4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine.
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Transcript of 4.1 Graphs of Sine and Cosine OBJ: Graph sine and cosine.
4.1 Graphs of Sine and Cosine
OBJ: Graph sine and cosine
1 DEF: Sine Graph
1
0 π π 3π 2π
-1 2 2
0 1 0 -1 0
1 DEF: Sine Graph
1
0 π π 3π 2π
-1 2 2
0 1 0 -1 0
y = d + a(trig b (x + c))
a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2π
b
c (starting point)
d (vertical shift)
y = sin x
Ref.no
Amp.1
Per. 2π
¼ Per. π/2
St. Pt. 0
Vert. Sh.none
0 1 0 1 0
1 0
-1 π/2 3π/2 4π/2
2 DEF: Cosine Graph
0 π π 3π 2π
2 2
1 0 -1 0 1
2 DEF: Cosine Graph
- π 0 π π 3π 2π
2 2 2
1 0 -1 0 1
DEF: Periodic function
A function f with the property f(x) = f(x+p) for every real number x in the domain of f and for some real positive number p. The smallest possible positive value of p is the period of the function f.
3 EX: Graph y = 2 sin x
0 π π 3π 2π
2 2
0 1 0 -1 0
2(0 1 0 -1 0)
0 2 0 -2 0
3 EX: Graph y = 2 sin x
0 π π 3π 2π
2 2
0 2 0 -2 0
DEF: Amplitude of Sine and Cosine
The graph of y = a sin x or y = a cos x will have the same shape as y = sin x or y cos x, respectively, except with range - a y a . The number a is called the amplitude.
y = d + a(trig b (x + c))
a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2π
b
c (starting point)
d (vertical shift)
4 y = -2 cos x
1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2
2 1 0 -1 π/2 3π/2
4π/2 -2
4 y = -2 cos x
Ref.yes
Amp.- 2
Per. 2π
¼ Per. π/2
St. Pt. 0
Vert. Sh.none
1 0 -1 0 1 -2(1 0 -1 0 1) -2 0 2 0 -2
2 1 0
-1 π/2 3π/2 4π/2 -2
4 y = -2 cos x
2 1 0 -1 π/2 3π/2
4π/2 -2
DEF: Vertical Translation
A function of the form y =d + a sin b x or of the form y = d + a cos b x is shifted vertically when compared with y = a sin b x or y =a cos b x.
y = d + a(trig b (x + c))
a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2π
b
c (starting point)
d (vertical shift)
5 EX: Graph y = – 3 + 2 sin x
0 π π 3π 2π
2 2
1 DEF: Sine Graph
1
0 π π 3π 2π
-1 2 2
0 1 0 -1 0
3 EX: Graph y = 2 sin x
0 π π 3π 2π
2 2
2(0 1 0 -1 0)
0 2 0 -2 0
5 EX: Graph y = – 3 + 2 sin x
1
0 π π 3π 2π
-1 2 2
2(0 1 0 -1 0)
0 2 0 -2 0
5 EX: Graph y = – 3 + 2 sin x
1
0 π π 3π 2π
-1 2 2
2(0 1 0 -1 0)
0 2 0 -2 0
-3-3-3 -3 -3
5 EX: Graph y = – 3 + 2 sin x
1
0 π π 3π 2π
-1 2 2
-3 -1 -3 -5 -3
DEF: Phase Shift
The function y = sin (x + c) has the shape of
the basic sine graph y = sin x, but with a
translation c units: to the right if c < 0
and to the left if c > 0. The number c is
the phase shift of the graph. The cosine
graph has the same function traits.
y = d + a(trig b (x + c)
a (amplitude) multiply a times (0 |1 0 -1 0 1)b (period) 2π
b
c (starting point)
d (vertical shift)
EX: Graph y = sin (x – π/3)6 EX: Graph y = 4 – sin (x – π/3)
2 5 8 11 14 -1 6 6 6 6 6
0 1 0 -1 0
6 EX: Graph y = 4 – sin (x – π/3)
2 5 8 11 14 -1 6 6 6 6 6
0 1 0 -1 0
-1(0 1 0 -1 0
0 -1 0 1 0
6 EX: Graph y = 4 – sin (x – π/3)
2 5 8 11 14 -1 6 6 6 6 6
0 -1 0 1 0
+4 +4 +4 +4 +4
4 3 4 5 4
6 EX: Graph y = 4 – sin (x – π/3)
2 5 8 11 14 -1 6 6 6 6 6
4 3 4 5 4
EX: Graph y = 3cos (x + π/4)7 EX: Graph y =-3 + 3cos(x+π/4)
EX: Graph y = 3cos (x + π/4)7 EX: Graph y =-3 + 3cos(x+π/4)
- 3 5 7 4 4 4 4 4
1 0 -1 0 1
EX: Graph y = 3cos (x + π/4)7 EX: Graph y =-3 + 3cos(x+π/4)
- 3 5 7 4 4 4 4 4
1 0 -1 0 13(1 0 -1 0 1)
3 0 -3 0 3
7 EX: Graph y =-3 + 3cos(x+π/4)
- 3 5 7 4 4 4 4 4
3 0 -3 0 3
-3 -3 -3 -3 -3
0 -3 -6 -3 0
__ __ __ __ __ __ __ __ __
7 EX: Graph y =-3 + 3cos(x+π/4)
- 3 5 7 4 4 4 4 4
__ __ __ __ __ __ __ __ __
1 EX: Graph y = -2 +sin x
Ref, Amp
No, 1
Per
2 π
¼ Per 0 π π 3π 2π
π/2 2 2
St.Pt. 0
Vert. Shift
2
1 EX: Graph y = -2 +sin x
0 π π 3π 2π
2 2
0 1 0 -1 0
-2 -2 -2 -2 -2
-2 -1 -2 -3 -2
1 EX: Graph y = -2 +sin x
0 π π 3π 2π
2 2
0 1 0 -1 0
-2 -2 -2 -2 -2
-2 -1 -2 -3 -2
2 EX: Graph y = 3 – 2 cos x
Ref, Amp
Yes, -2
Per
2 π
¼ Per 0 π π 3π 2π
π/2 2 2
St.Pt. 0
Vert. Shift
3
2 EX: Graph y = 3 – 2 cos x
0 π π 3π 2π
2 2
1 0 -1 0 1
-2(1 0 -1 0 1)
-2 0 2 0 -2
2 EX: Graph y = 3 – 2 cos x
0 π π 3π 2π
-2(1 0 -1 0 1) 2 2
-2 0 2 0 -2
+3 +3 +3 +3 +3
1 3 5 3 1
2 EX: Graph y = 3 – 2 cos x
0 π π 3π 2π
-2(1 0 -1 0 1) 2 2
-2 0 2 0 -2
+3 +3 +3 +3 +3
1 3 5 3 1