Aim: What’s the a in y = a sin x all about?

1. Aim: What’s the a in y = a sin x all about? Graph the following: set your calculator window to following settings: Xmin= -1 Xmax=2.5 Xscl=/2 Ymin=-4 Ymax=4 Yscal=1 then graph the following y = sinx; y = 2 sinx; y = 3 sinx Do Now:

2. maximum 1 2π 2π 4π 4π 2π 4π -1 minimum 5π/2 3π/2 π/2 7π/2 5π/2 9π/2 9π/2 5π/2 7π/2 3π/2 π/2 7π/2 π/2 3π/2 9π/2 maximum 2 y = 2 sin x minimum 3π π 3π 5π 5π 3π 5π π π -2 3 maximum y = 3 sin x minimum -3 y = a sin x y = sin x amplitude - 1 amplitude - 2 amplitude - 3

3. y = cos x 1 radians 2π 4π 2π 4π -1 π/2 7π/2 5π/2 9π/2 9π/2 7π/2 π/2 5π/2 3π/2 3π/2 y = 2 cos x π 5π 5π 3π 3π π 3 y = 3 cos x 5π π 2π 3π 4π π/2 3π/2 5π/2 7π/2 9π/2 -3 y = a cos x amplitude - 1 2 amplitude - 2 -2 amplitude - 3

4. 6 – (-6) = 6 2 What the a is All About y = asin x y = acos x The amplitude of a periodic function is half the difference between the minimum and maximum values of the function. In general, for the functions y = asin x and y = acos x: amplitude = | a | max = 6 min = -6 ex. y = 6 sin x amplitude is 6

5. y = 2 cos x y = -2 cos x Why? amplitude = 2 What if a is negative? Sketch the graph of y = -2 cos x over the interval 0 ≤ x ≤ 2π: (w/o graphing calculator) 1. Table of values 2. Plot points & sketch 2 1 π 2π π/2 3π/2 5π/2 -1 -2 key points Min., Zero, and Max.

6. y = 2 cos x What if a is negative? 2 1 π 2π π/2 3π/2 5π/2 -1 -2 y = -2 cos x ? y = 2 cos x y = -2 cos x rx-axis y = 2 cos x y = -2 cos x y = a cos x & y = (-a)cos x are reflections of each other through the x-axis. y = a sin x & y = (-a)sin x are reflections of each other through the x-axis.

7. 1 4π 2π 2π 4π 4π 2π 4π 2π -1 1 1 1 9π/2 9π/2 π/2 7π/2 3π/2 π/2 7π/2 π/2 3π/2 5π/2 3π/2 9π/2 7π/2 5π/2 5π/2 7π/2 5π/2 π/2 9π/2 3π/2 -1 -1 -1 5π π 3π π 3π π 5π 3π 3π π 5π 5π What about the b? y = cos x y = cos 2x y = cos 1/2x y = cos 4x

8. 1 -1 y = sin x 2π 2π 2π 2π 4π 4π 4π 4π 1 1 1 3π/2 7π/2 5π/2 9π/2 5π/2 π/2 π/2 3π/2 9π/2 7π/2 5π/2 9π/2 5π/2 3π/2 7π/2 9π/2 7π/2 π/2 3π/2 π/2 -1 -1 -1 y = sin 2x 5π π π 3π 3π 5π 5π π 3π π 3π 5π y = sin 1/2x y sin 4x What about the b?

9. 1 -1 period – 2π y = sin x 2π 4π 4π 2π 1 π/2 7π/2 5π/2 9π/2 5π/2 π/2 7π/2 3π/2 9π/2 3π/2 -1 y = sin 2x period – π π 3π 3π π 5π 5π What about the b? 1 2 How often does the cycle repeat itself over the interval 0 ≤ x ≤ 2π? 1 y = sin x one time 2 y = sin 2x two times frequency (b) – of a periodic function is the number of cycles from 0 ≤ x ≤ 2π. (the number of times the function repeats itself).

10. 1 -1 1 cycle y = sin x 4π 2π 4π 2π 1 π/2 7π/2 9π/2 5π/2 5π/2 3π/2 9π/2 π/2 3π/2 7π/2 -1 2 cycles y = sin 2x In General: π 3π 3π π 5π 5π = 2π period |b| of a function = 6π = (2π) pd. 1/3 Length of cycle? 1 2 number of cycles from 0 to 2π • length of 1 cycle = 2π b • period = 2π ex. y = sin 1/3x b = 1/3

11. = 2π period |b| = 2π = 2π = π period |b| |2| of a function Understanding Sine/Cosine Curves y = a sin bx y = a sin bx amplitude = | a | frequency = |b| Sketch the graph of y = 3 sin 2x in the interval 0 ≤ x ≤ 2π. 1) Determine the amplitude & period a = 3, b = 2 amplitude = | a | = 3 max. = 3 min. = -3 divide the period π, into 4 equal intervals: π/4, π/2, 3π/4, and π. Repeat for second half: 5π/4, 3π/2, 7π/4, and 2π.

12. Graphing Sine/Cosine Curves Sketch the graph of y = 3 sin 2x in the interval 0 ≤ x ≤ 2π. max. = 3, min. = -3; period is π 2. Plot points & sketch y = 3 sin 2x 3 π 2π -3

13. amplitude = | a | = 1 amplitude = | a | = 2 max. = 2 min. = -2 max. = 1 min. = -1 = 2π = 2π = 2π period |b| |1| = 2π = 2π = 4π period |b| |1/2| divide the period 2π, into 4 equal intervals: π/2, π, 3π/2, and 2π. divide the period 4π, into 4 equal intervals: π, 2π, 3π, and 4π. Model Problem Sketch, on the same set of axes, the graphs of y = 2 cos x and y = sin 1/2 x in the interval 0 ≤ x ≤ 2π. 1) Determine the amplitudes & periods y = 2 cos x y = sin 1/2 x a = 2, b = 1 a = 1, b = 1/2

14. period = 2π y = 2 cos x max. = 2 min. = -2 period = 4π y = sin 1/2x max. = 1 min. = -1 y = 2 cos x y = sin 1/2x π 2π 2 cos x = sin 1/2x Model Problem (con’t) Sketch, on the same set of axes, the graphs of y = 2 cos x and y = sin 1/2 x in the interval 0 ≤ x ≤ 2π. 2. Plot points & sketch 2 1 -1 -2

15. Model Problems The amplitude of y = 2 sin 2x is 1)  2) 2 3) 3 4) 4 What is the range of the function 3 sin x? 1) y> 0 2) y< 0 3) -1 <y< 1 4) -3 <y< 3 What is the minimum value of f() in the equation f() = 3 sin 4 1) -1 2) -2 3) -3 4) -4 What is the period of sin 2x? 1) 4 2) 2 3)  4) 4

16. Model Problems 2 Which is the equation of the graph shown? • y = 2 sin ½x 2) y = 2 cos ½x • 3) y = ½ sin 2x 4) y = ½ cos 2x

17. Regents Prep 2 Which is the equation of the graph shown? • y = 2 sin ½x 2) y = 2 sin 2x • 3) y = ½ cos 2x 4) y = 2 cos 2x

18. Model Question Which function has a period of 4 and an amplitude of 8? • y = -8 sin 8x 2) y = -8 sin ½ x • 3) y = 8 sin 2x 4) y = 4 sin 8x The period of a sine function is 300 and its amplitude is 3. Write the function in y = a sin bx form. y = 3 sin 12x