Period and Amplitude - Iona Preparatory School



Name Period Alg2/Trig

Period and Amplitude

Period – the length of one cycle of a graph

A function whose graph repeats a basic pattern is called a periodic function. To find the period of a function, start from any point on the graph (usually the origin) and proceed to the right until the pattern begins to repeat. Go to pg 743: 29 – 32.

Set the mode to degrees, switch to dot mode (line 5)

Set window to:

← Xmin 0

← Xmax 360

← Xscl 30 (each tick mark on the x-axis represents 30()

← Ymin -4

← Ymax 4

← Yscl 0.5 (each tick mark on the y-axis represents .5)

A. Compare the graphs of y = sin x, y = sin 2x and y = sin ½ x (only graph one at a time). What is the period of each of these functions?

y = sin x: y = sin 2x: y = sin ½ x:

Graph more equations y = sin bx using other positive values of b What conjecture can you make about the effect of the value of b on the periods of the graphs of y = sin bx?

As b gets larger, the period gets

As b gets smaller, the period gets

(b determines what is known as the frequency of the graph.)

Conclusion: for y = sin bx period =

B. Repeat for y = cos x, y = cos 2x and y = cos ½ x. What is the period of each?

As b gets larger, the period of the graph gets

As b gets smaller, the period of the graph gets

(b determines what is known as the frequency of the graph.)

Conclusion: for y = cos bx period =

C. Repeat for y = tan x, y = tan 2x and y = tan ½ x. What is the period of each?

As b gets larger, the period of the graph gets

As b gets smaller, the period of the graph gets

(b determines what is known as the frequency of the graph.)

Conclusion: for y = tan bx period =

Amplitude – half the distance between the maximum and minimum y-values of the graph

A. Compare the graphs of y = sin x, y = 2 sin x and y = ½ sin x. What is the amplitude of each function?

y = sin x: y = 2 sin x: y = ½ sin x:

B. Graph more equations y = a sin x using other values of a (a tells the amplitude of the graph). What conjecture can you make about the effect of a on the graph of y = a sin x?

As a gets larger than 1, the height of the graph gets

As a gets smaller than 1, the height of the graph gets

Conclusion: the amplitude of y = a sin x is

C. Repeat for y = cos x, y = 3 cos x and y = ½ cos x. What is the amplitude of each function?

y = cos x: y = 3 cos x: y = ¼ cos x

Conclusion: the amplitude of y = a cos x is

D. Repeat for y = tan x, y = 4 tan x and y = ⅓ tan x. What is the amplitude of each function?

E. Example: Find the amplitude and period of each function and draw one period of its graph starting at the origin. The number of degrees in the period is the ending point of the cycle and should be labeled.

y = cos 3x

Period:

Xmax:

Ymax:

Ymin:

Practice

1. y = ¼ sin x

Period:

Xmax:

Ymax:

Ymin:

2. y = 2 sin ( ⅓ x)

Period:

Xmax:

Ymax:

Ymin:

3. y = 2.5 tan ( ½ x)

Period:

Xmax:

Ymax:

Xmin:

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