What is the proportionality constant?
The proportionality constant is a relational numerical element, used to define the pattern of similarity between 2 magnitudes that are altered simultaneously. It is very common to represent it as a linear function in a generic way using the expression F (X) = kX However, this is not the only possible representation of proportionality.
For example, the relationship between X and Y in the function Y = 3x has a proportionality constant equal to 3. It is observed that as the independent variable former.
The alterations applied to one variable have immediate repercussions on the other, so there is a value known as the proportionality constant. This serves to relate the different magnitudes that both variables acquire.
What does the proportionality constant consist of, and types
According to the trend in the change of the variables, proportionalities can be classified into 2 types.
- Direct proportionality. It suggests a unidirectional relationship between two magnitudes. In it, if the independent variable shows some growth, the dependent variable will also grow. Likewise, any decrease in the independent variable will cause a decrease in the magnitude of Y. For example, the linear function used in the introduction; Y = 3X corresponds to a direct proportionality relationship. This is because the increase in the independent variable The value of the proportionality constant “K” in a direct relationship is defined as K = Y/X.
- Inverse or indirect proportionality . In this type of functions, the relationship between the variables is presented antonymously, where the growth or decrease of the independent variable corresponds respectively to the decrease or growth of the dependent variable. For example, the function F (x) = k/x is an inverse or indirect relationship. Since the value of the independent variable begins to increase, the value of k will be divided by an increasing number, causing the dependent variable to decrease in value according to the proportion. Depending on the value taken by K, the trend of the inverse proportional function can be defined. If k>0, then the function will be decreasing on all real numbers. And its graph will be located in the 1st and 3rd quadrant. On the contrary, if the value of K is negative or less than zero, the function will be increasing and its graph will be in the 2nd and 4th quadrants.
How is the proportionality constant calculated?
There are different contexts where the definition of the proportionality constant may be required. In the different cases, different data about the problem will be shown, where the study of these will finally yield the value of K.
In a generic way the aforementioned can be recapitulated. The values of K correspond to two expressions depending on the type of proportionality present:
Direct: K = Y/X.
Inverse or indirect: K = YX
- According to your graph . Sometimes only the graph of a function will be partially or completely known. In these cases it will be necessary, through graphic analysis, to determine the type of proportionality. Then it will be necessary to define a coordinate that allows verifying the values of X and Y to apply to the corresponding K formula. The graphs referring to direct proportionalities are linear. On the other hand, graphs of inverse proportional functions usually take the form of hyperbolas.
- According to table of values . In some cases there is a table of values with the values corresponding to each iteration of the independent variable. Typically, this involves making the graph, as well as defining the value of K.
- According to analytical expression . Shows the expression that defines the function analytically. The value of K can be solved directly, or it can also be inferred from the expression itself.
- By direct or compound rule of three . In other exercise models, certain data are presented, which refer to the relationship between values. This makes it necessary to apply the direct or compound rule of three to define other data necessary in the exercise.
History of the proportionality constant
The concept of proportionality has always been present. Not only in the minds and work of great mathematicians, but in the daily lives of the population, due to its practicality and applicability.
It is very common to encounter situations that require a proportionality approach. These are presented in each case where it is necessary to compare variables and phenomena that have certain relationships.
Using a timeline we can characterize the historical moments in which mathematical advances regarding proportionality have been applied.
- 2nd century BC . Adoption of the fractions and proportions storage system in Greece.
- 5th century BC . Discovery, also in Greece, of the proportion that relates the side and the diagonal of a square.
- 600 BC . Thales of Miletus presents his theorem regarding proportionality.
- 900. Extension of the decimal system previously used by India in ratios and proportions. Contribution made by the Arabs.
- XVII century . Contributions arrive regarding the proportions in Euler’s calculus.
- XIX century . Gauss provides the concept of complex number and proportion.
- Twentieth century . Proportionality as a function model is defined by Azcárate and Deulofeo.
It is required to calculate the value of the variables x, y, z and g. Knowing the following proportional relationships:
3x + 2y – 6z + 8g = 1925
x/3 = y/8 = z/3 = g/5
We proceed to define the relative values of the proportionality constant. We can obtain these from the second relationship, where the value that divides each variable indicates a relationship or ratio referring to K.
X = 3k y = 2k z = 3k g = 5k
The values are substituted in the first expression, where the new system will be evaluated in a single variable k.
3(3k) + 2(2k) – 6(3k) + 8(5k) =1925
9k + 4k – 18k + 40k = 1925
35k = 1925
K = 1925/35 = 55
Using this value of the proportionality constant we can find the figure that defines each of the variables.
x = 3(55) = 165 y = 2(55) = 110
z = 3(55) = 165 g = 5(55) = 275
Calculate the constant of proportionality and the expression that defines the function, given its graph.
First, the graph is analyzed, its linear character being evident. This indicates that it is a function with direct proportionality and that the value of K will be obtained through the expression k= y/x
Then a determinable point on the graph is chosen, that is, one where the coordinates that make it up can be accurately seen.
For this case the point (2, 4) is taken. From where we can establish the following relationship.
K = 4/2 = 2
So the expression is defined by the function y=kx, which in this case will be
F(x) = 2x