The Nature of Graphs - Ch 03
Point Symmetry
Two distinct points P and P’ are symmetric
with respect to a point, M, if and only if
M is the midpoint of line segment PP’
_______________________________.
P
itself
Point M is symmetric with respect to ____.
M
P'
3-1
p. 106
The Nature of Graphs - Ch 03
Symmetry with Respect to the Origin
The graph of a relation S is symmetric with
respect to the origin if and only if
(-a,-b) is an element of S
_______________________whenever
(a,b) is an element of S.
A function f(x) has a graph that is
symmetric with respect to the origin
f(-x) = -f(x)
if and only if _____________.
3-1
p. 107
The Nature of Graphs - Ch 03
Line Symmetry
Two distinct points P and P’ are symmetric
with respect to a line L if and only
perpendicular bisector
if L is the ___________________of
the
line segment PP’.
A point P is symmetric to itself with respect
P is on line L
to line L if and only if ______________.
3-1
p. 108
The Nature of Graphs - Ch 03
Definition of Inverse Relations
Two relations are inverse relations
if and only if one relation contains
(b, a) whenever the
the element ______
other relation contains the
element (a, b).
-1(x)
f
The inverse of f(x) is written ________.
3-3
p. 126
The Nature of Graphs - Ch 03
Vertical Asymptote
The line x = a is a vertical asymptote for
infinity
the function f(x) if f(x) approaches ______
negative infinity
or f(x) approaches ________________
as x approaches a from either the
left or the right.
3-4
p. 135
The Nature of Graphs - Ch 03
Horizontal Asymptote
The line y = b is a horizontal asymptote for
the function f(x) if f(x) approaches b
x approaches infinity
as _______________________
x approaches negative infinity
or as _________________________.
3-4
p. 135
The Nature of Graphs - Ch 03
Slant Asymptote
The oblique line L is a slant asymptote for a
function f(x) if the graph of f(x)
x approaches infinity
approaches L as ___________________
x approaches negative infinity
or as ___________________________.
An oblique line is a line that is
neither vertical nor horizontal
_________________________.
3-4
p. 135
The Nature of Graphs - Ch 03
Rules for Derivatives
Constant Rule:
The derivative of a constant function is
constant
_________.
If f(x) = c, then f ‘ (x) = 0.
3-6
p. 151
The Nature of Graphs - Ch 03
Rules for Derivatives, continued
Power Rule:
If f(x) = xn, where n is a rational number,
then the derivative of f(x) is…
.
(n-1)
f ‘ (x) = n x
3-6
p. 151
The Nature of Graphs - Ch 03
Rules for Derivatives, continued
Constant Multiple of a Power Rule:
If f(x) = c xn, where c and n are both
rational numbers, then the derivative of f(x)
is…
f ‘ (x) = n c x(n-1)
3-6
p. 151
The Nature of Graphs - Ch 03
Rules for Derivatives, continued
Sum Rule:
If f(x) = g(x) + h(x), then the derivative
of f(x) is …
f‘ (x) = g‘ (x) + h‘ (x)
3-6
p. 151
The Nature of Graphs - Ch 03
Conditions for Continuity
A function f(x) is continuous at a point x = c
if it satisfies three conditions:
(1) the function is defined at c;
(2) the function approaches the same
y-value to the left and the right of x=c;
(3) the y-value that the function approaches
from each side is f(c).
3-8
p. 164
The Nature of Graphs - Ch 03
Continuity on an Interval
A function f(x) is continuous on an interval
if it is continuous for each value of x
in that interval.
3-8
p. 165
The Nature of Graphs - Ch 03
Increasing and Decreasing Functions
A function f(x) is increasing if and only if
f(x1) < f(x2) whenever x1 < x2.
A function f(x) is decreasing if and only if
f(x1) > f(x2) whenever x1 < x2.
3-8
p. 168