# CHAPTER 4 SOLVING INEQUALITIES by zhangyun

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```									CHAPTER 5: GRAPHS AND FUNCTIONS

5-1 RELATING GRAPHS TO EVENTS

You can use an equation, an inequality,
or a proportion to make a statement
about a variable. You can use a graph to
show the relationship between two
variables.
5-2 RELATIONS AND FUNCTIONS

A RELATION is a set of ordered pairs.
The (age, height) ordered pairs below
form a relation.

IDENTIFIYING RELATIONS AND
FUNCTIONS

Giraffes
Age (years)    18 20 21 14 18
Height(meters) 4.25 4.40 5.25 5.00 4.85

You can list the set of ordered pairs in a
relation using braces.

{(18, 4.25), (20, 4.40), (21, 5.25), (14,5.00),
(18, 4.85)}

Recall that a function is a relation that
assigns exactly one output (range) value
for each input (domain) value.
One way you can tell whether a relation
is a function is to analyze the graph of
the relation using the vertical line test. If
any vertical line passes through more
than one point on the graph, then for
some value of x there is more than one
value of y. Therefore the relation is not
a function.

EVALUATING FUNCTIONS
You can think of a function as an input-
output machine. If you know the input
values you can use the function rule to
find the output values. The output
values depend on the input values.

y = 3x + 4

INPUT                  OUTPUT
X                      Y
1                      7
2                      10
3                      13
Another way to write the function
y = 3x + 4 is f(x) = 3x + 4

A function is in FUNCTION NOTATION
when you use f(x) to indicate the
outputs.

Example:
Evaluate the function rule f(x) = -3x + 5
to find the range of functions for the
domain {-3,1,4}
5-3 FUNCTION RULES, TABLES, AND
GRAPHS

CONTINUOUS DATA – Data where
numbers between any two data values
have meaning. Use a solid line to
indicate continuous data.

DISCRETE DATA: Data that involve a
count of items, such as the number of
people, or the number of cars. For
discrete data indicate each item with a
point.

Graph the function: y = |x| + 1

Graph the function f(x) = x^2 + 1
5-4 WRITING A FUNCTION RULE

You can write a rule for a function by
analyzing a table of values. Look for a
pattern relating the independent and
dependent variables.

Write a function rule for the table:

x                      f(x)
1                      -1
2                      0
3                      1
4                      2

Answer: f(x) = x – 2
x                   y
1                   2
2                   4
3                   6
4                   8

x                   y
1                   3
2                   4
3                   5
4                   6

Answer: y = x + 2
5-5 DIRECT VARIATION

DIRECT VARIATION – A function in the
form y = kx, where k does not equal 0, is
a direct variation. The constant of
variation for direct variation k is the
coefficient x. The variables y and x are
said to vary directly with each other.

WRITING AN EQUATION OF DIRECT
VARIATION

Is the equation a direct variation? If it is
find the constant of variation.

Example:
5x + 2y = 0
2y = -5x
y = -5/2x
The equation is in the form of y = kx, so
the equation is a direct variation and the
constant of variation is -5/2.

Example:

5x + 2y = 9
2y = -5x + 9
y = -5/2x + 9/2

The equation can not be written in the
form of y = kx.

Write an equation given a point.

Write an equation of direct variation
given the point (4, -3)

y = kx
-3 = k(4)
-3/4 = k
y = -3/4x
PROPORTIONS AND EQUATIONS OF
DIRECT VARIATIONS

For the table, use the ratio y/x to tell
whether y varies directly with x.

x              y               y/x
-3             2.25            2.25/-3 = -.75
1              -.75            -.75/1 = -.75
4              -3              -3/4 = -.75
6              -4.5            -4.5/6 = -.75
5-6 INVERSE VARIATION

INVERSE VARIATION: An equation in
the form xy = k or y=k/x where k does
not equal 0, is an INVERSE VARIATION.

The CONSTANT OF VARIATION FOR
INVERSE VARIATION is k, the product of
x * y for an ordered pair (x,y).

Inverse variations have graphs with the
same general shape. The smaller the
constant of variation, the closer the
curve is to the axis.

Write an equation given a point.

Suppose y varies inversely with x and
y = 7 when x = 5. Write an equation for
the inverse variation.

xy = k
5(7) = k
35 = k
xy = 35

Find the missing coordinate.

The points (3,8) and (2,y) are two points
on the graph of an inverse variation.
Find the missing value.

x1 * y1 = x2 * y2
3(8) = 2(y2)
24 = 2(y2)
12 = y2

COMPARING DIRECT AND INVERSE
VARIATION
Does the data in each table represent direct
or inverse variation?

x      y
2      5
4      10
10     25
x    y
5    20
10   10
25   4

COPY THE GRAPHS FOR DIRECT AND
INVERSE VARIATIONS.
5-7 DESCRIBING NUMBER PATTERNS

INDUCTIVE REASONING AND NUMBER
PATTERNS

Inductive reasoning is making
conclusions based on patterns you
observe. A conclusion you reach by
inductive reasoning is a conjecture.

WRITING RULES FOR ARITHMETIC
SEQUENCE

A number pattern is also called a
sequence.

One kind of number sequence is an
arithmetic sequence. You form an
arithmetic sequence by adding a fixed
number to each previous term. This
fixed number is a common difference.
For example: -4,        5,        14,        23
+9        +9         +9

9 is the common difference.

Try…
11,23,35,47, ….
8,3,-2,-7, ….

Consider the sequence 7, 11, 15, 19, ….
Think of each term as the output of a
function. Think of the term number as
the input.

Term number       1  2 3 4              input
Term              7 11 15 19            output

You can use the common difference of
the terms of an arithmetic sequence to
write a function rule for the sequence.
For this sequence the common
difference is 4.
Let n = the term number of the
sequence.

Let A(n) = the value of the nth term of
the sequence.

A(1) = 7
A(2) = 7 + 4 = 7 + 1*4
A(3)= 7 + 4 + 4 = 7 + 2 * 4
A(4)= 7 + 4 + 4 + 4 = 7 + 3 * 4
A(n) = 7 + 4 + 4 + 4 + …. + 4 = 7 + (n - 1)*4

RULE:
A(n) = a     +    (n-1)       d
nth    first      term        common
term term         number      difference

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