section 3.3 properties of functions. testing the graph of a function for symmetry

Section 3.3

Properties of Functions

Testing the Graph of a Function for Symmetry

Symmetries

Definition: Two points (a , b) and (c , d) in the plane

are said to be symmetric about the x-axis if a c and b d symmetric about the y-axis if a c and b d symmetric about the origin if a c and b d

Symmetries

Schematically

In the above figure, P and S are symmetric about the x-axis, as are Q

and R; P and Q are symmetric about the y-axis, as are R and S; and P

and R are symmetric about the origin, as are Q and S.

Reflections

To reflect a point (x , y) about the:

x-axis, replace y with y. y-axis, replace x with x. origin, replace x with x and y with y.

Symmetries of Relations

Definition: Symmetry with respect to the x-axis.

A relation R is said to be symmetric with respect to

the x-axis if, for every point (x , y) in R, the point

(x , y) in also in R.

That is, the graph of the relation R remains the

same when we change the sign of the y-coordinates

of all the points in the graph of R.


Definition: Symmetry with respect to the y-axis.


the y-axis if, for every point (x , y) in R, the point

( x , y) in also in R.


same when we change the sign of the x-coordinates

of all the points in the graph of R.


Definition: Symmetry with respect to the origin.


the origin if, for every point (x , y) in R, the point

( x , y) in also in R.


same when we change the sign of the x- and the y-

coordinates of all the points in the graph of R.


In many situations the relation R is defined by an

equation, that is, R is given by

R {(x , y) | F (x , y) 0},

where F (x , y) is an algebraic expression in the

variables x and y.

In this case, testing for symmetries is a fairly

simple procedure as explained in the next slides.

Testing the Graph of an Equation for Symmetry

To test the graph of an equation for symmetry about the x-axis: substitute (x , y) into the

equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the x-axis.

about the y-axis: substitute ( x , y) into the equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the y-axis.

Testing the Graph of an Equation for Symmetry

To test the graph of an equation for symmetry about the origin: substitute ( x , y) into the

equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the origin.

Examples

Find the x- and y-intercepts (if any) of the graph of

(x 2)2 + y 2 = 1. Test for symmetry.

Examples


x 3 + y

33xy = 0. Test for symmetry.

Examples


x 4 = x

2 + y 2. Test for symmetry.

Examples


y 2 = x

3 + 3x 2. Test for symmetry.

Examples


(x 2 + y

2)2 = x 3 + y

3. Test for symmetry.

A Happy Relation


10/(x2 + y21) + 1/((x 0.3)2+(y 0.5)2) +

1/((x + 0.3)2 + (y 0.5)2) + 10/(100x2 + y2) +

1/(0.05x2 +10(y x2 + 0.6)2)=100

Test for symmetry.

A Happy Relation

Symmetries of Functions

When the relation G is a function defined by an

equation of the form y f (x) , then G is given by

G {(x , y) | y f (x) }.

In this case, we can only have symmetry with

respect to the y-axis and symmetry with respect to

the origin. A function cannot be symmetric with respect to the x-axis

because of the vertical line test.

Even and Odd Functions

Definition: A function f is called even if its graph

is symmetric with respect to the y-axis.

Since the graph of f is given by the set

{(x , y) | y f (x) }

symmetry with respect to the y-axis implies that

both (x , y) and ( x , y) are on the graph of f.

Therefore, f is even when f (x) f (x) for all x

in the domain of f.

Even and Odd Functions

Definition: A function f is called odd if its graph

is symmetric with respect to the origin.

Since the graph of f is given by the set

{(x , y) | y f (x) }

symmetry with respect to the origin implies that

both (x , y) and ( x , y) are on the graph of f.

Therefore, f is odd when f (x) f (x) for all x

in the domain of f.

Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.

Even function because it is symmetric with respect to the y-axis

Neither even nor odd. No symmetry with respect to the y-axis or the origin.

Odd function because it is symmetric with respect to the origin.

