At the end of this lesson, the learners are expected to find the equation/rule for a relation that shows a quadratic function.
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. Move the small circle on the slider going to the right then the you will see the lines formed by connecting two points. Use this applet to answer the succeeding questions below.
1. How many lines are formed by connecting 2 points?[br]2. How many lines are formed given 3 non - collinear points?[br]3. How many lines are formed given 4 non - collinear points?[br]4. How many lines are formed given 5 non - collinear points?[br]5. How many lines are formed given 6 non - collinear points?[br][br]Write your answer in a tabular form.[br][table][tr][td]No. of Points[/td][td]2[/td][td]3[/td][td]4[/td][td]5[/td][td]6[/td][/tr][tr][td]No. of Lines[/td][td][/td][td][/td][td][/td][td][/td][td][/td][/tr][/table]
In that activity that shows the relationship between the no. of points and no. of lines formed, which is the dependent variable and the independent variable?
Independent Variable : Number of Points[br]Dependent Variable : Number of lines formed
Show in the graph the relationship between the number of points and the number of lines formed. Use the GeoGebra App below for your graph.
Based from the graph, what kind of relation/function exist between the number of points and the number of lines formed?
How is the number of lines formed related to the number of points? As the number of points increases, what can you say about the number of lines formed?
In this part, you will find the first and second difference in the values from 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.[br][br][table][tr][td]No. of points[/td][td]Independent variable[/td][td](x)[/td][td]2[/td][td]3[/td][td]4[/td][td]5[/td][td]6[/td][td]7[/td][td]8[/td][/tr][tr][td]No. of lines[/td][td]Dependent variable[/td][td](y)[/td][td]1[/td][td]3[/td][td]6[/td][td][/td][td][/td][td][/td][td][/td][/tr][tr][td]First difference in (y)[/td][td][/td][td][/td][td]2[/td][td]3[/td][td][/td][td][/td][td][/td][td][/td][td][/td][/tr][tr][td]Second difference in (y)[/td][td][/td][td][/td][td]1[/td][td][/td][td][/td][td][/td][td][/td][td][/td][td][/td][/tr][/table]
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.[br][br]To get the equation of a quadratic function given three points, use the equation y = ax[sup]2[/sup] + bx + c.[br]Use the coordinates of the three points and substitute them in y = ax[sup]2[/sup] + 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 = ax[sup]2[/sup] + bx + c to get the equation of the quadratic function being described by the relation.[br][br]
Based from your answers in the given activity, get the value of the first ordered pair then substitute it in y = ax[sup]2[/sup] + bx + c to get the first linear equation.[br] first ordered pair ( x , y ) [math]\Longrightarrow[/math] ( 2, 1) [br][br] 1 = a (2)[sup]2[/sup] + b(2) + c[br] 1 = 4a + 2b + c[br] 4a + 2b + c = 1 First Equation
Based from your answers in the given activity, get the value of the second (2nd) ordered pair then substitute it in y = ax[sup]2[/sup] + bx + c to get the second (2nd) linear equation.[br] second ordered pair ( x , y ) [math]\Longrightarrow[/math] ( ____, _____ )[br][br]Follow the steps in the given example to get the second equation. You can use the GeoGebra Note below for your solution and answer.
Based from your answers in the given activity, get the value of the third (3rd) ordered pair then substitute it in y = ax[sup]2[/sup] + bx + c to get the third (3rd) linear equation.[br] third ordered pair ( x , y ) [math]\Longrightarrow[/math] ( ____, _____ )[br][br]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.
Write the three equations formed following the given format below :[br][br]First Equation:[br]Second Equation:[br]Third Equation:
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.
After determining the values of a , b and c, substitute them in y = ax[sup]2[/sup] + 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?
Using the same applet, determine the total number of diagonals formed in each polygon formed by moving the circle on the slider going to the right. Find also the general rule ( Equation ) in determining the number of diagonals. The applet is copied below for your easy reference.[br][br]Complete the entry of the table below. You can use the GeoGebra notes below fro your answer.[br][br][table][tr][td] No. of sides[/td][td]3[/td][td]4[/td][td]5[/td][td]6[/td][td]7[/td][td]8[/td][td]n[/td][/tr][tr][td]No. of Diagonals[/td][td]0[/td][td]2[/td][td][/td][td][/td][td][/td][td][/td][td][/td][/tr][/table]