Year 11 Maths A Investigation FINAL
A
Part of the Quadratics Topic
Year
11 Maths A
Name:
Noah Nishihara
Table of Contents
Introduction
The
purpose of this investigation will beto test the effectiveness of quadratic
functions in modelling arches.Accurate representations of pre-existing
structures are required, and by using parabolic functions, the modelling of archesis
expected to be successful. A wide concave
arch will be found within a photo taken from a point facing perpendicular to
the bridge to ensure it is a front-on view of the Anji Bridge in China. Toformulate
models, its imagewill be imported to Geogebra to assign a pair of axes,for
which the x-axis lies on the baseline of the bridge. Using the zoom function,
evenly spaced points will be highly accurately plotted along the outer ‘line’below
the arch bridge. The vertex of the arch will be estimated to be below the
sculptured faceof a lion and x-intercepts plotted where the bridge meets the
ground.
Next, an algebraic method will be chosen to create
functions based on points of the actual arch. To compare and determine the most
accurate function, three quadratic forms which are the factored form, vertex
form, and general form, are to be solved for using different points.The two
x-interceptswill determine factored form. The coordinates of a third point on
the arch will then be substituted into the function. The coordinates of an
estimated vertex into which the coordinates of a second point on the arch will
be substitutedwill givethe vertex form. Thereafter, instead of relying upon
comparisons of only a single different third or second point, three random
points on the arch will be selected and substituted into the function to
determine the general form. The three equations will be solved simultaneously
and with technology. Effects of different points on the resulting quadratic
model will bediscussed along with visual comparisons through direct graphing of
models in Geogebra.This involves observing the extent of the overlap between
the models and the actual arch.This will allow three best-fitting quadratic models
to be derived.
Because the intended outcome is also to determine the
most effective modelling method for arches, the three best-fitting models
derived from the algebraic method will be compared with the actual arch. A
prediction of the model with least variance will be made. To achieve greater
accuracy in this comparison, the variance between the modelled heights and the
actual heights of the arch will be found. This will becalculated for ten evenly
spaced points spanning the arch. The ten resulting differences will then be
added to give the overall variance. The variance will be derived for three
different equations and the prediction of the model with least variance will be
tested.
Regression modelling will be used to derive
quadratic modelsbased on points inputted. A comparison will be made between
regression models fitting fewer points with those fitting a greater number of
points. This is to test the impact of the amount of points onhow well the curve
fits to the arch.The r2 value will be recorded to give a measure of
accuracy for reference. The results will be analysed and discussed by taking
into consideration limitations and reasonableness of the process.
Results and Interpretation:
Figure
1 Initial Points Across Bridge
Several points were
accurately plotted across the outer ‘line’ below the arch bridge. X-intercepts
were plotted at the intersection of the bridge and the ground. The vertex of
the arch wascalculated with
Firstly, three points were
selected on the arch of which the coordinates were used to derive the equation
of the quadratic function in the factored
form of
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The most accurate parabola
obtained from the three points in factored
form was that of
For vertex form, which involved constructing
models of the chosen arch with only the coordinates of the vertex and a second
point on the arch, the steps followed are outlined below while the calculations
and results are beside the factored form results above. This has been done out
of page constraints.
It can be observed
that the coordinates of the vertexare first substituted
into thevertex form function
The first example
shows the coordinates of point S were substituted into the vertex form function
This
process was repeated for points P and Q. So, forvertex form the coordinates of
the vertex were used together with the coordinates of a second point on the
arch. The vertex turning point was located at (h,k) = (4.1,1.62). The chosen (x,y)
points on the arch were: S (3.8, 1.62), P (2.9, 1.54), and Q (3.2, 1.58) as
well as C(-0.4,0) and D(8.5,0) and M(2,1.3)included in Appendix 2. The
For the general form
quadratic y=ax2+bx+c, the first setof coordinates chosen included points O
(2.5, 1.45), W (4.9, 1.57), and F1 (6.9, 1.01) from which:
It was found following
the graphing of results that the selection
of certain points raised the level of accuracy more than others.The
factored form calculations required the selection of a third point. The points
were chosen from different sections of the bridge. Although the vertex gave
best results, point G1 on the far right, point V on the right of the
vertex and point I on the far left side of the bridge were also attempted. Of
these, point G1 had a greater
Again, in the vertex
form calculations, the second point was selected based on different regions of
the bridge. In this case, points situated relatively close to each other have
been included above. Points M, P, and Q have the slopes, -0.07, -0.06, and
-0.05 respectively. Only the graph of point M lies close to the actual arch
line. [Check P and Q] The other two (x-intercept) calculations are accurate. It
can be summarised that the closer to the vertex the second point is, the less
accurate the resulting model will become. This is proven by the calculation
involving point S.
