PSMT Term 1 Topic 2 Polynomials
Table of Contents
1.0 Intro
o 1.1 Aim
o 1.2 Method
2.0 Considerations
o 2.1 Assumptions
o 2.2 Observations
3.0 Analyses/ Solve
o 3.1 Quadratic model (algebraically, using tech, discuss)
o 3.2 Quartic Model (algebraically, using tech, discuss)
o 3.3 Combination Model (algebraically, using tech, discuss)
4.0 Evaluation
o 4.1 Strengths
o 4.2 Limitations
5.0 Conclusion
6.0 References
7.0 Appendices
1
1.0 Intro
1.1 Aim
o The aim of this investigation is to research and construct a suitable function that replicates the curved
section of the Mackinac Bridge in Michigan, The United States. The structure of the bridge’s curved
section will be modelled by a set of appropriate calculated and computed polynomial functions (𝑦 =
𝑎(𝑥 − 𝑏)2 + 𝑐, 𝑦 = 𝑎(𝑥 − 𝑏)4 + 𝑐) as well as the sum of even degree polynomials
(𝑦 = 𝑎0 + 𝑏2 (𝑥 – 𝑎2 )2 + 𝑏4 (𝑥 – 𝑎4 )4 + ⋯ + 𝑏2𝑛 (𝑥 – 𝑎2𝑛 )2𝑛 + ⋯ ). Algebraic procedures
as well as technological tools such as Desmos and the TI-84 Plus CE will be utilised in order to
evaluate the viability of models including the R2 values and analysis of residuals.
1.2
Method
During the formulation of the series of polynomials which display the curve of The Mackinac Bridge, a set of
procedures were followed.
o 1.2.1 Firstly, the blueprint of the Mackinac Bridge in The United States had been inserted along the virtual
grid on Desmos (Figure 1).
o 1.2.2 Next, the cartesian coordinates of each point on the curve had also been collected and recorded into
a table of values, as displayed in Figure 1 ‘Table of Plotted Points and Correlation on the Graph’.
o 1.2.3 Following this, through inputting the values of ‘x1’ and ‘y1’ (Figure 1) and calculating a unique ‘a
value’ (Figure 9), a quadratic (𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑐), quartic (𝑦 = 𝑎(𝑥 − 𝑏)4 + 𝑐) and combination (𝑦 =
𝑎(𝑥 − 𝑏)2 + 𝑏(𝑥 − 𝑏)4 + 𝑐) model were developed in attempt to model the curve of the plotted points on
the bridge. These models are presented in Figures 2, 4 and 6.
o 1.2.4 Subsequently, these same equations were remodelled in the development of a technical model,
created automatically by the graphing software ‘Desmos’ as illustrated in Figures 3, 5 and 7.
o 1.2.5 Then, the residuals of each created formula had been recorded into a table of values.
o 1.2.6 Lastly, a separate dot-plot graph had been created on the software ‘Desmos’ to efficiently compare,
contrast and evaluate these residuals (Figure 8).
2.0
2.1
Considerations
Observations
A range of observations had been detected during the progression of the generated polynomial function of the
bridge. It was observed that during the construction of each formulated polynomial function; the calculator which
had been used to determine each value had been limited to a certain number of decimal places, hence reducing the
accuracy of the results. Additionally, the initial Cartesian coordinates that had been marked and recorded into the
table of values in Figure 1, may have deviated from the correct points on the graph as a result of the quality of the
image and the orientation due to the zoom function on the Desmos program. Consequently, this may have
influenced some evident deviations in results. Furthermore, it was observed that the curve of the bridge had been
positive meaning that the calculated ‘a’ values of each model would be positive. As displayed in Image 3 when
using the zoom function in Desmos it was observed that the qualty of the image had decreased, creating
uncertainty in determining the expected coordinate of the curve. The translation of the bridge had been 4.99 units
to the left and 0.093 units up, producing a turning point (4.99, 0.093). It had also been observed that the domain of
the investigated curve had been 2.44 ≤ 𝑥 ≤ 7.54 while the range of the curve had been 0.093 ≤ 𝑦 ≤ 0.56. A
relevant observation that had been developed throughout the progression of the investigation had been that various
other bridges had represented an irregularly oriented shape or asymmetry.
