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Barnett/Ziegler/Byleen
College Algebra: A Graphing Approach
Chapter One
Functions and Graphs
Copyright © 2000 by the McGraw-Hill Companies, Inc.
The Cartesian Coordinate System
y axis
I
II
b
P(a,b)
Coordinates
{
Ordinate
Abscissa
Q(-9,5)
Origin
a
x axis
R(-10, -10)
III
IV
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-1-1
Distance Between Two Points
y
y2
P2 (x 2 , y 2 )
d(P1 , P2 )
x
0
x1
P1 (x 1 , y 1 )
y1
|yy22 –– y1y |1 |
x2
(x 2 , y 1 )
– xx1 |
||xx22 – 1|
d(P1, P2) =
(x2 – x1)2 + (y2 – y1)2
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-1-2
Circle
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
Standard Equation of a Circle
P(x, y)
Circle with radius r and center at (h,k):
r
(x – h)2 + (y – k) 2 = r 2
C(h, k)
r >0
x
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-1-3
Graphing Utility Screens
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
Definition of a Function
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
Functions Defined by Equations
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 = x2 – 4 defines a function.
•
The equation x2 + y2 = 16 does not define a function.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-3-6
Vertical Line Test for a Function
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
0
5
-5
x
-5
0
5
x
-5
(A) 4y – 3x = 8
(B) y2 – x2 = 9
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-3-7
Agreement on Domains and Ranges
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)
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
Increasing, Decreasing, and Constant Functions
f (x)
g(x)
5
10
f (x) = – x 3
–5
g(x) = 2x + 2
x
0
5
–10
–5
0
5
x
–5
(a) Decreasing on (–  )
(b) Increasing on (–,  )
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-4-9(a)
Increasing, Decreasing, and Constant Functions
h(x)
p(x)
5
5
h(x) = 2
–5
0
5
–5
x
–5
5
2
p(x) = x – 1
x
–5
(c) Constant on (–  )
(d) Decreasing on (–, 0]
Increasing on [0,  )
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-4-9(b)
Local Maxima and Local Minima
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)
Local maximum
f(c)
f(c)
a
c
b
x
Local minimum
a
c
b
Copyright © 2000 by the McGraw-Hill Companies, Inc.
x
1-4-10
Six Basic Functions
Absolute Value Function
g(x)
Identity Function
f(x)
5
5
–5
5
x
–5
5
g(x) = |x|
f(x) = x
–5
1.
x
2.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(a)
Six Basic Functions
Square Function
Cube Function
m(x)
h(x)
5
5
x
–5
2
h(x) = x
3.
–5
5
x
5
3
m(x) = x
–5
4.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(b)
Six Basic Functions
Cube-Root Function
Square-Root Function
p(x)
n(x)
5
5
x
5
n(x) =
5.
x
–5
5
–5
p(x) =
3
x
x
6.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1-5-11(c)
Graph Transformations
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:
A>1
y = A f(x)
0<A<1
Vertically expand graph of y = f(x) by
multiplying each ordinate value by A
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