Aim: Graphs of y = asin bx and y = acos bx Course: Alg. 2 & Trig. Aim: Whats the a in y = a sin x...

Aim: Graphs of y = asin bx and y = acos bx Course: Alg. 2 & Trig.

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

Do Now: Graph the following:set your calculator window to following settings:Xmin= -1Xmax=2.5Xscl=/2Ymin=-4Ymax=4Yscal=1then graph the following

y = sinx; y = 2 sinx; y = 3 sinx


y = a sin x

1

-1

y = 2 sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

2

-2

y = 3 sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

3

-3

y = sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

maximum

minimum

maximum

minimum

maximum

minimum

amplitude - 1

amplitude - 2

amplitude - 3


radians

y = a cos x

y = cos x1

-1

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

y = 2 cos xπ 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

2

-2

y = 3 cos x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

3

-3

amplitude - 1

amplitude - 2

amplitude - 3


What the a is All About

y = a sin x

y = a cos x

In general, for the functions y = a sin x and y = a cos x:

amplitude = | a |

ex. y = 6 sin x

The amplitude of a periodic function is halfthe difference between the minimum andmaximum values of the function.

amplitude is 6

max = 6min = -6

6 – (-6)2

= 6


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

x -2cos x = y

0 -2cos 0 = -2

π/2 -2cos π/2 = 0

π -2cos π = 2

3π/2 -2cos 3π/2 = 0

2π -2cos 2π = -2

1

2

-1

-2

π 2π 3π/2 π/2 5π/2

y = 2 cos x

y = -2 cos x

amplitude = 2

2. Plot points & sketch

Why? key points Min., Zero, and Max.


What if a is negative?

1

2

-1

-2

π 2π 3π/2 π/2 5π/2

y = 2 cos x

y = -2 cos x

y = 2 cos x y = -2 cos x?

y = 2 cos x y = -2 cos xrx-axis

y = a cos x & y = (-a)cos x are reflectionsof each other through the x-axis.

y = a sin x & y = (-a)sin x are reflectionsof each other through the x-axis.


What about the b?

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/21

-1

-1

1π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

y = cos x

y = cos 2x

y = cos 1/2x

y = cos 4x


What about the b?

1

-1 y = sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1

y = sin 1/2x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1

y = sin 2x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1

y sin 4x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2


What about the b?

1

-1 y = sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1

y = sin 2x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

1

2

How often does the cycle repeat itself overthe interval 0 ≤ x ≤ 2π?

period – 2π

period – π

y = sin x one time

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).

12


In General:

Length of cycle?

1

-1 y = sin x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

-1

1

y = sin 2x

π 2π 3π/2 π/2 3π 4π 7π/2 5π/2 5π 9π/2

1

2

ex. y = sin 1/3x b = 1/3

of a function= 2π

|b|period

1 cycle

2 cycles

number of cycles from 0 to 2π • length of 1 cycle = 2π

= (2π)1/3

pd. = 6π

• period = 2πb


Understanding Sine/Cosine Curves

of a function= 2π

|b|periodamplitude = | a | frequency = |b|

y = a sin bx y = a sin bx

Sketch the graph of y = 3 sin 2x in the interval 0 ≤ x ≤ 2π.

1) Determine the amplitude & period

amplitude = | a | = 3 = 2π|b|

period = 2π|2|

= πa = 3, b = 2

max. = 3min. = -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.


Graphing Sine/Cosine Curves

Sketch the graph of y = 3 sin 2x in the interval 0 ≤ x ≤ 2π.

π

2. Plot points & sketch

4

2

34

54

32

74

2π

3

-3

max. = 3, min. = -3; period is π

y = 3 sin 2x


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

a = 2, b = 1amplitude = | a | = 2

max. = 2 min. = -2

= 2π|b|

period = 2π|1|

= 2π

divide the period 2π, into 4 equal intervals: π/2, π, 3π/2, and 2π.

y = 2 cos xa = 1, b = 1/2

amplitude = | a | = 1

max. = 1 min. = -1

= 2π|b|

period = 2π|1/2|

= 4π

divide the period 4π, into 4 equal intervals: π, 2π, 3π, and 4π.

y = sin 1/2 x


Model Problem (con’t)

2. Plot p

oints &

sketch

π

4

2

34

54

32

74

2π

2

-2

y = 2 cos x max. = 2 min. = -2 period = 2π

y = sin 1/2x max. = 1 min. = -1 period = 4π

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 cos x = sin 1/2x

1

-1

y = 2 cos x

y = sin 1/2x


Model ProblemsThe 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


Model Problems

2

1

2 4 6

Which is the equation of the graph shown?

1) y = 2 sin ½x 2) y = 2 cos ½x

3) y = ½ sin 2x 4) y = ½ cos 2x

2


Regents Prep

2

1

-1

-2

2 4 6

Which is the equation of the graph shown?

1) y = 2 sin ½x 2) y = 2 sin 2x

3) y = ½ cos 2x 4) y = 2 cos 2x

2


Model Question

Which function has a period of 4 and an amplitude of 8?

1) 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

Aim: Graphs of y = asin bx and y = acos bx Course: Alg. 2 & Trig. Aim: Whats the a in y = a sin x...

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Transcript of Aim: Graphs of y = asin bx and y = acos bx Course: Alg. 2 & Trig. Aim: Whats the a in y = a sin x...