Version 2: Finding the Equation (Rule) for a Relation that Shows Quadratic Function

OBJECTIVE:

This lesson aims to help the learners to find the equation/rule for a relation that shows a quadratic function.

POINTS AND LINES RELATIONSHIP

The applet below contains points and lines that can be formed out of the given points. The value of n on the upper part of the slider corresponds to the number of points. By moving the small circle on the slider going to left and right direction you will see the lines formed by connecting two points. Represent in tabular form the relationship being described here by the number of points and the number of lines formed. Write your answer in the table provided below the applet.

VISUAL REPRESENATION OF POINTS AND LINES RELATIONSHIP

TABULAR REPRESENTATION OF THE POINTS AND LINES RELATION

INVESTIGATE DEEPLY, SOLVE HEURISTICALLY

Look at their relationship in the tabular from. How are the two consecutive values in the number of points related to the number of lines formed? What mathematical operations are involved? How are they related? If n represent the number of points and L represents the number of lines formed, then write their relationship in an equation using the said variables.

ALTERNATIVE SOLUTION

Another way to find the equation of a relation is by getting the value of the first, second, third and nth difference in the dependent variable and with the integration of solving system of linear equations. In the points and lines formed relationship, which is the dependent variable?

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FINDING THE DIFFERENCES IN THE VALUES OF THE DEPENDENT VARIABLE

In this part, you will find the first and second difference in the values of the dependent variable. Let x be the independent variable and y be the dependent variable. To get the first difference in the values form the dependent variable, subtract the two consecutive values making the first value as the subtrahend and the second value as the minuend. ( ex. 3 - 1 = 2 ; 6 - 3 = 3 and so on.). Follow the same process in finding the second difference in the values from the dependent variable.

No. of points	Independent variable	(x)	2	3	4	5	6
No. of lines	Dependent variable	(y)	1	3	6
First difference in (y)			2	3
Second difference in (y)			1

READING TASK : VALUE OF THE SECOND DIFFERENCE

What are the values in the second difference? Are the equal or not? If the second difference yield the same values, then the relation shows a quadratic function. In other words, the characteristic of the second difference in the values of the dependent variable is the determinant whether a relation is a quadratic function. To get the equation of a quadratic function given three points, use the equation y = ax² + bx + c. Use the coordinates of the three points and substitute them in y = ax² + bx + c to get three linear equation in three variables. After that, determine the value of a, b and c by solving system of linear equation in three variables. After determining the value of a, b and c, substitute them in y = ax² + bx + c to get the equation of the quadratic function being described by the relation.

CREATE FIRST LINEAR EQUATION : EQUATION 1 ( Done as an Example )

Based from your answers in the given activity, get the value of the first ordered pair then substitute it in y = ax² + bx + c to get the first linear equation. first ordered pair ( x , y ) ( 2, 1) 1 = a (2)² + b(2) + c 1 = 4a + 2b + c 4a + 2b + c = 1 First Equation

CREATE SECOND LINEAR EQUATION : EQUATION 2

Based from your answers in the given activity, get the value of the second (2nd) ordered pair then substitute it in y = ax² + bx + c to get the second (2nd) linear equation. second ordered pair ( x , y ) ( ____, _____ ) Follow the steps in the given example to get the second equation. You can use the GeoGebra Note below for your solution and answer.

CREATE THIRD LINEAR EQUATION : EQUATION 3

Based from your answers in the given activity, get the value of the third (3rd) ordered pair then substitute it in y = ax² + bx + c to get the third (3rd) linear equation. third ordered pair ( x , y ) ( ____, _____ ) Follow the steps in the given example to get the third ( 3rd) equation. You can use the GeoGebra Note below for your solution and answer.

CREATE A SYSTEM OF LINEAR EQUATION IN THREE VARIABLES

Write the three equations formed following the given format below . Equations written like the format below is called system of linear equation First Equation: Second Equation: Third Equation:

ADDITIONAL RESOURCES : SOLVING SYSTEM OF LINEAR EQUATION IN THREE VARIABLES

SOLVE THE SYSTEM OF LINEAR EQUATION FORMED

Find the values of a, b and c by solving the system of linear equation in three variables formed by the first, second and third equation. You can use the GeoGebra notes below for your solution.

STATE THE RULE (EQUATION) OF THE RELATION

After determining the values of a , b and c, substitute them in y = ax² + bx + c. The result will be the rule ( Equation. What is the rule / Equation for determining the number of lines formed based from the given points which are not collinear any three of them?

STATE IN WORDS WHAT YOU HAVE LEARNED

1. Based from the tabular value , even without graphing, how will you determine whether the relation is a quadratic function? 2. What are the ways to determine the the equation of a relation that shows quadratic function?

APPLY WHAT YOU HAVE LEARNED: NUMBER OF DIAGONALS IN A POLYGON

How is the number of diagonals related to the number of sides of a polygon. Using the same applet, try to investigate how the number of diagonals related to the number of sides of a polygon. The applet is copied below for your easy reference. Complete the entry of the table below then find the general rule ( Equation ) in determining the number of diagonals in relation to the number of sides of a polygon.