Post on 18-Jan-2016
Periodic Functions.A periodic function is a function f such the f(x) = f(x + np) for every real number x in the domain of f, every integer n, and some positive real number p. The least possible positive value of p is the period of the function.
2sinsin nxx 2coscos nxx
Graphing Sine and Cosine functions.
30
60
3
12
sin,cos
30
60
3
12
sin,cos
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy sin
x
y
1. Plot the 5 quadrant values for 1 period.
2. Repeat to the left.
3. For better accuracy, consider plotting when sin (x) = ½.
These points are one tick mark left and right of the x intercept. y = sin(x) is an odd function. f(-x) = – f(x)
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy cos
x
y
1. Plot the 5 quadrant values for 1 period.
2. Repeat to the left.
3. For better accuracy, consider plotting when sin (x) = ½.
These points are one tick mark left and right of the x intercept.
2
2
3 22
1
2
3
1
2
3
y = sin (x)
y = cos (x)y = 1
y = -1
The Amplitude of a Sine or Cosine curve is half the distance from the relative minimum value and relative maximum value. The distance between -1 to 1 is 2 and half of 2 is 1. The amplitude is 1. The Amplitude is located in the equation.
xAy cos xAy sin
1
1
The Amplitude of a Sine or Cosine curve is |A| .
x
y
2
2
3 22
1
2
3
1
2
3
x
yGraph by hand…
xy cos2
1
xy sin3
2
1
2
1A
33 A
The negative on the 3 will make the Sine curve flip over the x – axis.
2
2
3 32 4
1
2
3
1
2
3
x
yChanging the Period Length.
This is a Horizontal Stretch or Shrink. We need to multiply a constant to the x in the function.
Bxy cos
Bxy sin
BPeriod
2
Graph.
xy 2sin
2
2
2
Period
B
xy
2
1cos
4
212
2
1
Period
B
2
2
3 22
1
2
3
1
2
3
x
yPhase Shift.This is a Horizontal Shift left or right.
CBxy cos CBxy sin
B
Cx
CBx
0
Set Bx – C = 0 and solve for x.
x = + , shift Right.
x = – , shift Left.
xy sin
Graph.
x
x 0
Left pi units.
2
2
3 22
1
2
3
1
2
3
x
yVertical Shift.
Dxy cos
Dxy sin
+D = shift Up.
– D = shift Down.
2sin xy
Graph.
Up 2 units.
2
2
3 22
1
2
3
1
2
3
x
y
1cos2 xy
Graph.
Plot the first 5 points according to the Amplitude.
Shift the 5 points up 1 unit.
Draw in the Cosine curve for one period.
Fill the graph with as many Periods as needed.
22 A
Determine all the Transformations.
42cos3 xy 53
sin2
xy
Amplitude ________________
Period ___________________
Phase Shift _______________
Vertical Shift _____________
Amplitude ________________
Period ___________________
Phase Shift _______________
Vertical Shift _____________
33 A
radB
2
22
A B C D
2;
2;02
Rtxx
4Down
A B C D
22 A
rad63
2
3
2
03
x
3;33
Ltx
5Up
Graphing Tangent and Cotangent functions.
x
xy
cos
sin
1,0
1,0
0,1 0,1
2
2,
2
2
2
2,
2
2
x
xxy
sin
coscot
1,0
1,0
0,1 0,1
2
2,
2
2
2
2,
2
2
xy cot
x6
0
17.13 undefined
4
3
6.03
3
1512
7.3
536
4.11
2
0
1180
3.57
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy tan
x
y
1. Plot vertical asymptotes at every odd .
2. x – intercepts are at every . k is an integer. (halfway between V.A.)
3. Halfway between the asymptotes and x – int. the y values are -1 and 1 from left to right.
( Quarter points)
Make what looks like a cubic curve and don’t touch the asymptotes.
2
k
Domain:
Range:
kx
2
,
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy cot
x
y
1. Plot vertical asymptotes at every . k is an integer.
2. x – intercepts at every odd .
(halfway between V.A.)
3. Halfway between the asymptotes and x – int. the y values are 1 and -1 from left to right at the quarter points.
Make what looks like a cubic curve and don’t touch the asymptotes.
2
k
Remember, tangent is increasing and cotangent is decreasing.
Domain:
Range:
kx
,
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy sec
x
y
1. Plot cos(x) as a reference.
2. V.A. Through the x – int. of the cosine curve.
3. Plot the points at the relative maxs. and mins. of cos(x) curve.
4. Plot the reciprocal value of the y – values of ½ of cos(x) as a 2 for sec(x) curve.5. Draw the curves as wide parabolas near the vertex.
Domain:
Range:
kx
2
,11,
How to graph by hand.
2
2
3 22
1
2
3
1
2
3
xy csc
x
y
1. Plot sin(x) as a reference.
2. V.A. Through the x – int. of the sine curve.
3. Plot the points at the relative maxs. and mins. of sin(x) curve.
4. Plot the reciprocal value of the y – values of ½ of sin(x) as a 2 for csc(x) curve.5. Draw the curves as wide parabolas near the vertex.