2.2
Assumptions
Through the completion of the investigation a number of assumptions were considered, as a result of observed
considerations, in order to ensure efficiency and accuracy throughout the process. An assumption was made that
the functions generated by the calculator of limited decimal places, had been accurate in order to be able to draw
relevant conclusions. Also, it was assumed throughout the procedure that the plotted points of the original graph
displayed a correctly oriented image. As a result of an observation of other bridges being minorly irregular in
shape or asymmetry, it was also assumed that the bridge had been symmetrical in construction to maintain the
assumed relationship of the structure as an even degree polynomial. Moreover, in plotting the initial cartesian
coordinates of the graph as displayed in Figure 1, it had been observed that while using the zoom function on the
desmos program, the image had become blurred (Image 3). Due to this limitation, it had been assumed that the
plotted point had presumed an accurate location on the curve of the image.
2
3.0
Analysis/Solve
Figure 1: Table of Plotted Points and Imaged Correlation on the Graph:
𝐷𝑜𝑚𝑎𝑖𝑛: 2.44 ≤ 𝑥 ≤ 7.54
𝑅𝑎𝑛𝑔𝑒: 0.093 ≤ 𝑦 ≤ 0.56
3
3.1
o
Quadratic Model (algebraically, using tech, discuss)
Figure 2 and 3 represent the calculated and computed models of the quadratic function
(𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑐) which replicate the curve of the Mackinac Bridge, Michigan United States. The
467
formulated quadratic function is modelled by the equation ‘𝑦1 ~ 6500 (𝑥1 − 4.99)2 + 0.093′ as calculated
in Figure 9 (refer to appendix). To create this equation; the simultaneous solutions of 3 equations, with the
inputted values of 3 specific coordinates of the curve in Figure 1, was determined. It was then converted
from general form (y=𝑎𝑥 2 + 𝑏𝑥 + 𝑐) to turning point form 𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑐 where b and c reveals the
horizontal and vertical translations. The 𝑟 2 value of this function relative to the coordinates of the bridge
was represented by the value ‘0.9986′ which discloses a close correlation between the regression lines
and the Cartesian coordinates. Additionally, the residuals of this function is displayed in Figure 8 as the
orange dots which are clearly closely related or identical to the line ′𝑦 = 0′ meaning that there is a
minimum deviation between each coordinate displayed in Figure 1 and the line displayed in Figure 2.
Additionally, a technical model had also been constructed by the Desmos software, modelled as
‘′𝑦1 ~𝑎(𝑥! − 𝑏)2 + 𝑐. The 𝑟 2 value of this function relative to the coordinates of the bridge was
represented by the value ‘0.9996′ this reveals a close correlation between the regression lines and the
Cartesian coordinates. Additionally, the residuals of this function are also displayed in Figure 8 as the blue
dots which are definite in relation to the line ′𝑦 = 0′ meaning that there is also a minimum deviation
between each coordinate displayed in Figure 1 and the line displayed in Figure 3. Although, both of the
models displayed in Figures 2 and 3 are extremely accurate, the technical model formed by Desmos
clearly displays a slightly greater 𝑟 2 value being greater by 0.001.
Figure 2: The Formulated Quadratic Function
4
Figure 3: The Desmos Formulated Quadratic Function
3.2
Quartic Model (algebraically, using tech, discuss)
Figure 4 and 5 represent the calculated and computed models of the quartic function (𝑦 = 𝑎(𝑥 − 𝑏)4 + 𝑐)
which replicate the curve of the Mackinac Bridge, Michigan United States. The formulated quartic
function is modelled by the equation ‘𝑦1 ~0.0110441757(𝑥1 − 4.99)4 + 0.093′ as calculated in Figure 9
(refer to appendix). To create this equation; the known variables of the curve which were the vertical and
horizontal translations, as well as the points on the curve were substituted into the equation
(𝑦1 = 𝑏(𝑥1 − 𝑐)4 + 𝑑) as well as the point (7.54, 0.56) in which the unique ‘b’ value was to be
determined by solution. The 𝑟 2 value of this function relative to the coordinates of the bridge was
represented by the value ‘0.7621′ which discloses a moderate correlation between the regression line and
the Cartesian coordinates. Additionally, the residuals of this function were displayed in Figure 8 as the
black-filled dots (e5), slightly visible behind the red dots for ‘Formulated Combination (e6)’ as illustrated
in Image 1. The residual plots of this Quartic model line diverge from the line ′𝑦 = 0′ between the points
which had been initially inputted into the function, such as the coordinates of the greatest values of x
‘(7.54, 0.56)’ and the turning point ‘(4.99, 0.093)’ as displayed in Image 2. This confirms that there is a
relative deviation between each coordinate displayed in Figure 1 and the line displayed in Figure 4.
