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A circle is a set of points in the xy-plane that are a
fixed distance r from a fixed point (h, k). The
fixed distance r is called the radius, and the fixed
point (h, k) is called the center of the circle.
y
(x, y)
r
(h, k)
x
The standard form of an equation of a
circle with radius r and center (h, k) is
 x  h
2
 y  k  r
2
2
Graph (x 1)  ( y  3)  16 by hand.
2
2
( x  1)  ( y  3)  16
2
2
( x  ( 1))  ( y  3)  4
2
2
( x  h)  ( y  k )  r
2
2
h = -1, k = 3, r = 4
Center: (-1, 3), Radius: 4
2
2
(-1, 7)
y
(3,3)
(-5, 3)
(-1,3)
x
(-1, -1)
The general form of the equation of
a circle is
x  y  ax  by  c  0
2
2
Find the center and radius of
2
2
x  y  4 x  8 y  5  0.
x  4x  y  8y  5
2
2
x  4x_ y  8y_  5
2
2



2
  4  4
 
 2




2
 8  16
 
 2
x  4 x  4  y  8 y  16  5  4  16
2
2
 x  2   y  4
2
2
 25
Center: (2, -4), Radius: 5
f
x
y
x
y
x
X
DOMAIN
Y
RANGE
The domain of a function f is the set of
real numbers such that the rule makes
sense.
Find the domain of the following
functions:
g ( x )  3x  5x  1
3
Domain of g is all real numbers.
4
s( t ) 
t 1
Domain of s is t |t  1 .
h( z )  z  2
z20
z  2
Domain of h is z| z  2.
Determine the domain, range, and
intercepts of the following graph.
y
4
0
-4
(2, 3)
(1, 0)
(0, -3)
(4, 0)
(10, 0)
x
A function f is even if for every
number x in its domain the number
-x is also in the domain and
f(-x) = f(x)
A function f is odd if for every
number x in its domain the number
-x is also in the domain and
f(-x) = -f(x)
Theorem
A function is even if and only if
its graph is symmetric with
respect to the y-axis. A
function is odd if and only if its
graph is symmetric with respect
to the origin.
Symmetry examples
(a) g( z)   z  2
2
g(-z) = -(-z)2+2 =-z2+2
g ( z)  g (  z)
Even function, graph symmetric with
respect to the y-axis.
(b) f ( x)  4 x  3x
5
f (  x)  4(  x)5  3(  x)  4 x5  3x
f ( x)  f ( x)
Not an even function
 f ( x)  ( 4 x5  3x)  4 x5  3x
f ( x)   f ( x)
Odd function, and the graph is
symmetric with respect to the origin.