
The features of a role graph tin prove us many aspects of the relationship represented by the office. Permit’south take a look at the more popular graphical features. Be sure to pay attending to the vocabulary and the notation used in this section.
Intercepts are the locations (points) where the graph crosses (or touches) either the
xaxis or
yaxis.
• To find the
yintercept,
gear up
x
= 0, and solve for
y.
Call back: the
yintercept will have an
xcoordinate of 0.
y = f
(x) = 2x
+ 2
y
= two(0) + 2;
y
= ii
yintercept: (0,2)
(Yep, you lot can as well read the
yintercept,
b, from the function equation
if it is in
y = mx + b
form.)
• To find the
xintercept
, set
y
= 0, and solve for
ten.
Remember: the
xintercept will take a
ycoordinate of 0.
y = f
(ten) = iix
+ 2
0 = 2x
+ two; 2x
= 2;
ten
= 1
tenintercept: (i,0)
tenintercepts may also exist referred to every bit “roots” or “zeros”
since they are where
f
(x) = 0.




yvalues positive
or
yvalues negative


• The
positive regions
of a function are those intervals where the function is
in a higher place the
xaxis.
It is where the
yvalues are positive (non zero).
• The
negative regions
of a function are those intervals where the function is
beneath the
xaxis.
It is where the
yvalues are negative (not zippo).
•
yvalues that are
on the
xaxis
are neither positive nor negative. The
xaxis is where
y
= 0.

Some functions are positive over their entire domain
(All yvalues above the tenaxis.)
positive: ∞ < x < +∞
or “all Reals”, or (∞,+∞)

Some functions are negative over their entire domain.
(All yvalues below the tenaxis.)
negative: ∞ < x < +∞
or “all Reals”, or (∞,+∞)

Some functions have both positive and negative regions.
(yvalues above and below 10axis)
positive: 10 > 0
or (0,+∞)
negative: ten < 0
or (∞,0)
(practice not include nothing)


Undercover to Finding the Intervals!
The undercover to correctly stating the intervals where a role is positive or negative is to recall that the
intervals
ALWAYS
pertain to the locations of the
xvalues. Think of reading the graph from left to correct along the
xaxis.
Exercise Non read numbers off the
yaxis for the intervals.
Stay on the
xaxis!


The discussion on this page will refer to “strictly” increasing and “strictly” decreasing. 
When looking for sections of a graph that are increasing or decreasing, exist certain to wait at (or “read”) the graph
from left to right.
•
Increasing:
A function is
increasing, if as
x
increases (reading from left to correct), y
likewise increases . In plain English, as you look at the graph, from left to correct, the graph goes upwardlyhill. The graph has a
positive slope.
By definition:
A part is strictly
increasing
on an interval, if when
x
_{one}
<
x
_{two}, so
f
(x
_{1}) <
f
(ten
_{2}).
If the function note is bothering you, this definition can also be idea of as stating
x
_{1}
<
x
_{2
}implies
y
_{1}
<
y
_{2}. Equally the
x‘due south become larger, the
y‘south get larger.
Example:
The role (graph) at the correct is increasing from the point (five,three) to the indicate (2,1), which is described equally
increasing
when
5 <
x
< 2 .
Using interval notation, information technology is described as increasing on the interval
(five,two).


It as well increases from the point (1,i) to the point (3,4), described as
increasing
when
1 <
ten
< 3.
Using interval notation, it is described equally increasing on the interval
(1,three).

•
Decreasing:
A office is
decreasing, if equally
x
increases (reading from left to right), y
decreases. In evidently English, as yous look at the graph, from left to right, the graph goes downhill. The graph has a
negative gradient.
By definition:
A office is strictly
decreasing
on an interval, if when
x
_{ane}
<
x
_{2},
then
f
(10
_{ane}) >
f
(x
_{2}).
If the function notation is bothering you, this definition tin besides be thought of as stating
x
_{i}
<
ten
_{two
}implies
y
_{1}
>
y
_{2}.
Equally the
x‘s go larger the
y‘s get smaller.
Example:
The graph shown above is decreasing from the bespeak (3,4) to the betoken (5,5), described as
decreasing
when
three <
x
< five.
Using interval notation, it is described every bit decreasing on the interval
(3,5).
•
Constant:
A office is
constant, if as
10
increases (reading from left to correct), y
stays the same. In plain English language, as you lot look at the graph, from left to right, the graph goes flat (horizontal). The graph has a
slope of zero.
By definition:
A function is
constant, if for any
x
_{1}
and
x
_{2
}in the interval,
f
(x
_{1}) =
f
(x
_{two}).
Case:
The graph shown above is constant from the point (2,1) to the point (1,1), described as
constant
when
2 <
x
< 1
. The
yvalues of all points in this interval are “one”.
Using interval notation, it is described as constant on the interval
(2,i).

Intervals
of increasing, decreasing or abiding
Always
pertain to
xvalues.
Practice Not read numbers off the
yaxis.
Stay on the
10axis for these intervals!

•
Intervals of Increasing/Decreasing/Constant:
Interval notation
is a popular notation for stating which sections of a graph are increasing, decreasing or constant. Interval notation utilizes portions of the role’due south domain (
10intervals). For the graph shown to a higher place, nosotros would write:
The role is increasing on the
xintervals
(five,2)
and
(1,3)
.
The part is decreasing on the
xinterval
(3,5)
.
The function is abiding on the
xinterval
(ii,i)
.

This is “open” interval notation.

Differing notations for increasing intervals:
Regarding intervals of increasing or decreasing on a graph, it is a popular convention to use simply “open” interval notation. Notwithstanding, it is considered correct to use either “open” or “airtight” notation when describing intervals of increasing or decreasing. References to ± infinity, however, are always “open” note.
Accept a look at the point (2,four) in the graph at the correct. Does that signal belong to the increasing interval? The decreasing interval? Both intervals? Neither interval?


Well, the answer may be both, neither, or a combination, depending upon the convention you are following. Y’all may see “increasing on the interval (five,2) or the interval [5,two], or the interval (5,2], or the interval [five,two). Just be consequent with the convention you are using.
Functions increase on intervals, not at points. The graph is neither increasing nor decreasing at the point (ii,4). However, i
f a function increases on an “open up” interval, and so adding the endpoints will not change this fact (equally long equally the endpoints are in the domain).
This site volition be using “open” interval notation to represent intervals of increasing and decreasing. Enquire your teacher which notation convention is preferred in your classroom.

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