Chart Pattern Recognition | Importance of graph pattern

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 parabola-like 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 real-world 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 y-intercept. The y-intercept is the point at which the line crosses the y-axis, and in this example, the y-intercept 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 y-intercept. The y-intercept is the point where the line crosses the y-axis, and the slope of the line is defined by the linear term's coefficient.

For instance, the linear polynomial "2x + 3" has a y-intercept of (0,3), where the line crosses the y-axis, 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 y-intercept 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.

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 x-coordinate 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 "risk-return 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 x-coordinate 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.

One example of a quadratic polynomial graph is a cost-minimization or profit-maximization 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 x-coordinate 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 x-coordinate 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 x-coordinates 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 x-coordinates 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 "risk-return 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.

 Cubic
 Linear

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 3-1=2 2 3 5-3=2 3 5 7-5=2 4 7 9-7=2 5 9 2nd Difference Table n f(n) 1st difference 2nd difference 1 1 4-1=3 5-3=2 2 4 9-4=5 7-5=2 3 9 16-9=7 9-7=2 4 16 25-16=9 5 25 3rd Difference Table n f(n) 1st difference 2nd difference 3rd difference 1 1 5-1=4 9-4=5 7-5=2 2 5 14-5=9 16-9=7 9-7=2 3 14 30-14=16 25-16=9 11-9=2 4 30 55-30=25 36-25=11 5 55 91-55=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 1st 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

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) = an2 + 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 y-intercept. The slope of the line is determined by the coefficient of the linear term, and the y-intercept is the point at which the line crosses the y-axis.

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 x-coordinate 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 x-coordinates 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
• Cubic polynomial
• Parabola
• Linear relationship
• Cubic relationship