Additionally, a technical model had also been constructed by the Desmos software, modelled as
‘′𝑦1 ~𝑎(𝑥! − 𝑏)2 + 𝑐’. The 𝑟 2 value of this function relative to the coordinates of the bridge was
represented by the value ‘0.9278′ this reveals a relatively close correlation between the regression lines
and the Cartesian coordinates. Additionally, the residuals of this function are also displayed in Figure 8 as
the green dots which are definite in relation to the line ′𝑦 = 0′ meaning that there was a minimum
deviation between each coordinate displayed in Figure 1 and the line displayed in Figure 3. Furthermore,
it can be inferred that Figure 5 of the ‘Desmos Formulated Quartic’ had represented a more accurate curve
in relation to the coordinates of the bridges curve in Figure 1, than Figure 4 of the ‘Formulated Quartic
Function’, having a greater 𝑟 2 value by ‘0.1657’.
5
Image 1
Image 2:
7.54, 0.56
4.99, 0.093
Figure 4: The Formulated Quartic Function
6
No
inputted
coordinates
Figure 5: The Desmos Formulated Quartic Function
3.3
Combination Model (algebraically, using tech, discuss)
Figure 6 and 7 represent the calculated and computed models of the quadratic and quartic combination
function (𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑑(𝑥 − 𝑏)4 + 𝑐) which replicate the curve of the Mackinac Bridge, Michigan
United States. The formulated combination function was modelled by the equation
‘𝑦1 ~0.0000027623(𝑥1 − 4.99)2 + 0.0110490048(𝑥1 − 4.99)4 + 0.093′ as calculated in Figure 9
(refer to appendix) and displayed in Figure 6. To create this equation; 2 initial equations were constructed
in the form (𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑑(𝑥 − 𝑏)4 + 𝑐) in which the coordinates closest to the turning point
(4.94, 0.093) and of the greatest y value (7.54, 0.56) were substituted for the values of x and y, in order to
ensure the regression line’s intersection of the end and middle of the curve. The vertical and horizontal
translations were kept the same throughout the simultaneous solution of these equations in determining
the refined a and d values. The 𝑟 2 value of this function relative to the coordinates of the bridge was
represented by the value ‘0.7624′ which discloses a moderately weak correlation between the regression
lines and the Cartesian coordinates. Additionally, the residuals of this function are displayed in Figure 8 as
the red dots which clearly deviate from the line ‘𝑦 = 0′ related or identical to the line ′𝑦 = 0′ indicating a
great deviation between each coordinate displayed in Figure 1 and the line displayed in Figure 6.
Additionally, a technical model had also been constructed by the Desmos software, modelled as ′𝑦1 =
𝑎(𝑥1 − ℎ)2 + 𝑏(𝑥1 − ℎ)4 + 𝑘′ in which ‘h’ and ‘k’ are the horizontal and vertical translations (Figure 7).
The 𝑟 2 value of this function relative to the coordinates of the bridge was represented by the value
‘0.9999′ this reveals a strong correlation between the regression lines and the Cartesian coordinates.
Additionally, the residuals of this function are also displayed in Figure 8 as the purple dots which have an
almost direct relationship to the line ′𝑦 = 0′ revealing a very minimum level of deviation between each
coordinate displayed in Figure 1 and the regression line displayed in Figure 3. Moreover, Figure 7 ‘The
Desmos Formulated Combination of the Quadratic and Quartic Function’ clearly represents a greater
correlation between the regression line (Figure 7) and the coordinates of the bridge (Figure 1) than Figure
6 ‘The Formulated Combination of the Quadratic and Quartic Function’. The 𝑟 2 value had been greater by
‘0.2375’.
