Parabola investigation. The property that was investigated was the relationship between the parabola and two lines that intersected the parabola.

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                    PARABOLA INVESTIGATION



This task was carried out to investigate one of the properties of a parabola. The property that was investigated was the relationship between the parabola and two lines that investigated the parabola.

An outline of the task

  • First, a particular parabola was selected. This parabola had the equation : y = x2-6x+11
  • Then the two intersecting lines were drawn. They had the equation : “y = 2x” and “y = x”
  • The relation between these graphs were found out by performing a set of steps that would give a specific value that denoted the relationship between the graphs. This relation was named “D”
  • Then the value of “a” in the equation of the parabola was varied so as to find a conjecture in the varying values of “D”.
  • Then this conjecture was tested on a parabola with different intersecting lines and the conjecture was modified to make it suitable for the second parabola.

 The task in detail

The main aim of the task was to find the variations in the value of “D”. The following steps were used to find the value of D for a particular graph. The below figure is that of the graph that is used as an example in demonstrating how the value of D was found out.

The x values of the intersections were named x1, x2,x3 and x4 from left to right as shown in the above diagram.

Then the following values were found out : x2 - x1  and x3- x4 and they were named Sl and SR respectively. Finally the value of D is found by finding the modulus of Sl - SR.

The equation y = x2 – 6x + 11 is in the form y = ax2 + bx + c.  Now the aim is to change the values of a and find a conjecture that can predict the value of D when a varies.


If we draw the graph of y = ax2 + bx + c, we see that there are three conditions present in the graph. The three conditions are written below.

  • Both the two lines “y=x” and ‘y=2x” intersect the graph of y = ax2 + bx + c
  • Only one line that is “y=2x”
  • None of the lines intersects(in this case, the value of D is 0)

For each case there is a separate conjecture. The following is the conjecture for the first case that is “Both the lines intersect”

The conjecture is written below.

As “a” is increased by 0.01, D reduces by 0.01”. How this conjecture was found out is shown below.

First the graph of y = 1.01x 2 – 6x + 11 was drawn, then the value of D was found out with the method that was described on page 3. This was repeated for some more values of a. The table of a along with their values of D is shown below. The graphs of the different values of  “a” are shown below.

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Graph : y = 1.01x 2 – 6x + 11

X1= 1.77

X2 = 2.41        SL = 2.41 – 1.77 = 0.64    SR = 6.15 – 4.52 = 1.63

X3 = 4.52                   D = 0.99

X4 = 6.15

The graph : y = 1.02x 2 – 6x + 11

X1 = 1.78

X2 = 2.44              SL = 0.66  SR = 1.64

X3 = 4.43                  D = 0.98


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