The selection of the three
points altered the coefficient of x in the function of the model from 0.69 to
0.61. The value of 0.61 comes from the first set with two points on the right
side of the bridge. The value of 0.69 is derived with a set of two points on
the left side of the bridge. The result 0.68 also comes from two points on the
left and one on the right. It was found that using two points on the right side
caused the model to come closer to the actual arch on the right side, however,
it rose above the actual arch on the left side. With the function with the coefficient
of 0.68, the right side of the model rose above the actual arch. Also, the use
of points on the far left side caused the model to rise further (the white line
of Appendix 6).
Analysis of Models
Table1:A
comparison of models resulted in three different models deemed to best fit the
arch
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Method used |
Resulting Equation |
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y=-0.08x2+0.66x+0.28 y=-0.08x2+0.67x+0.28 y=-0.09x2+0.67x+0.28 |
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y=-0.07x2+0.574x+2.7967 y=-0.06x2-0.49x+0.6114 |
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Figure 2 Three best models
The
coloured parabolas were graphed and are shown in Figure2, respective to its
colour in the table above. Nine resulting equations of each of the three
methods were graphed and included in Appendix 6.Of the best three models
selected from a total of nine models, the graph with the least variance was
predicted to be y=-0.08x2+0.66x+0.28
as it lay below the others and was closest to the actual bridge
arch. It overlapped the actual arch more than the parabola of Y= -0.075x2+0.605x+0.41.
To
test this prediction, eighteen new points spaced at intervals of 0.5 were
chosen. These points are shown in the second diagram of Appendix 1. Firstly, using
the spreadsheet function of Geogebra, the absolute value of the difference in
height between the arch of the bridge and parabola
y=-0.08x2+0.66x+0.28 was calculated.Refer to the first diagram
of Appendix 5. Column A was first filled with x coordinates of all bridge
points from left to right. The y coordinates of the parabola werethen calculated
in column B by typing f(x) where x becomes the x coordinates of bridge points.
The y coordinates of the actual arch then needed to be input in place of the x
coordinates in column A whereby the difference of y coordinates of both the
arch and model could be calculated. Copying data from Geogebra to Excel revealed
that the y coordinates of the parabola became accurate (four decimal places
instead of two).
In the second diagram of Appendix5,
the A column is the height of the actual arch according to the y axis, the B
column is the height of the chosen model, and the differences are in the C
column. Because the variance was in decimals and cannot be interpreted, the
differences were added together and the overall variance found. This was 18.3
for A, 18.5116 for B giving the overall variance 0.4044. The chosen model waspositioned
higher than the original arch overall, but the comparison required other
parabolas, hence, the other two models were drawn on Geogebra. It was predicted
that the overall variance of the others would be greater than 0.4044. The parabola f(x)= -0.05x2+0.41x+0.7795 was
then drawn. The overall variance of this pair of graphs turned out to be
3.9795. Finally, the parabola f(x) = -0.075x2+0.605x+0.41was
drawn. The overall variance of this pair of graphs turned out to be 0.70325.Therefore,
the parabola with the least overall variance is parabola y=-0.08x2+0.66x+0.28
which proved the prediction made previously based on visual comparisons.
The
overall variance successfully reflected the difference in heights between the
chosen model and the actual arch. It was assumed that the model was positioned
on the actual arch. If the actual arch and the model were near, the variance
would be approaching zero. The calculations acted as proof for the visual
comparisons undertaken above.
Regression Modelling
To further clarify
the quadratic model that best fits the actual arch, regression modelling was
undertaken with the graphics calculator CG20AU and Excel. Inthe fx-CG20 AU
graphics calculator, the same eighteen points were first entered into the Stat
app in two columns for both x and y coordinates, and the quadratic function was
calculated. It was noticed that the graph had vertex (4.08, 1.64), which is
close to the estimated point T (4.1, 1.62). All other points were accurate to
the nearest hundredth decimal place. The result is included in the bottom right
of Appendix 4.
