Trigonometry: Graphing, the tweaks 1

y

0.5 x − 5π 4

−π

− 3π 4

−π 2

−π 4

π 4

π 2

−0.5

by: javier

3π 4

π

5π 4

3π 2

7π 4

2π

graphing just the basics

graphing: just the basics

y 8 6 4 2

1

−2π

− 3π 2

−π

− π2

π 2

π

3π 2

2π

−1

− π2

−2 −4 −6 −8

1

−2π

− 3π 2

−π

− π2

π 2

−1

π

3π 2

2π

x π 2

graphing: just the basics

y

y = sin(x)

8 6 4 2

1

−2π

− 3π 2

−π

− π2

π 2

π

3π 2

2π

− π2

−1

y = cos(x) y = tan(x)

1

−2π

− 3π 2

−π

− π2

π 2

−1

π

3π 2

2π

−2 −4 −6 −8

x π 2

graphing the shift

graphing: the shift

20

the tweak: □ the basic graph

15

y = x2 10 5 0 −4

−2

0

2

4

6

graphing: the shift

20

the tweak: □ the basic graph

15

y = x2 10

□ the tweak

5 0 −4

−2

0

2

4

6

x 7→ x − 2

graphing: the shift

20

the tweak: □ the basic graph

15

y = x2 10

□ the tweak

x 7→ x − 2

5

□ the new equation 0 −4

−2

0

2

4

6

y = (x − 2)2

graphing: the shift

the tweak: □ the basic graph

8 6

y = x2 4 2 0 −2

−1

0

1

2

graphing: the shift

the tweak: □ the basic graph

8 6

y = x2 4

□ the tweak

2 0 −2

−1

0

1

2

y 7→ y − 2

graphing: the shift

the tweak: □ the basic graph

8 6

y = x2 4

□ the tweak

2

y 7→ y − 2

□ the new equation 0 −2

−1

0

1

2

y − 2 = x2

graphing: the shift

□ the basic graph

1.0

y = sin (x)

0.5

1 6

π

1 3

π

−0.5

−1.0

the tweak:

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

graphing: the shift

□ the basic graph

1.0

y = sin (x)

0.5

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y − 2

−0.5

−1.0

the tweak:

1 x 7→ − π + x 2

graphing: the shift

□ the basic graph

1.0

y = sin (x)

0.5

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y − 2

−0.5

1 x 7→ − π + x 2

□ the new equation −1.0

the tweak:

( ) 1 y − 2 = sin − π + x 2

graphing: the shift

□ the basic graph

3.0

y = sin (x)

2.0

□ the tweak

1.0

y 7→ y − 2 1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

13 6

π

7 3

π

5 2

π

1 x 7→ − π + x 2

□ the new equation −1.0

the tweak:

( ) 1 y − 2 = sin − π + x 2

graphing: the shift

□ the basic graph

1.0

y = cos (x)

0.5

1 6

π

1 3

π

−0.5

−1.0

the tweak:

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

graphing: the shift

□ the basic graph

1.0

y = cos (x)

0.5

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y + 1

−0.5

−1.0

the tweak:

x 7→

1 π+x 4

graphing: the shift

□ the basic graph

1.0

y = cos (x)

0.5

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y + 1

x 7→

−0.5

1 π+x 4

□ the new equation (

−1.0

the tweak:

y + 1 = cos

) 1 π+x 4

graphing: the shift

□ the basic graph

1.0

0.5

y = cos (x) −16

1 6

π

π

−0.5

−1.0

−1.5

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y + 1

1 π+x 4

□ the new equation (

−2.0

the tweak:

x 7→

y + 1 = cos

) 1 π+x 4

graphing: the shift

□ the basic graph

20.0

y = tan (x)

10.0

1 6

π

1 3

π

−10.0

−20.0

the tweak:

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

graphing: the shift

□ the basic graph

20.0

y = tan (x)

10.0

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y − 3

−10.0

−20.0

the tweak:

x 7→

1 π+x 2

graphing: the shift

□ the basic graph

20.0

y = tan (x)

10.0

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y − 3

x 7→

−10.0

1 π+x 2

□ the new equation (

−20.0

the tweak:

y − 3 = tan

) 1 π+x 2

graphing: the shift

□ the basic graph

20.0

y = tan (x)

