x k
Graph of one cycle
Cotangent Graph: y = cot q
To graph the cotangent, you could start with its reciprocal:
tangent.
Cotangent Graph: y = cot q
Where tan Ѳ = 0, cot Ѳ is undefined.These are asymptotes. Where tan Ѳ is undefined, cot Ѳ = 0.
Cotangent Graph: y = cot q
Reciprocals points are graphed.
Cotangent Graph: y = cot q
Graph of one cycleTo graph the cotangent, it may be easier to remember the asymptotes and
change to a downward slope.
Cotangent Graph: y = cot q
Domain:
Range:
y-int:
Amplitude:
Period:
Asymptotes:
all , x k
all undefined
none x k
Graph of one cycle
Things to notice:•The period for tan and cot is π. For all other trig functions, it is 2π.• If the graph goes up or down to infinity, the amplitude is “undefined” or none.•Asymptotes, and domain restrictions, are the same for tan and sec (π/2 + πk).•Asymptotes, and domain restrictions, are the same for cot and csc (πk).•All trig asymptotes have + πk, even when the period is 2π.
The End.