# Ib math SL portfolio parabola investigation

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IBHL 2 Math

Parabola Investigation

In this investigation, relationships between the points of intersection of a parabola and two different lines were examined.

First, the parabola and the lines  and  were used as an example.

As seen on the adjacent graph, the points of intersection were labeled left to right as x1, x2, x3, and x4.

Using a graphing calculator, these values were found:

x1=1.764,  x2=2.382,  x3=4.618, and  x4=6.234.

At this point, x1 was subtracted from x2, and x3 from x4, and the resulting numbers were labeled SL and SR respectively:

After this, a value D was found:

To further investigate this relationship, I followed this same process with many different parabolas and lines.  These lines and their values are shown in the table below (Figure 2.1). A graph of the displayed functions is below this (Figure 2.2). An explanation for each significant situation is provided below Figure 2.2.

Figure 2.1

Situation 3: This situation caused me to hypothesize the conjecture that, where A is the A value from the equation.  In addition, g(x) was tangent to f(x) at the point (10,10). This meant that x2 and x3 were the same (10), and showed that the conjecture held true for tangents.

Situation 6: A concave down parabola in the 1st quadrant still holds the conjecture, provided that the conjecture is changed to

Situation 7: Irrational A values work for the conjecture.

Situation 8: A concave up parabola in the third quadrant works.

Situation 9: Anthat is concave ...