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Ib math SL portfolio parabola investigation

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Introduction

___________

IBHL 2 Math

Parabola Investigation

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

First, the parabola image00.pngand the lines image01.png and image20.png 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.

image26.png

  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:

image31.png

image39.png

After this, a value D was found:

image49.png

To further investigate this relationship, I followed this same process with many different parabolas and lines.

...read more.

Middle

image16.png

image17.png

image33.png

image34.png[b]

Situation 3: This situation caused me to hypothesize the conjecture thatimage35.png, where A is the A value from the equationimage36.png.  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 image37.png

Situation 7: Irrational A values work for the conjecture.

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

Situation 9: Animage38.pngthat is concave down in the third quadrant works.

Situation 10: Intersections in both 1st and 3rd quadrants work as long as the intersections of one line are x2 and x3, and the intersections of the other line are x1 and x4.[c]

After situation 10, I proved my conjecture that when the lines g(x)=x and h(x)=2x intersect the parabolaimage36.png, the value of D is image40.png

image41.png

Using the quadratic formula image42.png, I found the intersections of lines f(x) and g(x) : image43.pngand image44.png.

image45.pngimage46.png[d]

Using the same method, image47.png and image48.png could also be found.

image45.pngimage50.png

Here, g(x) and h(x) were changed to see if this changed the conjecture. The situations tested appear algebraically below in Figure 3.1, and graphically in Figure 3.2

Figure 3.1

Situation #

image38.png

image51.png

image14.png

image02.png

11

image52.png

x

3x

2

12

image53.png

2x

4x

1.2

13

image54.png

image55.png

4x

3.6

14

image52.png

image56.png

2x

1

15

image52.png

image56.png

image57.png

1

...read more.

Conclusion

Example 3 = The SM, SL, and SR values of this example were about half of those in example 1, leading me to change my conjecture to image12.png[f]

Example 5 = Here, as in the quadratic, I changed the image14.png equation to 3x to observe its affect on the SM, SL, and SR values. When this change showed a change in the S values I formed the conjecture thatimage15.png.[g]

[a]Adequate, but not ideal.

[b]This is an interesting way to show the variety of parabolas that you considered.

[c]Nicely done. Stating each of your situations illustrates your reasoning.

[d]Nice job pointing out x2 and x3.

[e]Great job!

[f]Well done. You basically came up with one of the missing pieces for D, which is, when you distribute and simplify, get x2+x3+x6 – (x1+x4+x5)

Why didn’t you get rid of the 1/A since the other factor, Sm-Sl-Sr = 0 ?

[g]Conclusion!

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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