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Maths HL Type 1 Portfolio Parabolas

Extracts from this document...

Introduction

Maths Portfolio 1 HL Type 1 Parabola Investigation 1. Consider the function To find the four intersections in the graph shown above using the GDC, i. Press the 2nd button and then the TRACE button to select the CALC function. Select the intersect function by pressing button 5. ii. Select the first curve of intersection and press ENTER. iii. Select the second curve of intersection and press ENTER. iv. Select the area of estimation of the intersection point and press ENTER. v. The first intersection point between Repeat the above steps i-iv, to obtain the other intersection points between . The second intersection point between The first intersection point of The second intersection point of The To find the values of , To calculate the value of , 2. To find other values of D for other parabolas of the form ,with vertices in quadrant 1, intersected by the lines Consider the parabola and the lines , The intersections of the parabola with the lines can be calculated using both the GDC and manual calculation. By manual calculation, To calculate the intersection between , Sub (2) into (1), Sub Sub The intersections between the parabola are (3,3) and (6,6). By using the GDC, To calculate the intersection between i. Key in equations of into the GDC, ii. Press the TRACE button to plot the graph on the GDC, iii. Press the 2nd button and the TRACE button to select the CALC function. Select the intersect function by pressing 5 to calculate the intersection between the parabola and the linear line. iv. Select the first curve which is the parabola by pressing ENTER and then the second curve which is the line by pressing ENTER, then estimate the location of the intersection by moving the cursor using the left and right directional buttons and then press ENTER. v. Hence, the first intersection of the parabola is . ...read more.

Middle

into (1), Let be , To find the imaginary intersection between the parabola and the line Substitute (2) into (1), Let Hence, the x-values are: Calculation of D, The conjecture still holds true when the intersections are not real numbers, for real values of a and a>0. To prove the conjecture using the general equation of , Let the roots of the general equation be . Hence, it can be deduced that, Since , where To find the x-values of intersections between the parabola and the lines Substitute (2) into (1), Hence, since the roots of the equation are The sum of the roots of the equation, i.e. Substitute (3) into (1), Hence, since the roots of the equation are The sum of the roots of the equation, i.e. , Hence, the conjecture is proven for all real values of a, . 4. Investigating the conjecture when the intersecting lines are changed. To investigate whether the conjecture still works when the intersecting lines are changed, I will be using the same parabola while varying the intersecting lines. To vary intersecting lines, the intersectings lines all follow the general equation of , where m is the gradient and c is the constant. Hence, for the two intersecting lines, I will be varying the m value and the c value. The equation of the two intersecting lines will be as follows: Line equation 1: Line equation 2: The values of will be varied. Consider the parabola and the intersecting lines of , The intersections between the parabola and the intersecting lines can then be found via Autograph software: The intersections between the parabola and the line are (-2,-8) and (3,12) The intersections between the parabola and the line are (-5.162,5.162) and (1.162,-1.162) Let the x-values of the intersections between the parabola and the line be . Let the x-values of the intersections between the parabola and the line be Hence, the x-values: Calculation of D: The conjecture does not hold when the intersecting lines are changed. ...read more.

Conclusion

To find the intersections between the cubic function and the first quadratic curve, Substitute (2) into (1), Since the roots of the equation are The sum of roots,i.e. To find the intersections between the cubic function and the first quadratic curve, Substitute (3) into (1), Since the roots of the equation are The sum of roots, i.e. Since This can be proven graphically, Consider the cubic function and the quadratic equations The x-values: Calculation of D: |3-3| Alternatively, , where Hence, the conjecture that can be made for cubic polynomials is that for all cubic polynomials that are intersected with linear lines, however, when the cubic polynomial is intersected with quadratic equations, , where 6. Consider whether the conjecture might be modifired to include higher order polynomials. To consider higher order polynomials, The general equation would be , where n is the highest degree. Alternatively, it can be written as . The roots of the equation would be Hence, The sum of roots will be as proven earlier. When the polynomial intersects with a line that is at least two degrees lower than the polynomial,i.e. , the two equations can then be equated to find the points of intersection which will be the roots of the new equation. Hence the sum of roots will then be Hence when the parabola is intersected with two linear lines, the value of D will be Therefore, the conjecture that will hold as long as the polynomial is intersected with a line that is at least two degrees lower than the polynomial. However, when the polynomial is intersected with a line that is one degree lower,i.e. Equation of polynomial: Equation of intersecting line: , The sum of roots of the new equation will be , as according to examples above. When the polynomial is intersected by two lines that are one degree lower than the polynomial, the value of D will be Hence when the higher order polynomials of degree n is intersected with lines that are one degree less, the conjecture will be . ?? ?? ?? ?? 2 ...read more.

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