Domain:
Range:
kx
,11,
2
2
1
2
3
1
2
3
x
y
How to graph y = Atan(Bx – C) + D by hand.
1. Determine the Vertical Asymptotes.
Solve for x.
2
CBx
2
CBx
2. The x – intercept is halfway between
the vertical asymptotes.
BPeriod
3. The quarter points have y – values of
– A and + A, from left to right.
There is no amplitude. If you have – A
in the equation, flip over x – axis.
**If you have – B in the equation, flip over y – axis.
4. D is still used for the Vertical Shift.
Bx – C = 0 is still used to find the Phase Shift.
Add this value to the V.A. equations.
0
1
2
3
1
2
3
x
y
How to graph y = Acot(Bx – C) + D by hand.
1. Determine the Vertical Asymptotes.
Solve for x.
0 CBx CBx
2. The x – intercept is halfway between
the vertical asymptotes.
BPeriod
3. The quarter points have y – values of
+ A and + – A, from left to right.
There is no amplitude. If you have – A
in the equation, flip over x – axis.
**If you have – B in the equation, flip over y – axis.
Bx – C = 0 is still used to find the Phase Shift.
Add this value to the V.A. equations.
4. D is still used for the Vertical Shift.
How to graph by hand.
DCBxAy sec
2
1
2
3
1
2
3
y
2
3x
1. Determine the Vertical Asymptotes.
Solve for x.
2
CBx
2
CBx
BPeriod
2
2
3 CBx
**If you have – B in the equation, flip over y – axis.
2. The vertex is halfway between the vertical
asymptotes and is located at A and - A. If A is
negative, then flip over x – axis.
3. The reciprocal points should be multiplied by A.
Bx – C = 0 is still used to find the Phase Shift.
Add this value to the V.A. equations.
2
4. D is still used for the Vertical Shift.
I recommend that we make the changes to the Cosine curve 1st and then draw in the Secant curve in between the asymptotes.
How to graph by hand.
2
2
3 2
1
2
3
1
2
3
DCBxAy csc
x
y
1. Determine the Vertical Asymptotes.
Solve for x.
0 CBx CBx
BPeriod
2
2 CBx
**If you have – B in the equation, flip over y – axis.
2. The vertex is halfway between the vertical
asymptotes and is located at A and - A. If A is
negative, then flip over x – axis.
3. The reciprocal points should be multiplied by A.
Bx – C = 0 is stilled used to find the Phase Shift.
Add this value to the V.A. equations.
4. D is still used for the Vertical Shift.
I recommend that we make the changes to the Sine curve 1st and then draw in the Cosecant curve in between the asymptotes.
2
2
3 22
1
2
3
1
2
3
x
y
142
tan
xy
Graph
Shift the curve down 1 unit.
Find the location of the VA’s
Draw the tangent curve.
The x-intercept is halfway between the VA’s
242
x
242
x
4
4
4
4
4
3
2
x
42
x
24
3
22
x2
422
x
2
3x
2
x
Copy the graph to the right and left.
The quarter points are at half of the halves with y coordinates of 1 and -1.
2
2
3 22
1
2
3
1
2
3
x
y
142
tan
xy
Graph
BPeriod
Draw one period of y = tan(x).
Find the period length.
21
2
1
2
Find the shift.
042
x
42
x
24
x2
2
Shift to the left .
Double the radians for the points.
Down 1.
2
Copy the graph to the right and left.
2
2
3 22
1
2
3
1
2
3
x
y13
cot2
xy
Graph
Multiplying 2 to every y-coordinate.
Vertical shift up 1 unit.Copy to the right and left.
Find the location of the VA’s
03
x
3x
3
3
3
3
3
x 3
4x
The x-intercept is halfway between the VA’s
The quarter points are at half of the halves with y coordinates of 1(A) and -1(A). A = 2, so 2 & -2.
Draw the cotangent curve.
2
2
3 22
1
2
3
1
2
3
x
y13
cot2
xy
Graph
Draw one period of y = cot(x).
Find the period length.
BPeriod
1
No change.
Find the shift.
03
x
3
x
Shift to the right . 3
Multiplying 2 to every y-coordinate.
Vertical shift up 1 unit. Copy to the right and left.
2
2
3 22
1
3
1
2
3
x
y
1sec2
1 xy
Graph
No changes inside the secant function.Start with the basic curve of Cosine from Quadrant 4 to Quadrant 3
Vertical Asymptotes at the x-intercepts.
A = ½, multiply all y-coordinates by ½.
D = 1, vertical shift up one unit.
Draw in the Secant Curve…remember that the y-coordinates of ½ for Cosine are now flipped and the value is 2 for the y-coordinate.
2
Copy to the right and left.
2
2
3 22
1
2
3
1
2
3
x
y
Graph xy 2csc
BPeriod
2
2
2
2,,0 CBx
2
1. V.A. Solve for x.
2,,02 x
3,2,2 x
2
2
3,,
2
x
Draw the Sine Curve for reference.
Draw in the Cosecant Curve.
Copy the curve into the rest of the graph.
Change the MODE to DegreesEnter the equation into Y =
ZOOM 7 to activate the Trig window
2
2
3 22
1
2
3
1
2
3
x
y