Using theMicrosoft Excelsoftware
on a computer, a simple scatter plot was drawn using the eighteen points, and
then a trend linewas put onto it. The polynomial function that resulted was y =
-0.0814x2 + 0.6646x + 0.285 with a coefficient of determination of r²
= 0.9993. This was the same outcome as CG20AU and indicated that most of the
points fit the modelled regression. This model was deemed to be reasonably
accurate as the points were plotted suitably accurately.In an attempt to find a
higher r2 value, the previous points used for algebraic calculations
(total of 32) were input into Excel and the same process was followed to plot
an accurate line. Some of these points had the same y-intercept (e.g. 1.62).
The resulting function of the trend line was y = -0.0819x2 + 0.6682x
+ 0.2779 with an r² value of 0.9991. The accuracy of the trend line decreased
by close to 0.0002. Increasing the number of points hadlowered the accuracy of
the resulting model.The r2 value was found to be very close to 1 for
all models, showing that the accuracy was high. The residuals would have
increased as the number of points increased thus giving a lowerr2
value. Following this, the prediction quality of Excel was then compared with
Eureqa.
For the regression
model generated with Eureqa Pro Academic, the two sets of coordinate points were
combined and inputted to model a more precise arch with negligible errors. The
49 coordinates resulted in y= -0.08143x2+0.6648x+0.2822 without the
sorting of x-coordinate data into ascending order. Refer to the first diagram
of Appendix 7. On the second attempt, the data resulted in y= -0.08165x2+0.6664x+0.2833.
This is the result after the sorting and all the coefficients have become
larger.This function, y=-0.08x2+0.67x+0.28 was inputted into
Geogebra (due to the decimal place restriction) and the parabolic function was
found to be generally overlapping the actual arch, but would cross over to the
outer side at the vertex and at the x-intercepts.
The final smoothed
arch quadratic function was y= -0.08035x2+0.6574x+0.2896.The
smoothing capabilitywas found to allow the final model to better fit the
points. This function had an r2 value of 0.994.The Mean Absolute
Error (average of all absolute errors) came as 0.028699197. The maximum error
was 0.13932556. Refer to the third diagram of Appendix 7. Additionally, this function,
y= -0.08x2+0.66x+0.29, was graphed in Geogebra, showing a much
improved overlap between the model and the actual arch at the vertex. The
x-intercepts were not met because quadratic functions are limited in their
ability to curve excessively to meet the ends of the bridge arch. This was the
best model derived with Eureqa.
The higher power functions were believed to model the arch better, however, the cubic functions andother results were complex and graphing was not attempted. Thus, the two cubic functions with similar accuracy to the quadratic were not tested. The residuals or the vertical distances between plotted coordinates and the quadratic regressionwere found to besimilar to the variance. Nine points lay above the model. These were positive residuals. Ten negative residuals were observed. However, these ten points below the model were closer to the line of the model than the positive residuals. Twenty-nine residuals were zero, as the points lay on the line.Thebalance of positive and negative residuals was also proof of the validity of this model.Based on the above discussion, the errors of regressions are necessary for all models. Perfect models fitting all inputted points cannot be created.It is important however to also examine the different regression models and how they are affected by the position of the points inputted.
Both y= -0.08035x2+0.6574x+0.2896
from Eureqa and y = -0.0814x2 + 0.6646x + 0.285 (or algebraically
derived y=-0.08x2+0.66x+0.28) from Excel were considered accurate
and effective parabolic functions modelling the bridge. The difference (between
the regression and algebraic models) of the constant c value was 0.01, and
resulted in the Eureqa function situated closer to the bridge than the algebraic
model. For reference, a diagram has been included in Appendix 9.
Limitations
The first limitation of this
investigation occurs as a parallax error in the photo of the chosen arch
bridge. This is because the camera taking the photo was not at a perpendicular
angle to the face of the bridge. If the lengthwise of the camera body is not
parallel with the face of the bridge, the viewing distance from the viewpoint
of the camera to the ends of the bridges (x-coordinates) will be skewed. The
photo of the Anji Bridge chosen was taken facing slightly to the right. Thus,
the disparity in the distance between the bridge to the viewer can be rectified
by acknowledging the problem and addressing it by using single-lens reflex
cameras in which the viewfinder uses the same lens as the lens used for taking
the photo. This can be done if the arch photo is taken locally. In the event
that this is not possible, increasing the distance between the camera and the
bridge can also reduce error.
