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
