Copyright © 2000 by the McGraw-Hill Companies, Inc.

y axis
II
Coordinates
Q(-9,5) b
Origin a
I
P(a,b)
Ordinate
Abscissa x axis
III
R(-10, -10)
IV
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-1-1

y y
2
P
2
( x
2
, y
2
) x
1
P
1
( x
1
, y
1
) d ( P
1
, P
2
)
0 y
1
|
| x x –  x x
1
|
1
| x
2 x
( x
2
, y
1
)
d(P1, P2) = (x2 – x1)
2 + (y
2 – y1)
2
Copyright © 2000 by the McGraw-Hill Companies, Inc.
| | y y –
–  y
1 y |
1
|
1-1-2

A circle is the set of all points in a plane equidistant from a fixed point. The fixed distance is called the radius, and the fixed point is called the center. y
C r
( h, k )
P ( x, y )
Standard Equation of a Circle x
Circle with radius r and center at (h,k):
(x – h)2 + (y – k) 2 = r 2 r > 0
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-1-3

An image on the screen of a graphing utility is made up of darkened rectangles called pixels. The pixel rectangles are the same size, and do not change in shape during any application. Graphing utilities use pixel-by-pixel plotting to produce graphs.
Image Magnification to show pixels
The portion of a rectangular coordinate system displayed on the graphing screen is called a viewing window and is determined by assigning values to six window variables: the lower limit, upper limit, and scale for the x axis; and the lower limit, upper limit, and scale for the y axis.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-2-4

Rule Form
A function is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set.
The first set is called the domain of the function, and the set of all corresponding elements in the second set is called the range.
Set Form
A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components.
The set of all first components in a function is called the domain of the function, and the set of all second components is called the range.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-3-5

In an equation in two variables, if to each value of the independent variable there corresponds exactly one value of the dependent variable, then the equation defines a function.
If there is any value of the independent variable to which there corresponds more than one value of the dependent variable, then the equation does not define a function.
• The equation y = x 2 – 4 defines a function.
• The equation x 2 + y 2 = 16 does not define a function.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-3-6

An equation defines a function if each vertical line in the rectangular coordinate system passes through at most one point on the graph of the equation.
If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function.
y y
5 5
-5 5 x
0 -5 0
-5 -5
(A) 4y – 3x = 8 (B) y 2 – x 2 = 9
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5 x
1-3-7

If a function is defined by an equation and the domain is not indicated, then we assume that the domain is the set of all real number replacements of the independent variable that produce real
values for the dependent variable.
The range is the set of all values of the dependent variable corresponding to these domain values.
The symbol f(x) represents the real number in the range of the function f corresponding to the domain value x. Symbolically, f: x
 f(x). The ordered pair (x, f(x)) belongs to the function f.
If x is a real number that is not in the domain of f, then f is not
defined at x and f(x) does not exist.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-3-8

–5 f ( x )
10
0
f (x) = – x 3
5 x
–5 g ( x )
5
0
g(x) = 2x + 2
5 x
–10
(a) Decreasing on ( –  
)
–5
(b) Increasing on ( – 
,

)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-4-9(a)

h ( x )
5
h(x) = 2
0 5 x p ( x )
5
–5 –5
p(x) = x
2
5
– 1 x
–5 –5
(c) Constant on ( –  
) (d) Decreasing on ( – 
, 0]
Increasing on [0,

)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-4-9(b)

The functional value f(c) is called a local maximum if there is an interval (a, b) containing c such that f(x)
 f(c) for all x in (a, b).
f(x)
The functional value f(c) is called a local minimum if there is an interval (a, b) containing c such that f(x)
 f(c) for all x in (a, b).
f(x) f(c) Local maximum f(c) Local minimum a c b x a c
Copyright © 2000 by the McGraw-Hill Companies, Inc.
b x
1-4-10

Six Basic Functions
Identity Function f(x)
5
Absolute Value Function g(x)
5
–5
–5
1.
f(x) = x
5 x
–5
2.
g(x) = |x|
5 x
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(a)

Six Basic Functions
–5
Square Function h(x)
5
3.
h(x) = x
2
5 x
Cube Function m(x)
5
–5
–5 m(x) = x
3
4.
5 x
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(b)

Six Basic Functions
Square-Root Function n(x)
5
5.
5 n(x) = x x
Cube-Root Function p(x)
5
–5
–5 p(x) =
3 x
6.
5 x
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(c)

Vertical Translation: y = f(x) + k k > 0 Shift graph of y = f(x) up k units k < 0 Shift graph of y = f(x) down
 k
 units
Horizontal Translation: y = f(x+h) h > 0 Shift graph of y = f(x) left h units h < 0 Shift graph of y = f(x) right
 h
 units
Reflection: y = – f(x) Reflect the graph of y = f(x) in the x axis
Vertical Expansion and Contraction: y = A f(x)
A > 1 Vertically expand graph of y = f(x) by multiplying each ordinate value by A
0 < A < 1 Vertically contract graph of y = f(x) by multiplying each ordinate value by A
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-12