Examples Using the Graph

3) 5a f x x x 35f x x x 3 5x x f x

Odd function symmetric with respect to the origin

2) 2 3b g x x 232g x x 22 3 ( )x g x

Even function symmetric with respect to the y-axis

Examples Using the Equation

Determine whether each of the following functions is even, odd function, or neither.

3) 14c h x x 34 1h x x 34 1x

Since h (– x) does not equal h (x) nor – h (x), this function h is neither even nor odd and is not symmetric with respect to the y-axis or the origin.

General Function Behavior

Using the graph to determine where the function is increasing,

decreasing, or constant

INCREASING

DECREASING

CONSTANT

Find all values of x where the function is increasing.

Example Using the Graph

The function is increasing on 4 < x < 0. That is, on (4 , 0).


The function is decreasing on 6 < x < 4 and also on 3 < x < 6. That is, on (6 ,

4 ) (3 , 6 )

Find all values of x where the function is decreasing.


The function is constant on 0 < x < 3. That is, on (0 , 3).

Find all values of x where the function is constant.

Precise Definitions

Suppose f is a function defined on an interval I. We

say f is: increasing on I if and only if f (x1) < f (x2) for all

real numbers x1, x2 in I with x1 < x2.

decreasing on I if and only if f (x1) > f (x2) for all real numbers x1, x2 in I with x1 < x2.

constant on I if and only if f (x1) f (x2) for all real numbers x1, x2 in I.


Using the graph to find the location of local maxima and local minima

Local Maxima

The local maximum is f (c) and occurs at x c

Local Minima

The local minimum is f (c) and occurs at x c

Precise Definitions

Suppose f is a function with f (c) d.

We say f has a local maximum at the point (c , d) if

and only if there is an open interval I containing c

for which f (c) f (x) for all x in I.

The value f (c) is called a local maximum value of

f and we say that it occurs at x c.

Precise Definitions

Suppose f is a function with f (c) d.

We say f has a local minimum at the point (c , d) if

and only if there is an open interval I containing c

for which f (c) f (x) for all x in I.

The value f (c) is called a local minimum value of

f and we say that it occurs at x c.

There is a local maximum when x = 1.

The local maximum value is 2

Example

There is a local minimum when x = 1 and x = 3.

The local minima values are 1 and 0.

Example

(e) The region where f is increasing is.

(f) The region where f is decreasing is.

1,1 3,

, 1 1,3

Example


Using the graph to locate the absolute maximum and the

absolute minimum of a function

Absolute Extrema

Absolute Minimum

Let f be a function defined on a domain D and let c be in D.

Absolute Maximum

The number f (c) is called the absolute maximum value of f in D and we say that it occurs at c.

A function f has an absolute (global) maximum at x c in D if f (x) f (c) for all x in the domain D of f.

Absolute Maximum

Absolute Extrema

c

( )f c

Absolute Minimum

Absolute Extrema

A function f has an absolute (global) minimum at x c in D if f (c) f (x) for all x in the domain D of f.The number f (c) is called the absolute minimum value of f in D and we say that it occurs at c.

c

( )f c

Find the absolute extrema of f, if they exist.

The absolute maximum of 6 f (3) occurs when x 3.

The absolute minimum of 1 f (0) occurs when x 0.

Examples of Absolute Extrema

The absolute maximum of 3 f (5) occurs when x = 5.

There is no absolute minimum because of the “hole” at x = 3.



The absolute maximum of 4 f (5) occurs when x = 5.

The absolute minimum of 1 occurs at any point on the interval [1,2].



There is no absolute maximum.

The absolute minimum of 0 f (0) occurs when x = 0.



There is no absolute maximum.

There is no absolute minimum.



Extreme Value Theorem

If a function f is continuous* on a closed interval [a, b], then f has both, an absolute maximum and absolute minimum on [a, b]. Moreover each absolute extremum occurs at a local extrema or at an endpoint.

*Although it requires calculus for a precise definition, we will agree for now that a continuous function is one whose graph has no gaps or holes and can be traced without lifting the pencil from the paper.

a b a ba b

Attains both a max. and min.