10.0

−12

π

−13

π

−16

π

1 6

π

1 3

π

1 2

π

2 3

π

5 6

π

π

7 6

π

4 3

π

3 2

π

5 3

π

11 6

π

2π

□ the tweak y 7→ y − 3

x 7→

−10.0

1 π+x 2

□ the new equation (

−20.0

the tweak:

y − 3 = tan

) 1 π+x 2

graphing: the shift

□ the basic graph

4

x2 + y2 − 9 = 0

2

0

-2

-4 -4

the tweak:

-2

0

2

4

graphing: the shift

□ the basic graph

4

x2 + y2 − 9 = 0

2

□ the tweak

0

y 7→ y − 3

-2

-4 -4

the tweak:

-2

0

2

4

x 7→ x − 2

graphing: the shift

□ the basic graph

4

x2 + y2 − 9 = 0

2

□ the tweak

0

y 7→ y − 3

-2

x 7→ x − 2

□ the new equation

-4 -4

the tweak:

-2

0

2

4

(x − 2)2 + (y − 3)2 − 9 = 0

graphing: the shift

□ the basic graph

8 6

x2 + y2 − 9 = 0

4

□ the tweak

2 0

y 7→ y − 3

-2

x 7→ x − 2

□ the new equation

-4 -4

-2

the tweak:

0

2

4

6

(x − 2)2 + (y − 3)2 − 9 = 0

graphing the stretch

graphing: the stretch

□ the basic graph

4

x2 + y2 − 9 = 0

2

0

-2

-4 -4

the tweak:

-2

0

2

4

graphing: the stretch

□ the basic graph

4

x2 + y2 − 9 = 0

2

□ the tweak

0

y 7→ 3 y

-2

-4 -4

the tweak:

-2

0

2

4

x 7→ 2 x

graphing: the stretch

□ the basic graph

4

x2 + y2 − 9 = 0

2

□ the tweak

0

y 7→ 3 y

-2

x 7→ 2 x

□ the new equation

-4 -4

the tweak:

-2

0

2

4

4 x2 + 9 y2 − 9 = 0

graphing: the stretch

□ the basic graph

4

x2 + y2 − 9 = 0

2

□ the tweak

0

y 7→ 3 y

-2

x 7→ 2 x

□ the new equation

-4 -4

the tweak:

-2

0

2

4

4 x2 + 9 y2 − 9 = 0

graphing: the stretch

□ the basic graph

3

2

y = sin (x) 1

0

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

graphing: the stretch

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ 3 y

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

x 7→ 2 x

graphing: the stretch

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ 3 y

-1

-2

-3

x 7→ 2 x

□ the new equation 0

1

the tweak:

2

3

4

5

6

3 y = sin (2 x)

graphing: the stretch

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ 3 y

-1

-2

-3

x 7→ 2 x

□ the new equation 0

1

the tweak:

2

3

4

5

6

3 y = sin (2 x)

graphing: the stretch

□ the basic graph

3

2

y = cos (x) 1

0

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

graphing: the stretch

□ the basic graph

3

2

y = cos (x) 1

□ the tweak

0

y 7→

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

1 y 3

x 7→ 2 x

graphing: the stretch

□ the basic graph

3

2

y = cos (x) 1

□ the tweak

0

y 7→

-1

-2

-3

1 y 3

x 7→ 2 x

□ the new equation 0

1

the tweak:

2

3

4

5

6

1 y = cos (2 x) 3

graphing: the stretch

□ the basic graph

3

2

y = cos (x) 1

□ the tweak

0

y 7→

-1

-2

-3

1 y 3

x 7→ 2 x

□ the new equation 0

1

the tweak:

2

3

4

5

6

1 y = cos (2 x) 3

graphing: the stretch

□ try to graph

3

2

1 y = sin (3 x) 2

1

0

-1

-2

-3

0

1

YOU try it!

2

3

4

5

6

graphing: the stretch

□ try to graph

3

2

1 y = sin (3 x) 2

1

□ the tweak

0

-1

y 7→

-2

-3

0

1

YOU try it!

2

3

4

5

6

1 y 2

x 7→ 3 x

graphing: the stretch

□ try to graph

3

2

1 y = sin (3 x) 2

1

□ the tweak

0

-1

y 7→

-2

-3

0

1

YOU try it!

2

3

4

5

6

1 y 2

x 7→ 3 x

□ the new equation 1 y = sin (3 x) 2

graphing: the stretch

□ try to graph

3

2

1 y = sin (3 x) 2

1

□ the tweak

0

-1

y 7→

-2

-3

0

1

YOU try it!

