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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
x-axis or
y-axis.
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• To find the
y-intercept,
gear up
x
= 0, and solve for
y.
Call back: the
y-intercept will have an
x-coordinate of 0.
y = f
(x) = -2x
+ 2
y
= -two(0) + 2;
y
= ii
y-intercept: (0,2)
(Yep, you lot can as well read the
y-intercept,
b, from the function equation
if it is in
y = mx + b
form.)
• To find the
x-intercept
, set
y
= 0, and solve for
ten.
Remember: the
x-intercept will take a
y-coordinate of 0.
y = f
(ten) = -iix
+ 2
0 = -2x
+ two; 2x
= 2;
ten
= 1
ten-intercept: (i,0)
ten-intercepts may also exist referred to every bit “roots” or “zeros”
since they are where
f
(x) = 0.
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y-values positive
or
y-values negative
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• The
positive regions
of a function are those intervals where the function is
in a higher place the
x-axis.
It is where the
y-values are positive (non zero).
• The
negative regions
of a function are those intervals where the function is
beneath the
x-axis.
It is where the
y-values are negative (not zippo).
•
y-values that are
on the
x-axis
are neither positive nor negative. The
x-axis is where
y
= 0.
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Some functions are positive over their entire domain
(All y-values above the ten-axis.)
positive: -∞ < x < +∞
or “all Reals”, or (-∞,+∞)
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Some functions are negative over their entire domain.
(All y-values below the ten-axis.)
negative: -∞ < x < +∞
or “all Reals”, or (-∞,+∞)
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Some functions have both positive and negative regions.
(y-values above and below 10-axis)
positive: 10 > 0
or (0,+∞)
negative: ten < 0
or (-∞,0)
(practice not include nothing)
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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
x-values. Think of reading the graph from left to correct along the
x-axis.
Exercise Non read numbers off the
y-axis for the intervals.
Stay on the
x-axis!
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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.
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•
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 upwardly-hill. 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).
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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).
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•
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 down-hill. 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
y-values of all points in this interval are “one”.
Using interval notation, it is described as constant on the interval
(-2,i).
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Intervals
of increasing, decreasing or abiding
Always
pertain to
x-values.
Practice Not read numbers off the
y-axis.
Stay on the
10-axis for these intervals!
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•
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 (
10-intervals). For the graph shown to a higher place, nosotros would write:
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The role is increasing on the
x-intervals
(-five,-2)
and
(1,3)
.
The part is decreasing on the
x-interval
(3,5)
.
The function is abiding on the
x-interval
(-ii,i)
.
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This is “open” interval notation.
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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?
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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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