Introduction: Chart Pattern Recognition
Understanding a
graph's pattern is crucial because it enables you to comprehend how the
function it represents behaves. You can learn about a function's
characteristics, such as its shape, symmetry, and asymptotes, from the chart pattern.
For instance, a
linear polynomial graph's pattern, which is a straight line, indicates that the
function is a linear function. The parabolalike pattern of a quadratic
polynomial graph identifies the function as a quadratic function. A cubic
polynomial graph has a pattern that looks like a curve with a degree of three,
indicating that the function is a cubic function.
In fields like
finance and economics, where it is frequently required to use functions to
describe relationships between variables, understanding the structure of a
graph is particularly crucial. For instance, in finance, the pattern of a graph
can be used to comprehend the relationship between a stock's price and trading
volume, or the relationship between an investment's rate of return and the
degree of risk involved in the investment. In economics, the shape of a graph
can be used to comprehend the link between the cost of production and the
volume of output, as well as the relationship between the price of a thing and
the quantity desired by consumers.
Overall,
comprehending chart pattern recognition is crucial since it enables you to
evaluate and comprehend a function's behavior, which is helpful across many
fields of science, math, and other subjects.
What are Polynomials?
Polynomials are
mathematical expressions that solely use addition, subtraction, and
multiplication operations to combine variables and coefficients. The highest
exponent of the variable in an expression is the polynomial's degree.
Linear,
quadratic, and cubic polynomials are the three primary varieties. A degree of
one is assigned to a linear polynomial, a degree of two to a quadratic
polynomial, and a degree of three to a cubic polynomial. Each form of the polynomial has a unique graph pattern that serves as the equation's
representation on a coordinate plane.
We shall
examine the graph patterns for linear, quadratic, and cubic polynomials in this
article. Each form of the polynomial will be described, along with illustrations of
the corresponding graphs. Other mathematics as well as in disciplines like
physics and engineering, where polynomial functions are frequently employed to
simulate realworld occurrences, it is crucial to comprehend these patterns.
Definition of a polynomial
A polynomial is
a mathematical equation that solely uses addition, subtraction, and
multiplication to combine its variables and coefficients. A polynomial is, for
instance, the phrase "3x2 + 2x  1".
One or more
terms can be found in a polynomial. A term is a phrase with a plus or minus
sign between words. "3x2," "2x," and "1" are all
terms of the polynomial in the aforementioned case.
The highest
exponent of the variable in an expression is the polynomial's degree. For
instance, the polynomial "3x2 + 2x  1" has a degree of 2 because the
largest exponent is 2. According to their degree, polynomials are given
different names. For example, a polynomial with a degree of 1 is known as a
linear polynomial, a polynomial with a degree of 2 is known as a quadratic
polynomial, a polynomial with a degree of 3 is known as a cubic polynomial, and
so on.
Definition of a linear polynomial
A polynomial
having a degree of one is said to be linear. In other terms, it is a polynomial
in which the variable has the highest exponent of 1.
"2x +
3" is an illustration of a linear polynomial with a degree of 1
The graphs of
linear polynomials have the characteristic of being straight lines. A linear
polynomial's graph consists of a line with a slope and a yintercept. The
yintercept is the point at which the line crosses the yaxis, and in this
example, the yintercept is point (0,3). The slope of the line is determined by
the coefficient of the linear term (in this case, the coefficient of
"x" is 2).
In many
disciplines, including economics, linear polynomials are crucial because they
are used to simulate linear connections between variables
A linear
polynomial graph has a straight line for a shape. The graphs of linear
polynomials have the characteristic of being straight lines.
A linear polynomial's
graph consists of a line with a slope and a yintercept. The yintercept is the
point where the line crosses the yaxis, and the slope of the line is defined
by the linear term's coefficient.
For instance,
the linear polynomial "2x + 3" has a yintercept of (0,3), where the
line crosses the yaxis, and a slope of 2 (the linear term "x"s
coefficient is 2). This polynomial has a graph that is a straight line with a
slope of 2, and a yintercept of 0.
Usage of linear polynomial graphs in Economics
In order to
simulate linear correlations between variables, linear polynomial graphs are
frequently employed in economics. In a linear relationship, there is a direct
proportionality between the changes in the two variables, meaning that the
change in one is always equal to the change in the other. Accordingly, if one
variable rises, the other rises by a predetermined amount, and if one variable
falls, the other falls by a fixed amount.
For example, a
linear polynomial graph could be used to model the relationship between the
price of a good and the quantity of the good demanded by consumers. If the
price of the good increases, the quantity demanded by consumers may decrease by
a fixed amount, and if the price decreases, the quantity demanded may increase by
a fixed amount. The graph of this relationship would be a straight line with a
negative slope because as the price increases, the quantity demanded
decreases.
Linear
polynomial graphs are useful in economics because they allow analysts to make
predictions about how one variable will change based on changes in the other
variable. They can also be used to analyze the effect of changes in economic
policies on variables such as GDP, inflation, and employment.
Definition of a quadratic polynomial
A quadratic
polynomial is a polynomial with a degree of 2. In other words, it is a
polynomial in which the highest exponent of the variable is 2.
An example of a
quadratic polynomial is "x^2 + 2x + 1", which has a degree of 2 (the
exponent of the variable "x" is 2).
Quadratic
polynomials have the property that their graphs are parabolas. The graph of a
quadratic polynomial is a curve that is shaped like a parabola, with either a
single maximum or minimum point (if the coefficient of the quadratic term is
positive) or a single inflection point (if the coefficient of the quadratic
term is negative). The xcoordinate of the maximum or minimum point is the
solution to the equation obtained by setting the derivative of the quadratic
polynomial equal to zero.
Quadratic polynomials
are important in many fields, including physics, where they are used to model
phenomena such as the motion of objects under the influence of gravity.
Quadratic polynomials in Economics or Finance
Quadratic
polynomials are not as commonly used in economics and finance as linear
polynomials, but they can still be applied in certain circumstances.
For example, a
quadratic polynomial could be used to model the relationship between the price
of a good and the quantity of the good demanded by consumers, if the
relationship is more complex than a simple linear relationship. In this case,