2

3

4

5

6

1 y 2

x 7→ 3 x

□ the new equation 1 y = sin (3 x) 2

graphing: the stretch

1 0.5 0 -0.5 -1 0

1

2

3

4

YOU try it! □ try to graph

5

( 3 y = cos

1 x 2

)

6

graphing: the stretch

□ the tweak

1 0.5 0

y 7→ 3 y

-0.5 -1 0

1

2

3

4

YOU try it! □ try to graph

5

( 3 y = cos

1 x 2

)

6

x 7→

1 x 2

graphing: the stretch

□ the tweak

1 0.5 0

y 7→ 3 y

-0.5

x 7→

1 x 2

-1 0

1

2

3

4

YOU try it! □ try to graph

5

6

□ the new equation (

( 3 y = cos

1 x 2

)

3 y = cos

) 1 x 2

graphing: the stretch

1 0.5 0 -0.5 -1

□ the tweak 0

2

4

6

8

10

YOU try it! □ try to graph

12

y 7→ 3 y

x 7→

1 x 2

□ the new equation ( 3 y = cos

1 x 2

(

) 3 y = cos

) 1 x 2

graphing the flip

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

0

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ y

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

x 7→ −x

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ y

-1

-2

-3

x 7→ −x

□ the new equation 0

1

the tweak:

2

3

4

5

6

y = sin (−x)

graphing: the flip

3

□ the tweak

2 1

y 7→ y

0

x 7→ −x

-1

□ the new equation

-2 -3

-6

-4

-2

0

2

the tweak: □ the basic graph

4

6

y = sin (−x) y = sin (x)

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

0

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ −y

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

x 7→ x

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ −y

-1

-2

-3

x 7→ x

□ the new equation 0

1

the tweak:

2

3

4

5

6

−y = sin (x)

graphing: the flip

□ the basic graph

3

2

y = sin (x) 1

□ the tweak

0

y 7→ −y

-1

-2

-3

x 7→ x

□ the new equation 0

1

the tweak:

2

3

4

5

6

−y = sin (x)

graphing: the flip

□ the basic graph

3

2

y = cos (x) 1

0

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

graphing: the flip

□ the basic graph

3

2

y = cos (x) 1

□ the tweak

0

1 y 7→ − y 3

-1

-2

-3

0

1

the tweak:

2

3

4

5

6

x 7→ x

graphing: the flip

□ the basic graph

3

2

y = cos (x) 1

□ the tweak

0

1 y 7→ − y 3

-1

-2

-3

x 7→ x

□ the new equation 0

1

the tweak:

2

3

4

5

6

1 − y = cos (x) 3

graphing: the flip

□ the basic graph 5

y = cos (x) □ the tweak

0

1 y 7→ − y 3

-5

x 7→ x

□ the new equation 0 1 2 3 4 5 6

the tweak:

1 − y = cos (x) 3

graphing practice it all

graphing: practice it all

YOU try it!

4 3

□ try to graph

2 1 0

[ ( 1 π )] (y − 1) = sin 2 x − 3 2

-1 -2 -3 -4 0

1

2

3

4

5

6

graphing: practice it all

YOU try it!

4 3

□ try to graph

2 1 0

[ ( 1 π )] (y − 1) = sin 2 x − 3 2

-1 -2 -3 -4 0

1

2

3

4

5

6

□

graphing: practice it all

YOU try it!

4 3

□ try to graph

2 1 0

[ ( 1 π )] (y − 1) = sin 2 x − 3 2

-1 -2 -3 -4 0

1

2

3

4

5

6

□

graphing equation pre-graph preping

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = x2 + 6x − 7

1 0 -1 -2 -3 -4 0

1

2

3

4

5

6

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = x2 + 6x − 7

1 0

□

-1 -2 -3 -4 0

1

2

3

4

5

6

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = x2 + 6x − 7

1 0

□

-1 -2 -3 -4 0

1

2

3

4

5

6

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = 3 sin [2x + π] − 6

1 0 -1 -2 -3 -4 0

1

2

3

4

5

6

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = 3 sin [2x + π] − 6

1 0

□

-1 -2 -3 -4 0

1

2

3

4

5

6

graphing: equation pre-graph preping

YOU try it! □ try to graph

4 3 2

y = 3 sin [2x + π] − 6

1 0

□

-1 -2 -3 -4 0

1

2

3

4

5

6

graphing some sage help