section 1 3
Download
Skip this Video
Download Presentation
SECTION 1.3

Loading in 2 Seconds...

play fullscreen
1 / 24

SECTION 1.3 - PowerPoint PPT Presentation


  • 70 Views
  • Uploaded on

SECTION 1.3. PROPERTIES OF FUNCTIONS. EVEN AND ODD FUNCTIONS. Identifying Graphically: g(x) = x 3 - 6x See the graph. Odd functions are symmetric with respect to the origin. EVEN AND ODD FUNCTIONS. Identifying Graphically:. h(x) = 0.1x 4 - 2x 2 + 5 See the graph.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' SECTION 1.3' - nicole-finch


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
section 1 3
SECTION 1.3
  • PROPERTIES OF FUNCTIONS
even and odd functions
EVEN AND ODD FUNCTIONS

Identifying Graphically:

g(x) = x 3 - 6x See the graph.

Odd functions are symmetric with respect to the origin.

even and odd functions1
EVEN AND ODD FUNCTIONS

Identifying Graphically:

h(x) = 0.1x 4 - 2x 2 + 5 See the graph.

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

even and odd functions2
EVEN AND ODD FUNCTIONS

Identifying Algebraically:

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

g(x) = x 3 - 6x

g(- x) = (- x) 3 - 6(- x)

= - x 3 + 6x = - g(x)

even and odd functions3
EVEN AND ODD FUNCTIONS

Identifying Algebraically:

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

h(x) = 0.1x 4 - 2x 2 + 5

h(- x) = 0.1(- x) 4 - 2(- x) 2 + 5

= 0.1x 4 - 2x 2 + 5

= h(x)

example
EXAMPLE:

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

g(x) = x 3½x½

k(x) = ½ x 5½

h(x) = x 5+ 1

Use both graphical and algebraic methods of identification.

increasing and decreasing functions
INCREASING AND DECREASING FUNCTIONS

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

INCREASING FUNCTION

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

DECREASING FUNCTION

example1
EXAMPLE:
  • Determine the intervals on which the function is decreasing.
  • k(x) = x3 - 6x2 - 13x
  • - .89 < x < 4.89
local maximum
LOCAL MAXIMUM
  • When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then at c the value of f is called a local maximum of the function.
local minimum
LOCAL MINIMUM
  • When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then at c the value of f is called a local minimum of the function.
example2
EXAMPLE:
  • Use a graphing calculator to determine where f has a local maximum and a local minimum:
  • f(x) = 6x3 - 12x + 5 for - 2 < x < 2

max: (- 0.81, 11.53)

min: (0.81, -1.53)

average rate of change
AVERAGE RATE OF CHANGE
  • The average rate of change for a function is a measure of the change in the y-coordinate as the corresponding x-coordinate 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 function y = f(x), the average rate of change of f between c and x is defined as

This expression is also called the differencequotient of f at c.

example3
EXAMPLE:
  • Suppose you drop a ball from a cliff 1000 feet high. You measure the distance s the ball has fallen after time t using a motion detector and obtain the data in Table 3.
slide15

Seconds Feet

0 0

1 16

2 64

3 144

4 256

5 400

6 576

7 784

example4
EXAMPLE

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

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

Find the average rate of change between these two points.

example5
EXAMPLE

Rate of Change =

example6
EXAMPLE

Interpret the average rate of change found between the points.

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

example7
EXAMPLE

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

Find the average rate of change between these two points.

example8
EXAMPLE

Rate of Change =

example9
EXAMPLE

Interpret the average rate of change found between the points.

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

example10
EXAMPLE

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

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

ad