Attains a min. but no a max.

No min. and no max.

Open Interval Not continuous

Extreme Value Theorem

Using a Graphing Utility to Approximate Local Extrema

3Use a graphing utility to graph 2 3 1 for 2 2.

Approximate where has any local maxima or local minima.

f x x x x

f

Example

Example

3Use a graphing utility to graph 2 3 1 for 2 2.

Determine where is increasing and where it is decreasing.

f x x x x

f

Average Rate of Changeof a Function

Delta Notation

2 1q q q

If a quantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Example: If x is changed from 2 to 5, we write

2 1 5 2 3x x x

Delta Notation

2 1q q q

If a quantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Note: In the definition of q we do not assume that q1 < q2. That is, q may be positive, negative, or zero depending on the values of q1 and q2.

Delta Notation

Example - Slope: the slope of a non-vertical line that passes through the points (x1 , y1) and (x2 , y2) is given by:

2 1

2 1

y yym

x x x

change in

change in

f f

x x

( ) ( )f b f a

b a

The average rate of change of f (x) over the interval [a , b] is

The change of f (x) over the interval [a , b] is

Difference Quotient

( ) ( )f f b f a

Average Rate of Change

a) From 1 to 3, that is, over the interval [1 , 3].


b) From 1 to 5, that is, over the interval [1 , 5].


c) From 1 to 7, , that is, over the interval [1 , 7].


Geometric Interpretation of the Average Rate of Change

, ( )P a f a

, ( )Q b f b

secant line has slope mS

( ) ( )S PQ

f f b f am m

x b a

Is equal to the slope of the secant line through the points (a , f (a)) and (b , f (b)) on the graph of f (x)

x b a

( ) ( )f f b f a


Example: Compute the average rate of change of over [0 , 2].2( ) 3 2f x x x

( ) ( )

(2) (0)

2 0

S PQf f b f a

m mx b a

f f

8 0

2 0

= 4

This means that the slope of the secant line to the graph of parabola through the points (0 , 0) and (2 , 8) is mS 4.


Geometric Interpretation

8f

2x

Q

P

The units of f , change in f , are the units of f (x).

The units of the average rate of change of f are units of f (x) per units of x.

units of ( )is given in

units of

f f x

x x

Units for f x

Example

The graph on the next slide shows the numbers of

SUVs sold in the US each year from 1990 to 2003.

Notice t 0 represents the year 1990, and N(t)

represents sales in year t in thousands of vehicles.

a) Use the graph to estimate the average rate of change of N(t) with respect to t over [6 , 11] and interpret the result.

b) Over which one-year period (s) was the average rate of N(t) the greatest?

Example

a) The average rate of change of N over [6 , 11] is given by the slope of the line through the points P and Q.

Example

P

Q

N

t

P

Q

N

t

thousands SUVsyear

(11) (6) 20011 6

N N Nt

Compute

Sales of SUVs were increasing at an average rate of 200,000 SUVs per year from 1996 to 2001.

P

Q

N

t

Interpret

Suppose that g (x) 2x2 + 4x 3.

a) Find the average rate of change of g from 2 to 1.

b) Find the equation of the secant line containing the points on the graph corresponding to x 2 and x 1.

c) Use a graphing utility to draw the graph of g and the secant line on the same screen.

Equation of the Secant line

Suppose that g (x) 2x2 + 4x 3.a) Find the average rate of change of g from 2 to 1.

Solution

1 19(1) ( 2)

1 2 3

18 6

3

y g g

x

b) Find the equation of the secant line containing the points on the graph corresponding to x 2 and x 1.

From part a) we have m 6. Using the point-slope

formula for the line we have


1 1

( 1

9)

6( ( 2))

19 6 12

( )y y m x

x

y x

x

y

6or 7y x

c) Use a graphing utility to draw the graph of g and the secant line on the same screen.


section 3.3 properties of functions. testing the graph of a function for symmetry

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