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Testing the Graph of a Function for Symmetry. Symmetries. Definition: Two points (a , b) and (c , d) in the plane are said to besymmetric about the x-axis if a ? c and b ? ?dsymmetric about the y-axis if a ? ?c and b ? dsymmetric about the origin if a ? ?c and b ? ?d. Symmetries. Schematically
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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 ? x2 + 0.6)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.
24. General Function Behavior Using the graph to determine where the function is increasing, decreasing, or constant
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
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.
38. General Function Behavior Using the graph to locate the absolute maximum and the absolute minimum of a function
39. Absolute Extrema
40. Absolute Extrema
41. Absolute Extrema
47. Extreme Value Theorem
48. Extreme Value Theorem
49. Using a Graphing Utility to Approximate Local Extrema
52. Average Rate of Change of a Function
53. Delta Notation
54. Delta Notation
60. Geometric Interpretation of the Average Rate of Change
63. Geometric Interpretation
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?
70. 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. Equation of the Secant line
71. Suppose that g (x) ? ?2x2 + 4x ?3.
Find the average rate of change of g from ?2 to 1. Solution
72. 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 Equation of the Secant line
73. Use a graphing utility to draw the graph of g and the secant line on the same screen. Equation of the Secant line