The
zoom function was utilised during the plotting of points along the arch.
However, the coordinates of the points could only be determined to 2 decimal
places. This limited the precision of the later derived quadratic models and
resulted in functions which inaccurately reflect the actual arch to some
degree. To address this issue, a software package with greater capabilities and
exactness of points could have been used. In addition, the imported arch image
size was smaller than expected. Indeed the small size meant greater reliance on
detailed coordinates. Due to the two decimal place restriction, there existed
an ambiguity, namely involving the points situated too closely to one another.
These occupy different positions but have the same coordinates. During the
graphing, it was found that because Geogebra rounds numbers to two decimal
places, the graphs of some algebraically derived models may not accurately
reflect the function it is representing. In this case, an
In
choosing the two x-intercepts for factored form, the x-axis provided greater
accuracy to the positioning by ensuring that both points are horizontally on
the same level. However, the distance between them may not be accurate. This is
because the intersection of the outer rim (actual arch) of the bridge with the
ground varies. For instance the left side intersection may have been covered
with more sand than the other, thus resulting in a straight x-axis touching
both points. It may be straight but the distance is not correct. For vertex
form, the accuracy of the vertex was determined by the position of the animal
face in the middle of the bridge. The algebraically constructed models of the
arch resulted in functions with
This
systematic error also affects the y coordinates of points plotted on the actual
arch during the variance based comparison. Not only were the three derived
models based on approximate values; every calculation was. Adding the results
together resulted in an overall variance. Comparisons of the overall variance
were mostly accurate but not definite.
Finally
in the regression modelling, the points chosen in Geogebra (of which two sets
were used) again presented a minor problem. The two types of technology used
found unnecessarily precise numbers for the function fitting the inputted points.
If precise coordinates had been inputted in the first place, this function
would better fit the actual arch, which is the sole aim of this investigation.
Any inaccurately plotted points would affect the final function. It cannot be
deemed certain that no mistakes were made in the copying of coordinate points
as values into the spreadsheet and calculator either. The results from the two
methods could be cross-referenced with Eureqa. In terms of greatest actual arch
resemblance, the Eureqa results were proven by visual and numerical comparison
to be highly accurate regardless of the error that was created by the plotted
points. Functions which model the chosen arch bridge better were found however
they were difficult to model.
Conclusion
From this investigation
the most effective arch fitting the Zhaozhou Bridge was found. The process
taken to achieve this outcome began with the selection of the ZhaozhouBridge in
China and its placement on the x-axis in graphing software. Points were plotted
along the bottom of the bridge arch and used to mathematically construct
models. Subsequently, various selections of points and methods were used to
determine the most effective method. To consider the model which fit the arch
best, three models (functions) were found via algebraic means. The results are
summarised below.
Two
x-intercepts and another random point gave the functions:
§ y=-0.08x2+0.66x+0.28
§ y=-0.08x2+0.67x+0.28
§ y=-0.09x2+0.67x+0.28
The
calculations using the vertex and one random point gave:
Ø y=-0.05x2+0.41x+0.7795
Ø y=-0.07x2+0.574x+2.7967
Ø y=-0.06x2-0.49x+0.6114
Three
random points substituted into the general form gave:
·
y=
-0.075x2+0.605x+0.41
·
y=
-0.08x2+0.69+0.27
·
y=-0.08x2+0.68x+0.27
Upon graphing all
obtained models, the most accurate
algebraically solved model was found to be:
y=-0.08x2+0.66x+0.28
This was determined
bythe factored form of y=a(x-α)(x-β).The final
smoothed arch quadratic functionof
Eureqawas y= -0.08035x2+0.6574x+0.2896, which also supports this.
Both were considered accurate and effective parabolic functions modelling the
bridge. The difference of the constant c value was 0.01, and resulted in the Eureqa
function situated closer to the bridge than the former.Thus the Eureqa function
models the actual arch more accurately.
This investigation
proves that parabolic functions can successfully reflect actual arch models
seen in our everyday lives. It has determined an accurate and effective arch
for the Anji arch bridge. The parabolic
model:
y=-0.08x2+0.66x+0.28
was found to have
the greatest accuracy.It could be stated that the vertex and the x-intercepts brought
greater accuracy than other points. The vertex and x-intercepts were non-random
points which will always allow a model to roughly fit the actual arch if
substituted into the factored form quadratic function. To solve the problem, it
was necessary to include as many different methodologies and technology types
to ensure that results were consistent throughout. Issues that were overcome
included entering the coordinate points correctly, software inconsistencies,
and the understanding of limitations.
