• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Ib math SL portfolio parabola investigation

Extracts from this document...

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 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.

Middle    [b]

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: An that 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 parabola , the value of D is  Using the quadratic formula , I found the intersections of lines f(x) and g(x) : and .  [d]

Using the same method, and could also be found.  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 #    11 x 3x 2 12 2x 4x 1.2 13  4x 3.6 14  2x 1 15   1

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 [f]

Example 5 = Here, as in the quadratic, I changed the 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 that .[g]

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

[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!

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Extended Essay- Math

ï¿½~QBpï¿½"%ï¿½Rï¿½)ï¿½ï¿½ï¿½ <(r)&x @ï¿½F(ï¿½ï¿½G!ï¿½--+ï¿½ï¿½ï¿½.ï¿½iMï¿½ "ï¿½-uï¿½\(Kï¿½2GJDXï¿½ï¿½ ï¿½}Bï¿½5ï¿½ ï¿½p ...ï¿½yï¿½"GT{ï¿½QmgY>ï¿½,ï¿½ï¿½f22ï¿½ 7ï¿½BXï¿½ï¿½Zï¿½;ï¿½ï¿½eï¿½ï¿½Fï¿½ï¿½!ï¿½"1/4gLï¿½A:0ï¿½?ï¿½Orï¿½>ï¿½'ï¿½ï¿½ï¿½:Dï¿½ï¿½ï¿½&Yï¿½vGï¿½ï¿½É°:#"hï¿½ï¿½ ï¿½6ï¿½p"ï¿½[}Vï¿½vï¿½ï¿½{\$}ï¿½"ï¿½ï¿½(yï¿½gï¿½ï¿½eï¿½ï¿½ï¿½Ü¦fzï¿½(tm)ï¿½ï¿½>ï¿½ï¿½Yï¿½z"ï¿½|[Qyï¿½2ï¿½qzï¿½ï¿½|ï¿½ï¿½#ï¿½A,ï¿½3ï¿½(3ï¿½,bï¿½lï¿½ï¿½~e*^qa''{ï¿½.7ï¿½-ï¿½-ï¿½A-9ï¿½ï¿½7Xï¿½ï¿½ï¿½Aï¿½Vï¿½ï¿½ï¿½~9ï¿½Rï¿½(tm) dï¿½?ï¿½ï¿½ï¿½(c)"Ò·|ï¿½"ï¿½Oï¿½Oyï¿½c:ï¿½ï¿½ï¿½M(tm)@"(Fï¿½mï¿½"zï¿½ï¿½ï¿½ï¿½ Q-ï¿½ï¿½tï¿½ï¿½ï¿½ï¿½B-Mï¿½ï¿½Cï¿½ï¿½r8ï¿½ï¿½&ï¿½ï¿½2ï¿½sï¿½n3/4jï¿½Q~ï¿½ï¿½ï¿½ÌgbÌHXØï¿½"ï¿½ï¿½uï¿½ ï¿½'ï¿½ï¿½[email protected]ï¿½+:oï¿½ï¿½F!ï¿½p aï¿½ï¿½cp0h(r)ï¿½ï¿½Iï¿½*ï¿½ï¿½...'Õ7ï¿½`ï¿½ï¿½Tï¿½Pï¿½&Lï¿½.ï¿½'ï¿½"Ûï¿½Nï¿½o'ï¿½ï¿½lï¿½m8ÍMfï¿½-ï¿½>Kkï¿½ï¿½=ï¿½(c)ï¿½ï¿½ï¿½'ï¿½Òï¿½u:8ï¿½ï¿½bï¿½ï¿½ï¿½ï¿½Nï¿½#?ï¿½jï¿½7Rb1ï¿½z ï¿½Q\$ï¿½=[]n^ï¿½ï¿½ï¿½Zï¿½#~ï¿½Óµï¿½^"Gï¿½eï¿½krï¿½ï¿½ ï¿½N/ï¿½y>ï¿½_ï¿½ï¿½+ï¿½&ï¿½ï¿½'}ï¿½ç³'ï¿½cï¿½au\ï¿½ï¿½hï¿½ï¿½a;-ï¿½tï¿½ï¿½vc?