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

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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 and the line and.

- 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).

- 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
- x2≈2.382
- x3≈ 4.618
- x4≈ 6.236

- Find the values of and 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, calculate.

2. Find values of D for other parabolas of the form, with vertices in quadrant 1, intersected by the lines and. Consider various values of a, beginning with. Make a conjecture about the value of D for these parabolas.

Middle

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

Calculate.

Consider parabola, the lines and.

By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.

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

Calculate.

Table shows the values of D for parabolas of the form, with vertices in quadrant 1, intersected by the lines and.

Parabolas | a | b | c | D |

1 | -6 | 11 | 1 | |

1 | -7 | 13 | 1 | |

2 | -8 | 9 | ||

6 | -12 | 7 | ||

-5 | 14 | 2 | ||

-6 | 17 |

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

The values of D for parabolas of the form, with vertices in quadrant 1, intersected by the lines and, are inversely proportional to the values of a.

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

Conclusion

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

- 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

Calculate.

In this case, the conjecture holds true.

Consider parabola and the lines and

.

By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.

- 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

Calculate.

Consider parabola and the lines and

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

- 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

Calculate.

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

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

- 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

Calculate.

Proof:

ax2 + (b− m1) x + c= 0

ax2 + (b− m2) x + c= 0

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

, with a R.

Cao Huu Anh Khoa – 5X

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

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