the graph of the relationship would be a parabola, with either a single maximum
or minimum point (if the coefficient of the quadratic term is positive) or a
single inflection point (if the coefficient of the quadratic term is negative).
A quadratic
polynomial could also be used to model the relationship between the rate of
return on investment and the level of risk associated with the investment.
In this case, the graph of the relationship might be a "riskreturn
tradeoff" curve, with a single maximum point indicating the level of risk
at which the highest rate of return is achieved.
Overall, while
quadratic polynomials are not as commonly used as linear polynomials in
economics and finance, they can still be useful in certain situations where a
more complex relationship between variables is needed.
The graph of a quadratic polynomial: a parabola
The graph of a quadratic polynomial is a curve that is shaped like a
parabola.
A parabola is a
curve that is defined by a quadratic equation of the form "y = ax^2 + bx +
c", where "a", "b", and "c" are constants.
The graph of this equation is a curve that has either a single maximum or
minimum point (if the coefficient "a" is positive) or a single
inflection point (if the coefficient "a" is negative). The
xcoordinate of the maximum or minimum point is the solution to the equation
obtained by setting the derivative of the quadratic equation equal to zero.
The shape of
the parabola is determined by the value of the coefficient "a". If
"a" is positive, the parabola will open upwards and have a single
minimum point. If "a" is negative, the parabola will open downwards
and have a single maximum point. If "a" is zero, the parabola will be
a horizontal line.
Quadratic
polynomials are important in many fields, including physics, where they are
used to model phenomena such as the motion of objects under the influence of
gravity.
Examples of quadratic polynomial graphs
One example of
a quadratic polynomial graph is a costminimization or profitmaximization
curve.
In economics
and finance, a common goal is to maximize profit or minimize cost. This can
often be done by finding the optimal level of a variable that depends on one or
more other variables. For example, a firm might want to find the optimal level
of production that maximizes profit, given the cost of production and the price
of the goods being produced.
In these cases,
the relationship between the variables can be modeled using a quadratic
polynomial. For example, the profit of a firm might be modeled as a quadratic
function of the level of production, with the cost of production and the price
of the goods being produced as constants. The graph of this function would be a
parabola, with a single maximum point indicating the level of production at
which the maximum profit is achieved.
Quadratic
polynomial graphs are useful in economics and finance because they allow
analysts to find the optimal level of a variable that depends on one or more
other variables. They can also be used to analyze the effect of changes in
economic policies on variables such as profit, cost, and production.
Definition of a cubic polynomial
A cubic
polynomial is a polynomial with a degree of 3. In other words, it is a
polynomial in which the highest exponent of the variable is 3.
An example of a
cubic polynomial is "2x^3  4x^2 + x + 1", which has a degree of 3
(the exponent of the variable "x" is 3).
Cubic
polynomials have the property that their graphs are curves with a degree of 3.
The graph of a cubic polynomial is a curve that can have either one or three
turning points, depending on the coefficients of the polynomial.
Cubic
polynomials are important in many fields, including mathematics and physics,
where they are used to model phenomena such as the motion of objects under the
influence of forces.
The graph of a
cubic polynomial: a curve with a degree of 3
A cubic
polynomial is a polynomial with a degree of 3, which means that it has the form
"ax^3 + bx^2 + cx + d", where "a", "b",
"c", and "d" are constants. The graph of a cubic polynomial
is a curve that can have either one or three turning points, depending on the
values of the coefficients "a", "b", "c", and
"d".
For example, the cubic polynomial "2x^3  4x^2 + x + 1" has a graph that is a curve with a single turning point. The xcoordinate of the turning point is the solution to the equation obtained by setting the derivative of the cubic polynomial equal to zero.
Cubic
polynomials are important in many fields, including mathematics and physics,
where they are used to model phenomena such as the motion of objects under the
influence of forces. They are also used in engineering and other fields where
it is necessary to model complex relationships between variables.
Examples of cubic polynomial graphs
Here are some
examples of cubic polynomial graphs:
The cubic
polynomial "2x^3  4x^2 + x + 1" has a graph that is a curve with a
single turning point. The xcoordinate of the turning point is the solution to
the equation obtained by setting the derivative of the cubic polynomial equal
to zero.
The cubic
polynomial "x^3  3x^2 + 3x + 1" has a graph that is a curve with
three turning points. The xcoordinates of the turning points are the solutions
to the equation obtained by setting the derivative of the cubic polynomial
equal to zero.
The cubic
polynomial "x^3 + x^2 + x + 1" has a graph that is a curve with three
turning points. The xcoordinates of the turning points are the solutions to
the equation obtained by setting the derivative of the cubic polynomial equal
to zero.
Cubic polynomials
are important in many fields, including mathematics and physics, where they are
used to model phenomena such as the motion of objects under the influence of
forces. They are also used in engineering and other fields where it is
necessary to model complex relationships between variables.
Is there any use of cubic polynomial graphs in finance & economics
Cubic
polynomial graphs are not as commonly used in finance and economics as linear
and quadratic polynomial graphs, but they can still be applied in certain
circumstances.
For example, a
cubic polynomial could be used to model the relationship between the return on investment and the level of risk associated with the investment, if the
relationship is more complex than a simple linear or quadratic relationship. In
this case, the graph of the relationship might be a "riskreturn
tradeoff" curve with three turning points, indicating the levels of risk
at which the highest and lowest rates of return are achieved.
A cubic
polynomial could also be used to model the relationship between the price of a
good and the quantity of the good demanded by consumers if the relationship is
more complex than a simple linear or quadratic relationship. In this case, the
graph of the relationship might be a curve with three turning points,
indicating the levels of the price at which the highest and lowest levels of demand
are achieved.
Overall, while
cubic polynomial graphs are not as commonly used as linear and quadratic
polynomial graphs in finance and economics, they can still be useful in certain
situations where a more complex relationship between variables is needed.
Can we recognize cubic, quadratic, and linear patterns from raw data?
Yes, it is
possible to recognize cubic, quadratic, and linear patterns in raw data. One
way to do this is by plotting the data and visually inspecting the pattern of
the points.