ï¿½ï¿½G'ï¿½W,3ï¿½ï¿½mï¿½ï¿½ï¿½alo8 fAï¿½y{'ï¿½ï¿½"oï¿½ï¿½ï¿½Aï¿½{Dï¿½ï¿½ï¿½Eï¿½ï¿½ï¿½&\$ï¿½(r)ï¿½uï¿½z"Y:ï¿½x aT;MMï¿½ï¿½:zÉï¿½ï¿½1/2ï¿½d+ï¿½ï¿½Tï¿½ï¿½2.q\$D1/2Gï¿½ï¿½ NT2ï¿½[email protected]ï¿½ï¿½"GN\$ï¿½ï¿½ ï¿½e<\$ï¿½ï¿½ ï¿½wï¿½S'>Lï¿½<ï¿½9pU.ï¿½[email protected]ï¿½G8 @ï¿½ã£Iï¿½dXaï¿½ï¿½ ï¿½ï¿½y_-ï¿½:ï¿½ï¿½ï¿½]ï¿½-ï¿½[email protected]ï¿½ï¿½dï¿½ï¿½/=oï¿½" "ï¿½Édâ6ï¿½ï¿½ï¿½ï¿½ï¿½.ï¿½ï¿½]-~ï¿½ï¿½?å1/4"cï¿½Mï¿½ï¿½i-ï¿½{ï¿½{ï¿½ï¿½ï¿½(tm)p9&ázï¿½ï¿½|ï¿½^-ï¿½'Ô¤Baï¿½ï¿½0ï¿½Gï¿½"-ï¿½ï¿½=ï¿½ï¿½ï¿½lï¿½cEï¿½||ï¿½!+ï¿½ï¿½ï¿½"Gnï¿½ï¿½ Y\$ï¿½ï¿½ï¿½ï¿½felï¿½aVï¿½)ï¿½ï¿½ï¿½ï¿½ï¿½Ihï¿½G<ï¿½#ï¿½:ï¿½/8.ï¿½ï¿½ï¿½:ï¿½;"ï¿½ï¿½ ï¿½nï¿½ï¿½ZN>ï¿½"ï¿½ï¿½:ï¿½ï¿½ï¿½5ï¿½ï¿½ï¿½ï¿½vM gï¿½...ï¿½4Ë­4Zï¿½3/4ï¿½ï¿½T7%ï¿½O-ÓxWï¿½Fï¿½ï¿½cJï¿½Nï¿½ï¿½WHï¿½! |_]8*ï¿½9Fï¿½ï¿½ï¿½ï¿½ ï¿½ï¿½fï¿½!ï¿½4ï¿½vï¿½Rï¿½ï¿½'l"6ï¿½gLï¿½"ï¿½ï¿½0ï¿½ï¿½ï¿½`ï¿½|Pï¿½.3/47Waï¿½qï¿½6ï¿½ ï¿½ï¿½hï¿½ï¿½ï¿½ï¿½T2ï¿½8ï¿½0ï¿½2ï¿½&Þ¬ï¿½Jï¿½zOï¿½ï¿½_>@qV!ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½9...[~/ï¿½ï¿½ï¿½! ï¿½-ï¿½^Ø¹kï¿½Oï¿½ZØµHï¿½`J ï¿½(tm)ï¿½ï¿½r(r)ï¿½ï¿½? ï¿½aA...ï¿½XM..."Uï¿½(r)ï¿½ï¿½...ï¿½ï¿½-pï¿½-ï¿½ï¿½tÙ¦Wï¿½_TW5,-ï¿½ï¿½ï¿½-3/4ï¿½ï¿½7ï¿½3...ï¿½ï¿½3[_ï¿½PL ï¿½fTï¿½2ï¿½ï¿½ï¿½ï¿½xï¿½ï¿½?ï¿½Ed3/4 Zï¿½ï¿½Ó­Ä¬ï¿½:Bï¿½ï¿½@ï¿½rï¿½1/4%ï¿½ï¿½Z?ITï¿½'s!"ï¿½+Vï¿½ï¿½Òµ[mï¿½G ...ï¿½~ï¿½'ï¿½abï¿½oOIï¿½'@ï¿½!ï¿½jQWï¿½"ï¿½fï¿½ï¿½(r) y"ï¿½YÝµ"K-mï¿½Sï¿½9ï¿½'ï¿½ï¿½_sï¿½dï¿½vï¿½ï¿½}k-ï¿½ï¿½(c)ï¿½(r)ï¿½-ï¿½z-oV&ï¿½[email protected]ï¿½'ï¿½ï¿½ï¿½pï¿½ ï¿½\$&ï¿½MddnOï¿½ï¿½ï¿½/!ï¿½ )ï¿½ ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Oï¿½ï¿½ï¿½\$ï¿½"ï¿½4Uï¿½\$ï¿½"c]j;ï¿½ )jï¿½ï¿½3.6ï¿½ ks3/4ï¿½ï¿½u ï¿½ï¿½ï¿½.ï¿½ï¿½ï¿½|wJΚï¿½ï¿½ï¿½(r)iï¿½"%{ï¿½ï¿½Tï¿½ï¿½@ï¿½y5[ï¿½ï¿½1/25ï¿½ï¿½0ï¿½ï¿½^~"ï¿½ï¿½ï¿½@Kï¿½ï¿½Uï¿½O(r)ï¿½\ ï¿½xï¿½ï¿½ï¿½< yS1/4"pï¿½ï¿½:oï¿½0Cï¿½ï¿½ï¿½ï¿½erï¿½% mï¿½ï¿½>?ï¿½ï¿½Nï¿½ï¿½ï¿½eï¿½*ï¿½ï¿½\$d8ï¿½ï¿½SxJjï¿½"%ï¿½ï¿½Fï¿½ï¿½w...ï¿½ï¿½#[email protected]~ï¿½dï¿½sï¿½:ï¿½ï¿½ï¿½fiï¿½ï¿½ï¿½Sï¿½"grï¿½ï¿½Çï¿½gØ¿0ï¿½ï¿½M5S'%xï¿½)1/4Äµ Âï¿½{3/4ï¿½ï¿½ï¿½ ...ï¿½cï¿½ï¿½ï¿½ï¿½ï¿½&)-ï¿½ 6Ù²ï¿½]ï¿½%wï¿½Jï¿½Qï¿½ï¿½ o%ï¿½ï¿½ï¿½Wï¿½ã¢ï¿½+Iï¿½!`Cï¿½)ï¿½ï¿½ ï¿½ï¿½ï¿½ï¿½ï¿½6ï¿½gï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Wï¿½ï¿½´1/4ï¿½ï¿½ï¿½Ü±Ñï¿½VbXzGm7ï¿½ï¿½n<ï¿½3/4ï¿½mï¿½

