SECTION 1.3SECTION 1.3

PROPERTIES OF FUNCTIONSPROPERTIES OF FUNCTIONS

EVEN AND ODD FUNCTIONS

EVEN AND ODD FUNCTIONS

Identifying Graphically:Identifying Graphically:

g(x) = x g(x) = x 33 - 6x See the - 6x See the graph.graph.

Odd functions are symmetric Odd functions are symmetric with respect to the origin.with respect to the origin.

EVEN AND ODD FUNCTIONS

EVEN AND ODD FUNCTIONS

Identifying Graphically:Identifying Graphically:

h(x) = 0.1x h(x) = 0.1x 44 - 2x - 2x 2 2 + 5 See the + 5 See the graph.graph.

Even functions are symmetric Even functions are symmetric with respect to the y-axis.with respect to the y-axis.

EVEN AND ODD FUNCTIONS

EVEN AND ODD FUNCTIONS

Identifying Algebraically:Identifying Algebraically:

Odd function: f (- x) = - f (x)Odd function: f (- x) = - f (x)

g(x) = x g(x) = x 33 - 6x - 6x

g(- x) = (- x) g(- x) = (- x) 33 - 6(- x) - 6(- x)

= - x = - x 33 + 6x = - g(x) + 6x = - g(x)

EVEN AND ODD FUNCTIONS

EVEN AND ODD FUNCTIONS

Identifying Algebraically:Identifying Algebraically:

Even function: f (- x) = f (x)Even function: f (- x) = f (x)

h(x) = 0.1x h(x) = 0.1x 44 - 2x - 2x 2 2 + 5 + 5

h(- x) = 0.1(- x) h(- x) = 0.1(- x) 44 - 2(- x) - 2(- x) 2 2 + 5 + 5

= 0.1x = 0.1x 44 - 2x - 2x 2 2 + 5 + 5

= h(x)= h(x)

EXAMPLE:EXAMPLE:

Identify the following functions Identify the following functions as even, odd, or neither.as even, odd, or neither.

f(x) = x

x + 1

2

4 g(x) = xg(x) = x 3 3 xx

h(x) = xh(x) = x 5 5 + 1+ 1 k(x) = k(x) = xx 5 5

Use both graphical and Use both graphical and algebraic methods of algebraic methods of identification.identification.

INCREASING AND DECREASING FUNCTIONS

INCREASING AND DECREASING FUNCTIONS

INCREASING INCREASING FUNCTIONFUNCTION

A function is increasing A function is increasing when as x increases, y when as x increases, y also increases.also increases.

DECREASING DECREASING FUNCTIONFUNCTION

A function is decreasing A function is decreasing when as x increases, y when as x increases, y decreases.decreases.

Determine the intervals on Determine the intervals on which the function is which the function is decreasing.decreasing.

k(x) = xk(x) = x33 - 6x - 6x22 - 13x - 13x

- .89 < x < 4.89- .89 < x < 4.89

EXAMPLE:EXAMPLE:

When the graph of a function is When the graph of a function is increasing to the left of x = c and increasing to the left of x = c and decreasing to the right of x = c, then at decreasing to the right of x = c, then at c the value of f is called a c the value of f is called a local local maximummaximum of the function. of the function.

LOCAL MAXIMUMLOCAL MAXIMUM

When the graph of a function is When the graph of a function is decreasing to the left of x = c and decreasing to the left of x = c and increasing to the right of x = c, then at increasing to the right of x = c, then at c the value of f is called a c the value of f is called a local local minimumminimum of the function. of the function.

LOCAL MINIMUMLOCAL MINIMUM

Use a graphing calculator to Use a graphing calculator to determine where f has a local determine where f has a local maximum and a local minimum:maximum and a local minimum:

f(x) = 6xf(x) = 6x33 - 12x + 5 for - 2 < x < 2 - 12x + 5 for - 2 < x < 2

max: (- 0.81, 11.53)max: (- 0.81, 11.53)

min: (0.81, -1.53)min: (0.81, -1.53)

EXAMPLE:EXAMPLE:

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

The The average rate of changeaverage rate of change for a function is a measure for a function is a measure of the change in the y-of the change in the y-coordinate as the coordinate as the corresponding x-coordinate corresponding x-coordinate changes.changes.

AVERAGE RATE OF CHANGE OF f BETWEEN c

and x

AVERAGE RATE OF CHANGE OF f BETWEEN c

and x If c is in the domain of a If c is in the domain of a function y = f(x), the average function y = f(x), the average rate of change of f between c rate of change of f between c and x is defined asand x is defined as

c x)c(f )x(f

xy

This expression is also called This expression is also called the the differencedifference quotientquotient of of ff at at cc..

EXAMPLE:EXAMPLE:

Suppose you drop a ball Suppose you drop a ball from a cliff 1000 feet high. from a cliff 1000 feet high. You measure the distance You measure the distance ss the ball has fallen after time the ball has fallen after time t t using a motion detector using a motion detector and obtain the data in Table and obtain the data in Table 3.3.

SecondsSeconds FeetFeet

00 0 0

11 16 16

22 64 64

33 144144

44 256256

55 400400

66 576576

77 784784

EXAMPLEEXAMPLE

Draw a scatter diagram of the Draw a scatter diagram of the data treating time as the data treating time as the independent variable.independent variable.

Draw a line from (0,0) to Draw a line from (0,0) to (2,64).(2,64).

Find the average rate of Find the average rate of change between these two change between these two points.points.

EXAMPLEEXAMPLE

Rate of Change = Rate of Change =

ft/sec.32 0 - 20 - 64

ts

EXAMPLEEXAMPLE

Interpret the average rate of Interpret the average rate of change found between the change found between the points.points.

The ball is dropping at an The ball is dropping at an average speed of 32 ft/sec average speed of 32 ft/sec between 0 and 2 seconds.between 0 and 2 seconds.

EXAMPLEEXAMPLE

Now draw a line from (5,400) Now draw a line from (5,400) to (7,784).to (7,784).

Find the average rate of Find the average rate of change between these two change between these two points.points.

EXAMPLEEXAMPLE

Rate of Change = Rate of Change =

ft/sec192 5 - 7 400- 784

ts

EXAMPLEEXAMPLE

Interpret the average rate of Interpret the average rate of change found between the change found between the points.points.

The ball is dropping at an The ball is dropping at an average speed of 192 ft/sec average speed of 192 ft/sec between 5 and 7 seconds.between 5 and 7 seconds.

EXAMPLEEXAMPLE

What is happening to the What is happening to the average rate of change or the average rate of change or the speed as time passes?speed as time passes?

It is increasing as time passes It is increasing as time passes since the ball is accelerating since the ball is accelerating due to the effect of gravity.due to the effect of gravity.

FINDING AVERAGE RATES OF CHANGE

FINDING AVERAGE RATES OF CHANGE

Do Examples 6 and 8Do Examples 6 and 8

CONCLUSION OF SECTION 1.3CONCLUSION OF SECTION 1.3