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Parabola investigation. In this task, we will investigate the patterns in the intersections of parabola and the lines y =x and y=2x.

Extracts from this document...

Introduction

Parabola Investigation – Portfolio HL TYPE I

PARABOLA INVESTIGATION

Description

In this task, we will investigate the patterns in the intersections of parabola and the lines y =x and y=2x. Then we will prove and find the conjectures and to broaden the scope of the investigation to include other lines and other types of polynomials.

1. Consider the parabola image00.png and the line image01.png andimage03.png.

image96.png

  • Using Graphmatica software, we can find the four intersections. Below these points are illustrated.

We find out 2 intersection points of f(x) and h(x) which are (1.7639; 3.5279)

            and (6.2361; 12.4721).                                                                                                                                                                                                                                                                The other 2 intersection points between f(x) and g(x) are (2.382; 2.382) and    (4.618; 4.618).

image103.png

  • The x-values of these intersections as they appear from the left to right on the x-axis as x1, x2, x3, x4.
  • x1≈ 1.764
  • x22.382
  • x3≈ 4.618
  • x4≈ 6.236
  • Find the values of image83.png and image92.png and name them respectively SL and SR.

 SL = x2−x1≈ 2.382−1.764≈ 0.618

 SR = x4−x3≈ 6.236−4.618≈ 1.618

  • Finally, calculateimage21.png.

image21.png

image129.png

image02.png

image11.png

     2. Find values of D for other parabolas of the formimage15.png,image16.png with vertices in quadrant 1, intersected by the lines image01.png andimage03.png. Consider various values of a, beginning withimage39.png. Make a conjecture about the value of D for these parabolas.

...read more.

Middle

 andimage03.png.

image124.png

The x-values of these intersections from the left to the right on the x-axis:

  • x1≈ 2.417
  • x2≈ 3.172
  • x3≈ 8.828
  • x4≈ 11.583

Calculation of SL and SR:

SL = x2−x1≈ 3.172− 2.417≈ 0.755

SR = x4−x3≈ 11.583−8.828≈ 2.755

Calculateimage21.png.

image21.png

image125.png

image126.png

image127.png

Consider parabolaimage12.png, the linesimage01.png andimage03.png.

image128.png

By using Graphmatica software, we can obtain the four intersections of parabola image12.png , the linesimage01.png andimage03.png.

image130.png

The x-values of these intersections from the left to the right on the x-axis:

  • x1≈ 2.591
  • x2≈ 3.285
  • x3≈ 9.315
  • x4≈ 11.809

Calculation of SL and SR:

SL = x2−x1≈ 3.285− 2.591≈ 0.694

SR = x4−x3≈ 11.809−9.315≈ 2.494

Calculateimage21.png.

image21.png

image131.png

image132.png

image133.png

Table shows the values of D for parabolas of the formimage15.png,image16.png with vertices in quadrant 1, intersected by the lines image01.png andimage03.png.

Parabolas

a

b

c

D

image04.png

1

-6

11

1

image05.png

1

-7

13

1

image06.png

2

-8

9

image07.png

image08.png

6

-12

7

image09.png

image10.png

image07.png

-5

14

2

image12.png

image13.png

-6

17

image14.png

As can be seen from the table D is inversely proportional to the value of a.

The values of D for parabolas of the formimage15.png,image16.png with vertices in quadrant 1, intersected by the lines image01.png andimage03.png, are inversely proportional to the values of a.

image17.png

3. Investigate your conjecture for any real value of a and any placement of the vertex.

...read more.

Conclusion

image61.png andimage62.png.

image63.png

The intersection points then can be found by using Graphmatica software.

image64.png

  • x1≈ 1.258
  • x2≈ 1.459
  • x3≈ 7.541
  • x4≈ 8.742

Calculation of SL and SR:

SL = x2−x1≈ 1.459− 1.258≈ 0.201

SR = x4−x3≈ 8.742−7.541≈ 1.201

Calculateimage21.png.

image21.png

image65.png

image02.png

image11.png

  In this case, the conjecture holds true. image26.png

  Consider parabola image66.png and the lines image67.png andimage69.png

.

image70.png

By using Graphmatica software, we can obtain the four intersections of parabola image66.png , the lines image67.png andimage69.png.

image71.png

  • x1≈ 0
  • x2≈ 0.392
  • x3≈ 4.000
  • x4≈ 5.109

Calculation of SL and SR:

SL = x2−x1≈ 0.392− 0≈ 0.392

SR = x4−x3≈ 5.109−4.000≈ 1.109

Calculateimage21.png.

image21.png

image72.png

image73.png

image75.png

Consider parabola image76.pngand the lines image77.pngand image78.png

image79.png

Again, using Graphmatica software, we can obtain the four intersection points.

image80.png

  • x1≈ -2.000
  • x2≈ 0.209
  • x3≈ 1.500
  • x4≈ 4.791

Calculation of SL and SR:

SL = x2−x1≈ 0.209+2.000≈ 2.209

SR = x4−x3≈ 4.791−1.500≈ 3.291

Calculateimage21.png.

image21.png

image81.png

image82.png

image84.png

Consider parabola image85.pngand the line y = 5 and y =-3.

image86.png

By using Graphmatica software, we can obtain the four intersection points.

image87.png

  • x1≈ -6.123
  • x2≈ -5.000
  • x3≈ 1.000
  • x4≈ 2.123

Calculation of SL and SR:

SL = x2−x1≈ (-5.000) − (-6.123) ≈ 1.123

SR = x4−x3≈ 2.123−1.000≈ 1.123

Calculateimage21.png.

image21.png

image88.png

image89.png

Proof:

ax2 + (b− m1) x + c= 0

ax2 + (b− m2) x + c= 0

image90.png

image91.png

D=|SL−SR| = |(x2−x1) – (x4−x3)|= |x2−x1− x4+x3|

= |x2+x3−x1− x4|= |(x2+x3) −(x1+x4)|

x2 +x3 = 2[−(b− m1)/2a]= −(b− m1)/a

x1 +x4 = 2[−(b− m2)/2a]= −(b− m2)/a

image93.png, with a image94.pngR.

Cao Huu Anh Khoa – 5X

...read more.

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