If the data
points form a straight line, then the pattern is linear. If the data points
form a smooth curve that is shaped like a parabola and opens upwards, then the
pattern is quadratic. If the data points form a smooth curve that is shaped
like a parabola and opens upwards or downwards, then the pattern is cubic.
In addition to
visual inspection, it is also possible to fit a mathematical model to the data
and use the form of the model to determine the pattern. For example, a linear
model is of the form y = mx + b, a quadratic model is of the form y = ax^2 + bx
+ c, and a cubic model is of the form y = ax^3 + bx^2 + cx + d.
Constructing a difference table for the sequences
The difference
table is continued until a constant is obtained. We will use the following
notations
for
differences:
1st Difference Table 

n 
f(n) 
1st difference 

1 
1 
31=2 

2 
3 
53=2 

3 
5 
75=2 

4 
7 
97=2 

5 
9 


2nd Difference Table 

n 
f(n) 
1st difference 
2nd difference 

1 
1 
41=3 
53=2 

2 
4 
94=5 
75=2 

3 
9 
169=7 
97=2 

4 
16 
2516=9 


5 
25 



3rd Difference Table 

n 
f(n) 
1st difference 
2nd difference 
3rd difference 
1 
1 
51=4 
94=5 
75=2 
2 
5 
145=9 
169=7 
97=2 
3 
14 
3014=16 
2516=9 
119=2 
4 
30 
5530=25 
3625=11 

