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CHAPTER 8
Introduction
to Functions
SECTION 8-1
Equations in
Two Variables
DEFINITIONS
Open sentences in two
variables– equations and
inequalities containing
two variables
EXAMPLES
9x + 2y = 15
y = x2 – 4
2x – y ≥ 6
DEFINITIONS
Solution– is a pair of
numbers (x, y) called
an ordered pair.
Example
State whether each
ordered pair is a
solution of 4x + 3y = 10
(4, -2)
(-2, 6)
DEFINITIONS
Solution set– is the set of
all solutions satisfying
the sentence.
EXAMPLE
Solve the equation:
9x + 2y = 15
if the domain of x is
{-1,0,1,2}
SOLUTION
x
-1
0
1
(15-9x)/2
[15-9(-1)]/2
[15-9(0)]/2
[15-9(1)]/2
y
12
15/2
3
Solution
(-1,12)
(0,15/2)
(1,3)
2 [15-9(2)]/2 -3/2 (2,-3/2)
SOLUTION
the solution set is
{(-1,12), (0, 15/2), (1,3),
(2,-3/2)}
EXAMPLE
Roberto has $22. He buys
some notebooks costing $2
each and some binders
costing $5 each. If Roberto
spends all $22 how many of
each does he buy?
SOLUTION
n = number of notebooks
b = number of binders
(n and b must be whole
numbers)
2n + 5b =22
n = (22-5b)/2
SOLUTION
b
0
2
4
(22-5b)/2
[22-5(0)]/2
[22-5(2)]/2
[22-5(4)]/2
n
11
6
1
Solution
(0,11)
(2,6)
(4,1)
6
[12-5(6)]/2
-4
Impossible
SOLUTION
the solution set is
{(0,11), (2, 6), (4,1)}
SECTION 8-2
Points, Lines
and Their
Graphs
COORDINATE PLANE
consists of two
perpendicular number
lines, dividing the plane
into four regions called
quadrants
X-axis (abscissa)- the
horizontal number line
Y-axis (ordinate) - the
vertical number line
ORIGIN - the point where
the x-axis and y-axis cross
DEFINITION
ORDERED PAIR - a unique
assignment of real
numbers to a point in the
coordinate plane
consisting of one xcoordinate and one ycoordinate
DEFINITION
GRAPH – is the set of all
points in the coordinate
plane whose coordinates
satisfy the open
sentence.
LINEAR EQUATION
is an equation whose
graph is a straight
line.
Standard Form
Ax + By = C where A, B, C are
real numbers with A and B
not both zero. If A, B, C are
integers, the equation is in
standard form.
Example
Are these equations in
standard form?
2x – 5y = 7
0.5x + 4y = 12
2
x y + 3y = 4
1/x + 3y = 1
Graph the following lines
2x – 3y = 6
x = -2
y=3
SECTION 8-3
The Slope of
a Line
Property
A basic property of a
straight line is that its
slope is constant.
SLOPE
is the ratio of vertical
change to the
horizontal change.
The variable m is used
to represent slope.
FORMULA FOR SLOPE
m = change in y-coordinate
change in x-coordinate
m = rise
run
Or
SLOPE OF A LINE
m = y2 – y1
x2 – x1
Find the slope of the line
that contains the given
points.
M(4, -6) and N(-2, 3)
Find the slope of the line
that contains the given
points.
M(-2, 3) and N(4, 8)
HORIZONTAL LINE
a horizontal line
containing the point
(a, b) is described by
the equation y = b and
has slope of 0
VERTICAL LINE
a vertical line
containing the point
(c, d) is described by
the equation x = c and
has no slope
SECTION 8-4
The Slope
Intercept Form
of a Linear
Equation
SLOPE-INTERCEPT FORM
y = mx + b
where m is the slope and
b is the y -intercept
Y-Intercept
is the point where the
line intersects the y axis.
X-Intercept
is the point where the
line intersects the
x -axis.
Find the slope and
y-intercept and use
them to graph each
equation
1. y = -3/4x + 6
2. 2x – 5y = 10
THEOREM
Let L1 and L2 be two
different lines, with
slopes m1 and m2
respectively.
1.
L1 and L2 are parallel
if and only if m1=m2
THEOREM
and
2. L1 and L2 are
perpendicular if
and only if m1m2 = -1
Find the slope of a line
parallel to the line
containing points M and
N.
M(-2, 5) and N(0, -1)
Find the slope of a line
perpendicular to the line
containing points M and N.
M(4, -1) and N(-5, -2)
SECTION 8-5
Determining an
Equation of a
Line
Write an equation of a line
with the given y-intercept
and slope
m=3 b = 6
Remember: y=mx+b
THEOREM
Let P(x1,y1) be a point and m
a real number. There is one
line through P having slope
m. An equation of the line is
y – y1 = m (x – x1)
Write an equation of a line
with the given slope, passing
through the given point.
m = 1/2; (-8, 4)
Write an equation of a line
passing through the given
points
A(1, -3) B(3,2)
SECTION 8-6
Function Defined
by Equations
MAPPING
DIAGRAM
A picture
showing a
correspondence
between two
sets
MAPPING – the
relationship between the
elements of the domain
and range
FUNCTION
A correspondence
between two
sets, D and R,
that assigns to
each member of D
exactly one
member of R.
DOMAIN – the set of all
possible x-coordinates
RANGE – the set of all
possible y-coordinates
RANGE
The set R of the
function
assigned to at
least one
member of D.
SECTION 8-7
Function Defined
by Equations
FUNCTION
A correspondence
between two
sets, D and R,
that assigns to
each member of D
exactly one
member of R.
DOMAIN – the set of all
possible x-coordinates
RANGE – the set of all
possible y-coordinates
FUNCTIONAL
NOTATION
f (x) denotes the
value of f at x
ARROW
NOTATION
f:x
x+3
Is read the
function f that
pairs x with x + 3
VALUES of a
FUNCTION
The members of
its range.
EXAMPLE 1
Given f : x→4x – x2 with
domain D= {1,2,3,4,5}
Find the range of f.
Example 2
Given g:x
4 + 3x –
with domain D={-1, 0, 1, 2}
2
x
Find the range of g.
SECTION 8-8
Linear and
Quadratic
Functions
Linear Function
Is a function f that can be
defined by f(x) = mx + b
Where x, m and b are real
numbers. The graph of f
is the graph of y = mx
+b, a line with slope m
and y-intercept b.
FUNCTION
is a relation in which
different ordered pairs
have different first
coordinates.
RELATION
Is any set of ordered pairs.
The set of first coordinates
in the ordered pairs is the
domain of the relation, and
RELATION
and the set of second
coordinates is the range.
VERTICAL
LINE TEST
a relation is a function if and
only if no vertical line
intersects its graph more
than once.
Constant
Function
If f(x) = mx + b and
m = 0, then f(x) = b for
all x and its graph is a
horizontal line y = b
Determine if Relation
is a Function
{2,1),(1,-2), (1,2)}
{(x,y): x  + y = 3}
SECTION 8-9
Direct
Variation
SECTION 8-10
Inverse
Variation
END