Mathematics 2204

Unit 3Sinusoidal functions

Section 1

Periodic behaviour

INTERACTIVE

• characteristics of graphs - math 2204 - unit 3 - sinusoidals.swf

4 MAIN TYPES OF FUNCTIONSLinear function Quadratic

Absolute value of the quadraticSinusoidal function

Example 1 done in class (write this down)

The above graph was constructed by measuring the height above the floor of a point on the rim of the wheel of an exercise bicycle at certain times during its revolution. Use it to answer the following questions: (a) What is the maximum height of the point above the floor?(b) What is the minimum height of the point above the floor?(c)) What is the height of the axle above the floor?(d) What is the diameter of the wheel of the bike?(e) How long does it take the wheel to make one revolution?(f) Within the first 10 seconds, how often is the point 50 cm above the ground?

Example 1 solutiona) What is the maximum height of the point above the floor?

Ans. 75 cmb) What is the minimum height of the point above the floor?

Ans. 5 cmc) What is the height of the axle above the floor?

Ans. 40 cm d) What is the diameter of the wheel of the bike? Ans. 35e) How long does it take the wheel to make one revolution?

Ans. 7 sf) Within the first 10 seconds, how often is the point 50 cm

above the ground? Ans. Approximately 9.75 s

Problems from text

• Do CYU Questions 8 - 12 on pages 93 - 95.

INTERACTIVE

• sinusoidal graphs - cdli interactive - math 1204.swf

HOW TO DETERMINE IF A FUNCTION IS PERIODIC

A function is periodic if:

• their graph can be divided into intervals, called the period, such that the graph in any interval can be translated horizontally onto the graph in any other interval.

HOW TO DETERMINE IF A FUNCTION IS SINUSOIDAL

A sinusoidal graph has:

1. Only one local maximum and one local minimum in each periodic interval.

2. A sinusoidal axis which is a horizontal line half way between the local maximum and local minimum in each periodic interval.

Is this graph periodic and sinusoidal?

•The graph in the interval from A to O can be translated to match exactly the graph in the interval from O to B. Therefore, the graph is periodic.•the above graph has two local maxima in each period, for example in the interval A to O the local maxima are at P and R. Therefore the graph is not sinusoidal.

Is this graph periodic and sinusoidal?

The function above is a sinusoidal function. Its period is approximately 3.14, it has only one local maximum and minimum in each periodic interval, and its sinusoidal axis is y = 1.5 .

Is this graph periodic sinusoidal?

Is this graph periodic and sinusoidal?

Problems from text

• Do CYU Questions 14, 15, 17 on pages 99 - 101

Section 2

Transformations and Sinusoidal Functions

INTERACTIVE

• graphing using transformations - math 2204.swf

What is the transformational form of a function and mapping notation?

• The transformational form of the equation of the function

y = f (x)becomes

1/a(y - k) = f [1/b(x - h)]

• Mapping notation (x, y) (bx+h, ay+k)

when the graph of the original function is:

stretched vertically by a factor of a stretched horizontally by a factor of b translated vertically k units translated horizontally h units reflected in the x-axis if a is negative

EXAMPLE 2.1: Transformations of y = sin x

• Graph the following function:

EXAMPLE 2.1 SOLUTION

• Step 1: Start with the given equation which has already been written in transformational form (if it isn't you must put it in this form before you begin).

EXAMPLE 2.1 SOLUTION

• Step 2: Use the information in the equation to write in words what the transformations to the basic graph are to get this new function. In this example we have:

A vertical stretch of 2 A vertical translation of 3 A horizontal translation of 90o

EXAMPLE 2.1 SOLUTION

• Step 3: Use the word description of the transformations to write the mapping rule which maps a point on the original function into its corresponding point on the new function.

EXAMPLE 2.1 SOLUTION

• Step 4: Use the mapping rule to generate a table of values for the new function.

x y x + 90 2y + 3 0 0 90 3 90 1 180 5180 0 270 3270 -1 360 1360 0 450 3

EXAMPLE 2.1 SOLUTION

• Step 5: Use the points in the table to graph the function.

EXAMPLE 2.1 SOLUTION

• Now connect the points with a smooth curve.

Problems from text

• Do CYU Questions 14, 15 on page 111.

PROPERTIES OF SINUSOIDAL FUNCTIONS

• As a class of functions, we can say that all sinusoidal functions:

are periodic. have graphs that look like waves. have a sinusoidal axis have local maxima and minima. display symmetry about these local maxima and local minima

and these symmetries are equivalent. can be graphed as transformations of y = sin x and y = cos x.

EXAMPLE 2.2: Transformations of y = cos x

• Graph the following function:

EXAMPLE 2.2: solution

• Reflection in the x-axis• Vertical stretch of -½• Vertical translation of 5• Horizontal stretch of 3

Mapping rule: (x, y) (3x, -1/2y+5)

EXAMPLE 2.2: solution

x y

0 1

90 0

180 -1

270 0

360 1

x y

0 4.5

270 5

540 5.5

810 5

1080 4.5

Mapping rule: (x, y) (3x, -1/2y+5)

Problems from text

• Do the CYU Questions 25 - 33 on pages 115 - 117.

DETERMINING TRANSFORMATIONAL FORMS FROM GRAPHS

• To determine the sinusoidal axis which corresponds to the vertical translation k

• To determine the Amplitude that corresponds to Vertical stretch a.

• Determine the Horizontal stretch b.

• Determine Horizontal translation h for cos graph

• Determine horizontal translation h for sin graph

• Now plug values into corresponding transformational forms

max + min 2

Sinusoidal axis

= max - sinusoidal axisamplitude

stretch = 360periodHorizontal

Horizontal translation = x-coordinate of max.

1 1( ) cos ( )y k x ha b

1 1( ) sin ( )y k x ha b

Horizontal translation = x intercept

Problems from text

• Do Practice Exercises 1 to 9 on pages 127 & 128.

End of unit 3