Due to the
necessity of accurate architectural plans or diagrams, the modelling of
real-life architectural features involving arches was attempted with parabolic
functions. Through this investigation, the extent of the effectiveness of parabolic
functions in modelling arches has been determined. This includes an
understanding of the ability of parabolas to model arches very well but less accurately
for certain curves such as the edges of the Anji bridge. The method that gave
the best result was the regression modelling with Eureqa. Moreover, regardless
of the method used, non-random points including vertex and x-intercept points
were found to have the best results.
The investigation results
indicated a method of increasing the accuracy of all modelling of arches. To
increase the accuracy of the resulting models, more points were chosen. This
was particularly apparent with the vertex form calculations with only a
selection of two points. The results for vertex form calculations were highly
inaccurate, while those of factored and general form, due to the nature of it
allowing three points to be chosen, were more likely to derive a close-fitting
arch. Factored form only allowed the choosing of one third point, as the
x-intercepts were necessary for the calculation. This allowed most attempts to
derive a close-fitting arch, whereas general form was still likely to derive
skewed arches depending upon the choice of points. This further suggests the
use of non-random points as surely the easiest way to increase accuracy with
fewer points. It was certain that in the regression modelling, the inputting of
49 coordinates allowed the most accurate function to be derived. The use of
more points raises the overall accuracy to a substantial extent.
Further research
into the Zhaozhou Bridge could involve a physics investigation of how the
overall structure, including the arch, was able to maintain its strength for
centuries.
Bibliography
河北赵州桥图片-城市-(n.d.),
一起悦读网, Photograph, accessed
25 March 2017, <http://img.171u.com/image/1205/2709214375982.jpg>.
Appendices
Appendix 1: Anji/Zhaozhou
Bridge
[("Safe crossing bridge") is the world's oldest open-spandrel
segmental arch bridge of stone construction; Length:
50.82 m;Width: 9.6 m; Height: 7.3 m. Constructed in the years 595-605 CE]
Points for initial calculations
Points for variance calculations
Appendix 2: Further
Calculations
Appendix 3: Line of y=1.62
Appendix 4: CG20 General form
equation method and regression method
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Appendix 5: Spreadsheets for
Variance calculations
Note: Pictures are in order
followed in discussion.
For y= -0.08x² + 0.66x + 0.28
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For -0.05x² + 0.41x + 0.7795
For -0.075x² + 0.605x + 0.41
Appendix 6: Nine Models of All
Algebraic Methods
Note: Pictures are in order
followed in discussion.
Appendix 7:Eureqa Regressions
Note: Pictures are in order
followed in discussion. Third picture is the best model.
Appendix 8: Further results of
calculations finding leading coefficient values
Note: Points in middle give
more accurateleading coefficient values; Points on the left give smaller leading
coefficient values; Points on the right give larger leading coefficient values
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Table of Values Derived from Points
Factored Form |
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Leading Coefficient |
Third Point |
Position of Point |
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-0.078 |
E |
Left |
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-0.09 |
H1 |
Right |
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-0.0826 |
R |
Middle |
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-0.0839 |
J |
Left |
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-0.0823 |
W |
Middle |
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-0.0819 |
H |
Left |
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-0.0848 |
D1 |
Right |
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-0.103 |
J1 |
Right |
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-0.0865 |
F1 |
Right |
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-0.0818 |
T |
Middle |
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-0.0825 |
B1 |
Right |
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-0.0833 |
M |
Left |
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-0.0833 |
K |
Left |
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-0.0833 |
I |
Left |
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-0.0786 |
G |
Left |
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-0.0568 |
F |
Left |
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Table of Values Derived from Points
Vertex Form |
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Leading Coefficient |
Second Point |
Position of Point |
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-0.08 |
C |
Left |
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-0.08 |
D |
Right |
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-0.0726 |
M |
Left |
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-0.0796 |
H |
Left |
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-0.0778 |
F1 |
Right |
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-0.078125 |
W |
Right |
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-0.0787 |
I |
Left |
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Note: Points further from the
vertex are more likely to have accurateleading coefficient values
Appendix 9: Comparison of
Regression and Calculated Functions
Dotted red line: Eureqa
regression function; Blue line: algebraically derived function
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