Maths HL Type 1 Portfolio Parabolas
Maths Portfolio 1
HL Type 1
Parabola Investigation
. 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 .
vi. Repeat step iii and iv to find the second intersection by moving the cursor closer to the second intersection,
vii. Hence, the second intersection of the parabola is (7.646,15.292)
The values from left to right are:
Calculation of ,
Calculation of ,
Consider the parabola .
By using the software Autograph to calculate the intersection points,
i. Select the parabola and the line y=x to calculate the intersection points. Select the option of "Solve Intersections"
ii. Repeat step i. to solve for intersections for the parabola and line y=2x by highlighting the line y=2x instead of the line y=x.
iii. Label the intersection points.
The values from left to right along the x-axis are:
Calculation of ,
Calculation of D,
Consider the parabola
By using the software Autograph,
The intersections between the parabola are shown in the graph below:
The intersections between the parabola and line y=x are (2,2) and (3.333,3.333).
The intersections between the parabola and line y=2x are (1.667,3.333) and (4,8).
The x-values from left to right along the x-axis are:
Calculation of ,
Calculation of D,
Consider the parabola
By using Autograph software, the four intersections between the parabola and the lines can be found.
The intersections between the parabola and line y=x are (2.586,2.586) and (5.414,5.414)
The intersections between the parabola and line y=2x are (1.683,3.367) and (8.317,16.63)
The x-values from left to right on the x-axis:
Calculation of
Calculation of D
Consider the parabola and the lines
The four intersections between the parabola and the lines y=x and y=2x can be found via Autograph software.
The intersections between the parabola and line are (1.268,1.268) and (4.732,4.732)
The intersections between the parabola and line are (0.7251,1.45) and (8.275,16.55)
The x-values from left to right on the x-axis:
Calculation of ,
Calculation of D,
Consider the parabola and the lines ,
The intersections between the parabola and the lines y=x and y=2x can be found using Autograph software,
The intersections between the parabola and the line y=x are (3.4810724, 3.4810724) and (4.7007458,4.7007548)
The intersections between the parabola and the line y=2x are (2.4724083, 4.9448166) and (6.6185008,13.237002)
The x-values from left to right on the x-axis:
Calculation of ...
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The intersections between the parabola and line are (0.7251,1.45) and (8.275,16.55)
The x-values from left to right on the x-axis:
Calculation of ,
Calculation of D,
Consider the parabola and the lines ,
The intersections between the parabola and the lines y=x and y=2x can be found using Autograph software,
The intersections between the parabola and the line y=x are (3.4810724, 3.4810724) and (4.7007458,4.7007548)
The intersections between the parabola and the line y=2x are (2.4724083, 4.9448166) and (6.6185008,13.237002)
The x-values from left to right on the x-axis:
Calculation of
Calculation of D:
Table showing the values of D for various parabolas of the form
Parabola equation
a
b
C
-6
1
-8
8
2
-10
3
3
-15
20
-3
7
2
-1
2
3
-8
8
From the table, there is no relationship between b and c with . However, a is inversely proportional to the value of .
Conjecture: For parabolas with the form with vertices in quadrant 1, intersected by the lines y=x and y=2x, the value of a is inversely proportional to the value of , i.e. .
3. Investigation of the conjecture for any real value of a and any placement of the vertex.
To investigate my conjecture for any real value of a and any placement of the vertex, the conditions earlier attached to my conjecture that a>0 and vertices in quadrant 1 and intersected by the lines y=x and y=2x should be discarded, but the conjecture that the value of a is inversely proportional to the value of should be kept.
First, consider the parabola and the lines
The four intersections between the parabola and the two lines y=x and y=2x can be found once again by Autograph software.
The intersection between the parabola and the line y=x are (0.6492,0.6492) and (3.851,3.851)
The intersection between the parabola and the line y=2x are (0.7752,1.55) and (3.225,6.45)
The x-values:
Calculation of ,
The conjecture does not hold when , because the value of a is -2 and according to my conjecture made in part 2, the value of D should be
Hence, I would modify my conjecture to make always positive,i.e. ,
Consider the parabola and the lines
The intersections between the parabola and the lines :
The intersection between the parabola and the line are (-3.646,-3.646) and (1.646,1.646)
The intersection between the parabola and the line are (-3,-6) and (2,4)
x-values :
Calculation of D,
In this case, the conjecture holds true when the vertex is in the third quadrant,, and the parabola intersects the lines at two distinct points each.
Consider the parabola ,
The intersections of the parabola and the lines
The intersections between the parabola and the line y=x are (0.3625,0.3625) and (4.137,4.137)
The intersections between the parabola and the line y=2x are (0.3206,0.6411) and (4.679,9.359)
The x-values:
Calculation of D
The conjecture holds for the case when the vertex is in quadrant 4, , and the parabola intersects the lines at two distinct points each.
