1. Properties of Functions
Section 3.3
Properties of Functions 1

2. Testing the Graph of a Function for Symmetry

3. 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

4. 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.

5. 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.

6. 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.

7. Symmetries of Relations
Definition: Symmetry with respect to the y-axis.
A relation R is said to be symmetric with respect to
the y-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 x-coordinates
of all the points in the graph of R.

8. Symmetries of Relations
Definition: Symmetry with respect to the origin.
A relation R is said to be symmetric with respect to
the origin 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 x- and the y-
coordinates of all the points in the graph of R.

9. Symmetries of Relations
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.

10. 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.

11. 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.

12. Examples Find the x- and y-intercepts (if any) of the graph of
(x  2)2 + y 2 = 1. Test for symmetry.

13. Examples Find the x- and y-intercepts (if any) of the graph of
x 3 + y 33xy = 0. Test for symmetry.

14. Examples Find the x- and y-intercepts (if any) of the graph of
x 4 = x 2 + y 2. Test for symmetry.

15. Examples Find the x- and y-intercepts (if any) of the graph of
y 2 = x 3 + 3x 2. Test for symmetry.

16. Examples Find the x- and y-intercepts (if any) of the graph of
(x 2 + y 2)2 = x 3 + y 3. Test for symmetry.

17. A Happy Relation Find the x- and y-intercepts (if any) of the graph of
10/(x2 + y21) + 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  x )2)=100
Test for symmetry.

18. A Happy Relation

19. 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.

20. 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.

21. 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.

22. Examples Using the Graph
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.

23. Examples Using the Equation
Determine whether each of the following functions is even, odd function, or neither.
Odd function symmetric with respect to the origin
Even function symmetric with respect to the y-axis
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.

24. General Function Behavior
CONSTANT
Using the graph to determine where the function is increasing, decreasing, or constant
INCREASING
DECREASING

25. Example Using the Graph
Find all values of x where the function is increasing.
The function is increasing on 4 < x < 0. That is, on (4 , 0).

26. Example Using the Graph
Find all values of x where the function is decreasing.
The function is decreasing on 6 < x < 4 and also on 3 < x < 6. That is, on (6 , 4 )  (3 , 6 )

27. Example Using the Graph
Find all values of x where the function is constant.
The function is constant on 0 < x < 3. That is, on (0 , 3).

28. 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.

30. General Function Behavior
Using the graph to find the location of local maxima and local minima

31. The local maximum is f (c) and occurs at x  c
Local Maxima
The local maximum is f (c) and occurs at x  c

32. The local minimum is f (c) and occurs at x  c
Local Minima
The local minimum is f (c) and occurs at x  c

33. 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.

34. 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.

35. Example There is a local maximum when x = 1.
The local maximum value is 2

36. Example There is a local minimum when x = 1 and x = 3.
The local minima values are 1 and 0.

37. Example (e) The region where f is increasing is.
(f) The region where f is decreasing is.

38. General Function Behavior
Using the graph to locate the absolute maximum and the absolute minimum of a function

39. Absolute Extrema
Let f be a function defined on a domain D and let c be in D.
Absolute Maximum
Absolute Minimum

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

41. 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.
Absolute Minimum

42. Examples of Absolute Extrema
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.

43. Examples of Absolute Extrema
Find the absolute extrema of f, if they exist.
The absolute maximum of 3  f (5) occurs when x = 5.
There is no absolute minimum because of the “hole” at x = 3.

44. Examples of Absolute Extrema
Find the absolute extrema of f, if they exist.
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].

45. Examples of Absolute Extrema
Find the absolute extrema of f, if they exist.
There is no absolute maximum.
The absolute minimum of 0  f (0) occurs when x = 0.

46. Examples of Absolute Extrema
Find the absolute extrema of f, if they exist.
There is no absolute maximum.
There is no absolute minimum.

47. 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.

48. Extreme Value Theorem Attains both a max. and min.
a b
a b
a b
Attains both a max. and min.
Attains a min. but no a max.
No min. and no max.
Open Interval
Not continuous

49. Using a Graphing Utility to Approximate Local Extrema

50. Example

51. Example

52. Average Rate of Change of a Function

53. Delta Notation
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

54. Delta Notation
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.

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

56. Average Rate of Change
The change of f (x) over the interval [a , b] is
The average rate of change of f (x) over the interval [a , b] is
Difference Quotient

57. Average Rate of Change
a) From 1 to 3, that is, over the interval [1 , 3].

58. Average Rate of Change
b) From 1 to 5, that is, over the interval [1 , 5].

59. Average Rate of Change
c) From 1 to 7, , that is, over the interval [1 , 7].

60. Geometric Interpretation of the Average Rate of Change

61. Average Rate of Change
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)
secant line has slope mS

62. Average Rate of Change
Example: Compute the average rate of change of over [0 , 2].
= 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.

63. Geometric Interpretation

64. Units for f x
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.

65. 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.
Use the graph to estimate the average rate of change of N(t) with respect to t over [6 , 11] and interpret the result.
Over which one-year period (s) was the average rate of N(t) the greatest?

66. Example

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

68. Compute

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

70. Equation of the Secant line
Suppose that g (x)  2x2 + 4x 3.
Find the average rate of change of g from 2 to 1.
Find the equation of the secant line containing the points on the graph corresponding to x  2 and x  1.
Use a graphing utility to draw the graph of g and the secant line on the same screen.

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

72. Equation of the Secant line
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

73. Equation of the Secant line
Use a graphing utility to draw the graph of g and the secant line on the same screen.