Learning Target #6
Graph sin x, cos x, tan x and connect the graphs to their reciprocal functions along with harmonic motion
1
Sine Curve
θ (x)
y
keeps going
keeps going
2
Cosine Curve
keeps going
θ (x)
keeps going
y
3
Transformations of Trig Graphs
vertical shift reflection
over x­axis
d ± a sin( ±bx ± c)
reflection
over y­axis
"amplitude"
or vertical
stretch/shrink
"period"
or horizontal
stretch/shrink
horizontal
shift
* Amplitude is |a|.... how high the graph is from the x­axis
* Period .... let b > 0. The period of y = a sin bx is 2π
b
4
ASSIGNMENT
p. 307 #1­14
5
ex: Sketch the graph of y = 2 sin x
6
1
ex: Sketch f(x) = cos 2x
2
7
ex: Sketch the graph of y = ­3 cos(2
πx)
8
ex: Sketch the graph of f(x) = 2 + 3 cos 2x
9
ex: Sketch the graph of y = 4 + sin ( ­ x
2
π)
10
ASSIGNMENT
p. 307 #35­42, 45, 46, 51, 52
11
12
­2 π
­π
π
2π
f(x) = cos x
13
ex: Sketch the graph of y = 3 + sec 2x
14
f(x) = sin x
­2 π
­π
π
2π
15
ex: Sketch the graph of y = 2 csc (x + π/ 4 )
16
ASSIGNMENT
p. 318 #11­15, 21, 25
17
18
19
ASSIGNMENT
p. 318 #16, 17, 26, 27, 31, 35, 36, 38
20
Tangent Curve
θ (x)
y
keeps going
keeps going
21
x 2
To find asymptotes of tangent, use the equation bx = ± π 2
ex: Sketch the graph of y = tan 22
ex: Sketch the graph of y = ­3 tan 2x
23
ASSIGNMENT
p. 318 #8­10, 22­24
24
Cotangent Curve
­to graph, use points on the circle
θ (x)
y
keeps going
keeps going
25
ex: Sketch the graph of y = 2 cot x 2
To find asymptotes of cotangent, use the equation bx = 0 and bx = ±π 26
ASSIGNMENT
p. 318 #19, 20, 28
27
ASSIGNMENT
To be turned in tomorrow. Graph these.
1. y = ­2cos (4x ­ π)
1
2. y = 1 + sin 2
3. y = ­cot πx
( )x2
4. y = 2tan (x ­ )
π
2
5. y = ­csc (3x)
6. y = ­2 + 3sec (2x)
28
Harmonic Motion
10 cm
10 cm
0 cm
0 cm
­10 cm
Equilibrium
­10 cm
Max. Negative
Displacement
Max. Positive
Displacement
29
Simple Harmonic Motion
d = a sin ωt
or
d = a cos ωt
where d = distance, t = time away from the origin, a = amp. 2 π = period, ω = frequency
Note: |a| = amplitude, 2 π ω
frequency = # of cycles per unit time
ex: From the last slide, write the eq'n for the S.H.M. of the ball where the period is 4 seconds. What is the frequency?
30
ex: Given the equation for S.H.M.
d = 6 cos t 3 π
4
find: a.) max. displacement
b.) frequency
c.) value of d when t = 4 sec.
31
ASSIGNMENT
p. 341 #47­56
*for 47­50, parts a & b only
32