Consider the parabola and the lines
The intersections of the parabola and the lines y=x and y=2x are:
The intersections between the parabola and the line are (-3.531,-3.531) and (4.531,4.531)
The intersections between the parabola and thel ine are (-3.123,-6.246) and (5.123,10.25)
The x-values :
Calculation of D
Consider the parabola and the lines
The intersections of the parabola and the lines y=x and y=2x:
The intersection between the parabola and the line y=x is (1,1)
The intersections between the parabola and the line y=2x are (0.382,0.7639) and (2.618,5.236)
The x-values:
Since there is only one real intersection between the parabola and , the values of will be repeated.
Calculation of D,
The conjecture still holds true for when the intersect between the parabola and the line is repeated.
Consider the parabola and the lines .
The intersection between the parabola and the lines y=x and y=2x.
There is only one point of real intersection between the parabola and the line y=2x at (1,2).
Hence, to find the intersections between the parabola and the line i.e. to find the imaginary intersections between the parabola and the line .
Substitute (2) into (1),
Let be ,
Therefore, x-values:
Calculation of D,
Therefore, the conjecture of holds true when the parabola only intersects one line.
Consider the parabola and the lines
There are no real intersections between the parabola and the lines
There are two distinct imaginary intersections between the parabola and the line
To find the imaginary intersections between the parabola and the line ,
Substitute (2) 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. The D value was suppose to be 1 with the conjecture that . However, I cannot modify my conjecture yet as there is insufficient cases to be able to come up with a conjecture that can suit the purpose.
Consider the parabola and the lines and .The intersections of the parabola and the lines can then be found by the Autograph software.
The intersections between the parabola and the line are (-6,12) and (1,-2).
The intersections between the parabola and the line are (-4.702,2) and (1.702,2).
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 .
The x-values:
Calculation of D:
Once again the conjecture does not hold when the intersecting lines are changed as the D value was suppose to be 1 according to the conjecture that .
Consider the parabola and the lines and .
The intersections between the parabola and the lines can be found via Autograph software and is shown below:
The intersections between the parabola and the line are (-5.541,8.083) and (0.5414,-4.083).
The intersections between the parabola and the line are (-2.193,-7.77) and (3.193, 13.77).
Let the x-values of the intersections of the parabola and the line be
Let the x-values of the intersections of the parabola and the line be
The x-values :
Calculation of D:
The conjecture does not hold once again as the value is supposed to be 1 according to the conjecture of .
Consider the parabola and the line .
The intersections between the parabola and the lines can then be found via Autograph software and its shown below:
The intersections between the parabola and the line are (-4.702, 2) and (1.702,2).
The intersections between the parabola and the line are (-3.791, -3) and (0.7913, -3).
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
The x-values:
Calculation of D
0|
Hence the conjecture of is once again not proved with this case. The D value is supposed to be 1 according to the conjecture.
Consider the parabola .
The intersections between the parabola and the intersecting lines can then be found via Autograph software and is shown below:
The intersections between the parabola and the line are (-5.275,6) and (2.275,6).
The intersections between the parabola and the line are (-5,4) and (2,4).
Let the x-values of the intersections between the parabola and the line to be
Let the x-values of the intersections between the parabola and the line to be
The x-values :
Calculation of D:
Hence the conjecture of is once again proven wrong in this case.
I will now form a table to show the parabolas and different values of and the value of D.
In this table, I will also include two cases from part 3 to show when the intersecting lines were
Parabola
Equation of first intersecting line
Equation of second intersecting line
-1
4
0
0
5
-2
0
0
2
2
-2
4
-3
6
0
0
2
-3
0
0
0
6
4
0
2
0
0
2
0
0
Observing the table above, it can be observed that conjecture from earlier is non-applicable when the intersecting lines are changed. It can also be observed when the values are altered, the D value changes but when the values are altered, there is no change in the D value. Hence, I will try to find the relationship between the values and the D value and from there modify my conjecture.
It can be observed that the D value is the value of when the value of as shown in the first 6 cases in the table. Hence, I would modify my conjecture to be:
Conjecture : D is the absolute value of the difference of divided by a in the general equation of ,i.e. , where .
To prove this conjecture algerbraically using 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 and
5. Determine whether a similar conjecture can be made for cubic polynomials.
The general equation of a cubic polynomial is
Consider the cubic function ,
The cubic function has three distinct roots which will be labelled as respectively.
From the above equation:
To find the intersections between the cubic equation and the linear lines of ,
Let the intersections between the cubic function and the line to be
Let the intersections between the cubic function and the line to be .
To find the intersections between the cubic equation and the line
Substitute (2) into (1),
Hence, since the are roots to the equation of ,
The sum of roots, i.e.
The sum of the product of two roots, i.e.
The product of all three roots, i.e.
To find the intersections between the cubic equation and the line
Substitute (3) into (1),
Hence, since are roots to the equation of
The sum of the roots, i.e.
The sum of the product of two roots, i.e.
The product of all three roots, i.e.
To apply the conjecture to cubic polynomials,
I will also prove this conjecture graphically.
Consider the parabola and the lines
The x-values of the intersections :
Calculation of D:
However, the intersecting line may be quadratic in the case of cubic polynomials.
......(1)
.............(2)
.............(3)
Let the intersections between the cubic function and the line to be
Let the intersections between the cubic function and the line to be .
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 .
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