5 
55 
9155=36 


6 
91 



The following
three difference tables are for the general linear, quadratic, and cubic
functions. We
will use these tables throughout the remainder of this section.
These tables
may be used to determine the rules for a given sequence. This is illustrated by
the following
example. We note that they do not constitute proof that the rule is the one
for
the given
sequence:
Linear:f(n)=an+b 

n 
f(n) 
1st difference 

1 
a+b 
a 

2 
2a+b 
a 

3 
3a+b 


Quadratic: f(n) = an2 + bn + c 

n 
f(n) 
1st difference 
2nd difference 

1 
a+b+c 
3a+b 
2a 

2 
4a+2b+c 
5a+b 
2a 

3 
9a+3b+c 
7a+b 
2a 

4 
16a+4b+c 
9a+b 


5 
25a+5b+c 



Cubic: f (n) = an3 + bn2 + cn + d 

n 
f(n) 
1st difference 
2nd difference 
3rd difference 
1 
a+b+c+d 
7a+3b+c 
12a+2b 
6a 
2 
8a+4b+2c+d 
19a+5b+c 
18a+2b 
6a 
3 
27a+9b+3c+d 
37a+7b+c 
24a+2b 
6a 
4 
64a+16b+4c+d 
61a+9b+c 
30a+2b 

5 
125a+25b+5c+d 
91a+11b+c 


6 
216a+36b+6c+d 



For constructing linear, quadratic, and cubic equations, we will above tables
such as:
If the function is linear
The function is
a linear of form f (n) = an+ b
From the
difference table for the linear
a = 2 and
therefore a = 2 (using 1^{st} difference column)
a+b = 1 and
therefore b = 1 (using f(n) column)
Therefore, f(n) = an+b = (for
example when n=4) 2(4) + (1) = 7
If the function is Quadratic
The function is a quadratic of the
form f (n) = an2 + bn + c.
From the difference table for the
quadratic
2a = 2 and therefore a = 1 (using
column of 2nd difference)
3a + b = 3 and therefore b = 0
(using column of 1st difference)
a + b + c = 1 and therefore c = 0
(using column f (n)).
∴
f(n) = an^{2} + bn + c = 1(5)^{2} +0+0= 25
Therefore f (n) = an3 + bn2 + cn +
d.
6a = 2 ∴ a = 1/3 (from 3rd difference column)
12a + 2b = 5 ∴ b = 1/2 (from column 2nd difference)
7a + 3b + c = 4 ∴ c = 0.166 (from column 1st difference)
a + b + c + d = 1 ∴ d = 0 (from column f (n))
∴f
(n) = an3 + bn2 + cn + d = 1/3(3)3 +1/2(3)2 +0.167 ~ 14
Conclusion
Linear
polynomials: Linear polynomials are polynomials with a degree of 1, which means
that the highest exponent of the variable is 1. The graph of a linear
polynomial is a straight line with a slope and a yintercept. The slope of the
line is determined by the coefficient of the linear term, and the yintercept
is the point at which the line crosses the yaxis.
Quadratic
polynomials: Quadratic polynomials are polynomials with a degree of 2, which
means that the highest exponent of the variable is 2. The graph of a quadratic
polynomial is a curve that is shaped like a parabola, with either a single
maximum or minimum point (if the coefficient of the quadratic term is positive)
or a single inflection point (if the coefficient of the quadratic term is
negative). The xcoordinate of the maximum or minimum point is the solution to
the equation obtained by setting the derivative of the quadratic polynomial
equal to zero.
Cubic
polynomials: Cubic polynomials are polynomials with a degree of 3, which means
that the highest exponent of the variable is 3. The graph of a cubic polynomial
is a curve that can have either one or three turning points, depending on the
coefficients of the polynomial. The xcoordinates of the turning points are the
solutions to the equation
The
significance of comprehending these patterns in math and other disciplines like
finance and economics.
Comprehending
the patterns of graphs for linear, quadratic, and cubic polynomials are
important in math and other disciplines because it allows you to understand the
properties of these types of functions and how they behave. This understanding
is useful in many areas, including finance and economics, where it is often
necessary to model relationships between variables using these types of
functions.
For example, in
finance, linear polynomial graphs can be used to model the relationship between
the price of a stock and the volume of the stock traded. Quadratic polynomial
graphs can be used to model the relationship between the rate of return on investment and the level of risk associated with the investment. Cubic
polynomial graphs can be used to model more complex relationships between
variables, such as the relationship between the price of a good and the
quantity of the good demanded by consumers.
In economics,
linear polynomial graphs can be used to model the relationship between the
price of a good and the quantity of the good demanded by consumers. Quadratic
polynomial graphs can be used to model the relationship between the level of
production and the cost of production. Cubic polynomial graphs can be used to
model more complex relationships between variables, such as the relationship
between the level of employment and the level of GDP.
Overall,
understanding chart patterns recognition for linear, quadratic, and
cubic polynomials are important because it allows you to analyze and understand
complex relationships between variables, which is useful in many areas of math,
science, and other disciplines.
These are the
keywords for the article on chart patterns recognition:
 Polynomial
 Linear polynomial
 Quadratic polynomial
 Cubic polynomial
 Parabola
 Linear relationship
 Quadratic relationship
 Cubic relationship
0 Comments