periodic functions. a periodic function is a function f such the f(x) = f(x + np) for every real...

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)


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


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

,


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

,


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,


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.


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.






DCBxAy sec

2

1

2

3

1

2

3

y

2

3x


Solve for x.

2

CBx

2

CBx

BPeriod

2

2

3 CBx


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.



2


I recommend that we make the changes to the Cosine curve 1st and then draw in the Secant curve in between the asymptotes.


2

2

3 2

1

2

3

1

2

3

DCBxAy csc

x

y


Solve for x.

0 CBx CBx

BPeriod

2

2 CBx


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.



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

periodic functions. a periodic function is a function f such the f(x) = f(x + np) for every real...

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