chapter 3: the nature of graphs section 3-1: symmetry point symmetry: two distinct points p and p’...

Chapter 3: The Nature of Graphs

Section 3-1: Symmetry

Point Symmetry: Two distinct points P and P’ are symmetric with respect to point M if M is the midpoint of

Symmetry with respect to the Origin: The graph of a relation S is symmetric with respect to the origin if (-a, -b) are elements of S whenever (a, b) are in S. (they are called odd relations).

'PP


Example: Determine if is symmetric with respect to the origin.

Algebraically, this can be verified if

f(-x) = - f(x) for all x in the domain.

52)( xxf


Line Symmetry: Two distinct points P and P’ are symmetric with respect to a line k if k is the perpendicular bisector of

Symmetry with respect to the x and y axes: The graph of a relation S is symmetric with respect to the x-axis if (a, -b) are elements of S whenever (a, b) are in S. The graph of relation S is symmetric with respect to the y-axis if (-a, b) are elements of S whenever (a, b) are in S. (they are

called even relations.)

'PP


Symmetry with respect to the lines y = x and y = -x: The graph of a relation S is symmetric with respect to the line y= x if (b, a) are elements of S whenever (a, b) are in S. The graph of relation S is symmetric with respect to the line y = -x if (-b,-a) are elements of S whenever (a, b) are in S.


All Polynomial Functions whose powers are all

Even are Even Functions (symmetric to y-axis).

All Polynomial Functions whose powers are all Odd are Odd Functions (symmetric to Origin).

354 26 xxy

xxxxy 359 27


Example: Determine if the graph of

is symmetric with respect to the…

x-axis

y-axis

y = x

y = -x

the origin

35 2 xy


Example: Complete the graph below so that it is symmetric with respect to…

a) x-axis

b) y-axis

c) y = x

d) y = -x

e) the origin

6

4

2

-2

-4

-6

-5 5

5-Minute Check Lesson 3-2A

5-Minute Check Lesson 3-2B

Section 3-2: Analyzing Families of Graphs

A family of graphs is a group of graphs that displays one or more similar characteristics. A parent graph is a basic graph that can be transformed to create the other members in the family.

Some of the functions that you should be aware of: constant, identity, polynomial, square root, absolute value, greatest integer, rational


A line reflection flips a graph over a line called the axis of symmetry.

The shifting of a graph either vertically or horizontally is called a translation.

A dilation is a stretching or compressing of a graph.


The graph of y = f(x) + a translates the graph of f up ‘a’ units

In general, for a function y = f(x), the following are true when a > 0 …

The graph of y = f(x) - a translates the graph of f down ‘a’ units

The graph of y = f(x + a) translates the graph of f to the left ‘a’ units


The graph of y = f(x- a) translates the graph of f right ‘a’ units


The graph of y = -f(x) reflects the graph of f over the x-axis

The graph of y = f(-x) reflects the graph of f over the y-axis


The graph of y = a*f(x) is stretched vertically if a > 1.


The graph of y = a*f(x) is compressed vertically if 0 < a < 1.


The graph of y = f(ax) is compressed horizontally if a > 1.


The graph of y = f(ax) is stretched horizontally if 0 < a < 1.


Example: if the graph of f(x) is shown, draw the graph of each function based on f(x)…

a) f(x) + 1

b) f(x – 2)

c) f(-x)

d) 2*f(x)

6

4

2

-2

-4

-6

-5 5

A

B C


Example: Describe the transformations that have taken place in each of the related graphs:

2)3(.

5.

)(.

)2(.

)(

4

4

4

4

4

xyd

xyc

xyb

xya

xxf

Section 1-7: Piecewise Functions

A Piecewise Function is one that is defined by different rules for different intervals of the domain.

When graphing piecewise functions, the various pieces do not necessarily connect.


One example of a piecewise function is the Absolute value function.

0

0)(

xifx

xifxxf

Lesson Overview 1-7B

Lesson Overview 1-7A


Graph the following piecewise function:

2

23)(

2 xifx

xifxxg

Section 3-5: Continuity and End Behavior

All Polynomial Functions are continuous.

A function is continuous if there are no ‘breaks’ or ‘holes’ in the graph (you can graph the function without ‘lifting your pencil’).


There are 3 types of discontinuity that can occur with a function: Infinite, Jump, and Point.

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

A

6

4

2

-2

-4

-6

-5 5

B

E

F

Infinite Point Jump

Discontinuity Discontinuity Discontinuity

Lesson Overview 3-5A

Lesson Overview 3-5B


Here are three examples of discontinuous functions:

Infinite

Jump

Point

3

8)(

xxf

2,5

2,)(

xifx

xifxxg

2

44)(

2

x

xxxh


Example: Determine whether each function has infinite, point, or jump discontinuity.

a)

b)

c)

4

2

xy

3,

3,5

2 xifx

xifx

y

x

xy

|| 3


Continuity Test

A function is continuous at x = c if it satisfies the following conditions:

1. The function is defined at c ( f(c) exists )

2. The function approaches the same y-value on the left and right sides of x = c

3. The y-value that function approaches from each side is f(c).


Continuity on an Interval

A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval.

Example: is not continuous everywhere, but is continuous for x < 0 and for x > 0

2

1)(

xxf


To determine the end behavior for a polynomial function, look at the term with the highest power of x.

The end behavior of a function refers to the behavior of f(x) as |x| gets large (as x tends towards positive infinity or negative infinity).