2. ## Math IB SL BMI Portfolio

graph follows a sinusoidal pattern when extrapolating data from the graph beyond 20 years of age. Based on common knowledge, girls are usually finished growing at � 20 years of age - yet one can see that had the domain not been restricted in the sinusoidal function, the BMI would

1. ## Investigating Parabolas

Since (D)(a) = 1, D has to be equal to 1/a. Consequently, the conjecture is D = 1/a It is not about the conjecture but I am introducing this rule in order to demonstrate the exceptions. g. Basic rule i.

2. ## Math SL Circle Portfolio. The aim of this task is to investigate positions ...

something cannot be 0 or have a negative value, for it is a measurement. The two general statements derived from this investigation are: , (n= ) and ' = (n= r). The first general statement is consistent with the second general statement.

1. ## Stellar Numbers Investigation Portfolio.

�2] x  S4 = 1 + 72 S4 = 73 The general statement makes it is possible to calculate that S4 of the 6- Stellar star has 73 dots in it. The Diagram of the 6 point Stellar Star it is possible to show the formula is true for S4.

2. ## Ib math HL portfolio parabola investigation

This implies h>k. I am going to keep the values of h at 3 and k at 2 constant in this part, and change the values of 'a' to see any emerging pattern. Thus our equation is : y = a(x-3)2 + 2 1.

1. ## Shadow Functions Maths IB HL Portfolio

When Thus, passes through the three points of intersection of our two quartic functions. When defined as the function passing through the three points of intersection between the original quartic and its shadow function, with equation, then the relationship between , and is: The relationship between , and is thus (expected to be)

2. ## MATH IB SL INT ASS1 - Pascal's Triangle

x-value (n) y-value (Xn) 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 Figure 5: Xn calculated with the general formula When we compare the two tables (Fig. 2 and Fig. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 