1. 4.1 and 4.2 Sine Graph Sine & Cosine are periodic functions, repeating every period of 2  radians: 0 x y 180  360 2  90  /2 270 3  /2 1 y = sin (x)x y 0  /2 1  0 3  /2 -1 2  0 What is the amplitude? (How High is it?) What about the graph of : y = 2sin(x) 2 -2 Does the period ever change? What about: y = sin (2x) Does the graph always start at zero? Try: y = sin (x -  /2)

2. Equation of a Sine Curve 0 x y 180  360 2  90  /2 270 3  /2 1 2 -2 y = A sin (Bx – C)amplitude = | A | period = 2  B phase shift = C (right/left) B y = sin (x) y = 2 sin (x) y = sin (2x) y = sin (x -  /2) Does the graph ever shift vertically? – YES y = A sin (Bx – C) + D [D is the vertical shift] What would y = sin (x -  /2) + 1 look like? Graphing Guidelines 1.Calculate period 2. Calculate interval (period / 4) 3.Calculate phase shift-‘start-value’ 4. Create X/Y chart w/ 5 key points from ‘start-value’

3. Equation of a Cosine Curve 0 x y 180  360 2  90  /2 270 3  /2 1 2 -2 y = A cos (Bx – C)amplitude = | A | period = 2  B phase shift = C (right/left) B y = cos (x) y = 2 cos (x) y = cos (2x) y = cos (x -  /2) x y=cos(x) 0 1  /2 0  -1 3  /2 0 2  1

4. 4.3 Equation of a Tangent Function x 1 y = A tan (Bx – C) 1. Find & draw two asymptotes Set Bx –C =  and -  (solve for x) 2 2. Find & plot the x-intercept (Midway between asymptotes) 3. Plot points: x = ¼ from 1 st asymptote, y = -A x = ¾ before 2 nd asymptote, y = A y = tan(x) 0 22 -  2  3  2 22

5. Graphing a Cotangent Function x 1 y = A cot (Bx – C) 1. Find and draw two asymptotes Set Bx –C = 0 and  (solve for x) 2. Find & plot the x-intercept (Midway between asymptotes) 3. Plot points: x = ¼ from 1 st asymptote, y = A x = ¾ before 2 nd asymptote, y = -A y = cot(x) 0 22 -  2  3  2 22

6. 4.4 Graphing variations of y=csc(x) 0 x y 180  360 2  90  /2 270 3  /2 1 2 -2 y = sin (x) y = csc (x) csc(x) = 1/sin(x). To graph the cosecant: 1) Draw the corresponding sine graph 2) Draw asymptotes at x-intercepts 3) Draw csc minimum at sin maximum 4) Draw csc maximum at sin minimum

7. Graphing variations of y=sec(x) 0 x y 180  360 2  90  /2 270 3  /2 1 2 -2 y = cos (x) y = sec (x) sec(x) = 1/cos(x). To graph the secant: 1) Draw the corresponding cosine graph 2) Draw asymptotes at x-intercepts 3) Draw sec minimum at cos maximum 4) Draw sec maximum at cos minimum -  /2 450 5  /2

8. 4.5 Harmonic Motion The position of a point oscillating about an equilibrium Position at time t is modeled by either S(t) = a cos  t or S(t) = a sin  t Where a and  are constants with  > 0. Amplitude of motion is | a | Period is 2  /  Frequency is  /2  oscillations per unit of time Example: An object attached to a coiled spring is pulled down 5 in from equilibrium position and released. The time for 1 complete oscillation is 4 sec. (a)Give an equation that models the position of the object at time t Time of object release is: t = 0 and distance is then 5 in below equilibrium so, S(0) = -5 and we use: S(t) = cos  t. Period = 4 sec  2  /  = 4   =  / 2 So, S(t) = -5 cos (  / 2)t (b) Determine the position at t = 1.5 sec: S(1.5) = -5 cos (  / 2)(1.5) = 3.54 in.