If the highest power term is positive with an even value of n:

If the highest power term is positive with an odd value of n:

In general, for a polynomial function P such that…02

21

10 ...)( xaxaxaxaxP nnnn

yx yx

yx yx


If the highest power term is negative with an even value of n:

If the highest power term is negative with an odd value of n:

In general, for a polynomial function P such that…02

21

10 ...)( xaxaxaxaxP nnnn

yx yx

yx yx


Example: Without graphing, determine the end behavior of the following polynomial function…

3825 32 xxxy


Increasing and Decreasing Functions. Given that f is the function and are elements of f

A function f(x) is increasing if

whenever

A function f(x) is decreasing if

whenever21 xx

)()( 21 xfxf

)()( 21 xfxf

21 xx

21, xx


Increasing and Decreasing Functions. Given that f is the function and are elements of f

A function f(x) is constant if

whenever

)()( 21 xfxf

21 xx

21, xx


Increasing and Decreasing Functions.

Example: Determine if the following function is increasing or decreasing… 0,3)( 4 xxxf


Increasing and Decreasing Functions.

Example: Determine if the following function is increasing or decreasing…

3

2)(

xxg

Section 3-6: Graphs and Critical Points of Polynomial Functions

Graphs of Polynomial Functions have smooth curves with no breaks or holes.

For a function f(x), f’(x) (known as the derivative) defines the slope of that function. When f’(x) = 0, the tangent line to the curve at that point is horizontal. Points when f’(x) = 0 are called critical points.


Critical points can occur on a graph in three different forms: relative maximum, relative minimum, or a point of inflection.

Rel. Min. Rel. Max. Pt. of Inflection6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5

6

4

2

-2

-4

-6

-5 5


By determining the critical points and the x and y intercepts of a polynomial function, you can create a much more accurate and complete graph of the function.


Example: Determine the critical points for the following function. Then, determine whether each point represents a minimum, maximum, or point of inflection.

32)( 2 xxxf


Example: Determine the critical points for the following function. Then, determine whether each point represents a minimum, maximum, or point of inflection.

23 68)( xxxf

Section 3-7: Rational Functions and Asymptotes

The graph of a rational function usually includes vertical and horizontal asymptotes-lines towards which the graph tends as x approaches a specific value or as x approaches positive or negative infinity.

A Rational Function is an equation in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions, and q(x) does not equal zero.


A Hole occurs at x = a whenever there is a common factor (x – a) in the numerator and denominator of the function.

Example of a function with a hole:4

16)(

2

x

xxf


A Vertical Asymptote is line x = a for a function f(x) if from either the left or the right.

axasxforxf )()(

A Horizontal Asymptote is line y = b for a function f(x) if xorxasbxf )(


To find Vertical Asymptotes, determine when the denominator of the function will be equal to zero (not including ‘holes’).

6

3)(

2

xx

xxf


To find Horizontal Asymptotes, look at the limit of the function as x approaches infinity (3 scenarios):

xx

xxf

34

18)(

2

25

9)(

x

xxf

54

63)(

2

x

xxxf

y = 0 is horizontal asymptote since degree of denominator is larger

y = 9/5 is horizontal asymptote since degrees are equal

No horizontal asymptote since degree of numerator is larger


Determine the asymptotes for the following function:6

4

2

-2

-4

-6

-5 5

Section 3-7: Rational Functions and AsymptotesExample: Determine any holes, horizontal asymptotes, or vertical asymptotes for the following functions:

44

42

2

xx

xy

42

42

xx

y

x

xxy

3

52

Section 3-7: Rational Functions and AsymptotesExample: Create a function of the form y = f(x) that

satisfies the given set of conditions:

a) Vertical asymptote at x = 2, hole at x = -3

b) Vertical asymptotes at x = 5 and x = -9, resembles 2xy


A Slant Asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator.

2

3 1)(

x

xxf

Example of a function with a slant asymptote:

The equation of the slant asymptote for this example is y = x.


To determine the equation of the slant asymptote, divide the numerator by the denominator, and then see what happens as x

Example: Determine the slant asymptote for the following function:

3

143)(

2

x

xxxf

A graph is said to be symmetric withrespect to the x-axis if, for every point(x, y) on the graph, the point (x, -y) isalso on the graph.

If a graph is symmetric with respect tothe x-axis and the point (3, 5) is onthe graph then (3, -5) is also on thegraph.

A graph is said to be symmetric withrespect to the y-axis if, for every point(x, y) on the graph, the point (-x, y) is alsoon the graph.

If a graph is symmetric with respect to they-axis and the point (3, 5) is on the graph then(-3, 5) is also on the graph.

A graph is said to be symmetric withrespect to the origin if, for every point(x, y) on the graph, the point (-x, -y) is alsoon the graph.

If a graph is symmetric with respect to the originand the point (3, 5) is on the graph then (-3, -5)is also on the graph.

Test 3 2 242x y for symmetry.

x-axis: 3 2 242x y 3 2 242x y ( )3 2 242x y

Not symmetric with respect to x-axis

y-axis: 3 2 242x y

3 2 242 x y

3 2 242x y Symmetric with respect to y-axis

origin: 3 2 242x y

3 2 242 x y

3 2 242x y

Not symmetric with respect to origin

Test 3 2 242x y for symmetry.

chapter 3: the nature of graphs section 3-1: symmetry point symmetry: two distinct points p and p’...

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