Warm UpUsing your unit circle find each value:

1. Sin 0°=

2. Sin =

3. Sin π =

4. Sin =

5. Sin 2π =

Chapter 4Graphs of the Circular Functions

Section 4.1 Graphs of the Sine and Cosine Functions

Objective:SWBAT graph the sine and cosine functions with variations in amplitude and periods.

Periodic Functions

Many things in daily life repeat with a predictable pattern, such as weather, tides, and hours of daylight.

This periodic graph represents a normal heartbeat.

A function that repeats itself after a specific period of time is called a Periodic Function. Sine and Cosine

functions are periodic functions.

Sine and Cosine FunctionsWe are going to deconstruct the Unit circle and graph the sine and cosine functions on graph… Remember: Cosine is in the “x” spot in an ordered pair

and Sine is in the “y” spot.

VocabularySin wave - is a repetitive change or motion which, when plotted as a graph, has the same shape as the sine function. Amplitude - is the maximum distance it ever reaches from zero.

Period - is the time it takes to perform one complete cycle. (2π for cosine and sin)

Sine Function To sketch the graph of y = sin x first locate the key points.

These are the maximum points, the minimum points, and the intercepts. (5 total)

y

2

3

2

22

32

1

1

x

sin x

0x2

2

32

0 1 0 -1 0

Sine Function f(x) = sin x The graph is continuous over its entire domain,

(–, ). Its x-intercepts are of the form n, where n is an

integer.

Its period is 2.

The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin(–x) = –sin(x).

The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.

amplitude = |a|

If |a| > 1, the amplitude stretches the graph vertically.

If 0 < |a| > 1, the amplitude shrinks the graph vertically.

If a < 0, the graph is reflected in the x-axis.

Amplitude

The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the

range will be [|a|, |a|]. The amplitude is |a|.

Example: Graph y = 3 sin x compare to y = sin x.

Make a table of values.

030303sin x

01010sin x

3/2/20x

Amplitude

The range of y = 3sin x is [–3, 3].

2

32

4

y

x

4

2

y = – 4 sin x

reflection of y = 4 sin x y = 4 sin x

y = sin x

2

1y = sin x

y = 2 sin x

Amplitude

Period

For b > 0, the graph of y = sin bx will resemble that of

y = sin x, but with

period

Divide the interval into four equal parts to

obtain the values for which sin bx equal –1, 0, or 1.

(These values give the minimum points, x-intercepts, and maximum points on the graph)

GRAPHING y = sin bx

Graph y = sin 2x and compare to the graph of y = sin x.

The coefficient of x is 2, so b = 2, and the period is

The endpoints are 0 and and the three points between the endpoints are

• The x-values are:

GRAPHING y = sin bx

Graph y = sin 2x and compare to the graph of y = sin x.

Y = sin2x X Y (x, y)

0

GRAPHING y = sin bx

Graph y = sin 2x and compare to the graph of y = sin x.

HomeworkPage 141

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