7
Figure 6: The Formulated Combination of Quadratic and Quartic Function
Figure 7: The Desmos Formulated Combination of Quadratic and Quartic
Function
8
Figure 8: Visual Comparison of Residuals of Each Constructed Function:
𝑦=0
Desmos Formulated Quadratic (e1)
Formulated Quadratic (e4)
Desmos Formulated Combination of Quadratic and Quartic (e3)
Formulated Combination of Quadratic and Quartic (e6)
Desmos Formulated Quartic (e2)
Formulated Quartic (e5)
4.0
4.1
Evaluation
Strengths
Throughout the process of investigating and determining a suitable function that replicates the curved section of
the Mackinac Bridge, a range of strengths had been identified which influenced the success, viability and
reasonableness of the models displayed through Figures 2 to 7. The large range of increments of the initial curve
had influenced the ability to correctly evaluate the viability of the model by impacting the r squared value,
increasing the reasonableness consequently. In calculating the formula for each model in Figure 9 (refer to
appendix), the values close to the turning point, and largest values of y were frequently used in simultaneous
solutions and in determining unknowns, this had increased the possibility of creating a successful and reasonable
model as it had inputted definite and important points of the curve, which resulted in receiving the most efficient
models in relation to the coordinates in Figure 1. It had essentially imposed the basic shape of the curve from the
initial stages of the procedure. The effect of this procedure is relevant in image 2 where residual values had only
deviated between the 2 initially inputted points. The use of residual plots and the ‘𝑦 = 0′ line (Image 2) in
evaluating the accuracy of the models in relation to the coordinates in Figure 1, had clearly strengthened the
capability to correctly and effectively test the reasonableness and viability of the models. Additionally, as a result
of some assumptions made during the procedure of the investigation, the values determined in calculation (Figure
9 refer to appendix) had remained consistent and efficient. Some assumptions included assuming the accuracy of
values which required many decimal places, the orientation of the image as well as the physical bridge. Also, the
use of technology such as the software ‘Desmos’ and the ‘TI-84 Plus CE’ Graphing Calculator had made the
procedure very efficient and simple despite the complication of the investigation, through its display of virtual
geometrics and quick calculation.
9
4.2
Limitations
Accordingly, throughout the completion of the investigation through Figures 2 to 7, a set of limitations had also
been recognised which had manipulated the success, viability and reasonableness of the achieved results.
Although, the assumptions such as ‘assuming the accuracy of values which required many decimal places, the
orientation of the image as well as the
physical bridge’ had provided consistency and efficiency in the
results, this may impact on the reasonableness of the models and ultimately the ability of applying these methods
to real world situations as a result of the inaccuracy in the results. As visible in Image 3, while using the zoom
function it had been noticed that the curves of the bridge had become blurred. Consequently, slight deviations in
the data could be due to the quality of the inserted blueprint on the desmos grid as a result of the ambiguous
location of the plot. In investigating the dimensions and gathering research of the Mackinac bridge, it was
assumed that the curve of the bridge had no irregularities or asymmetry in shape. In the case of irregularities or
asymmetry that the bridge may have in construction, the accuracy and reasonableness of the model may be
compromised. In addition to that, if the image of the bridge also had irregularities, the accuracy and
reasonableness of the models may be compromised.
Image 3:
5.0
Conclusion
The purpose of this investigation was to research and construct a suitable model that replicated the curved section
of the Mackinac Bridge in Michigan, The United States. To achieve this the structure of the bridge’s curved
section was modelled by a set of appropriate calculated and computed polynomial functions
(𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑐, 𝑦 = 𝑎(𝑥 − 𝑏)4 + 𝑐) as well as the sum of even degree polynomials
(𝑦 = 𝑎0 + 𝑏2 (𝑥 – 𝑎2 )2 + 𝑏4 (𝑥 – 𝑎4 )4 + ⋯ + 𝑏2𝑛 (𝑥 – 𝑎2𝑛 )2𝑛 + ⋯ ). Algebraic procedures as well as
technological tools such as Desmos and the TI-84 Plus CE had been utilised in the evaluation of the viability of
the models including the 𝑟 2 values and analysis of residuals. The Desmos constructed ‘Combination of Quartic
and Quadratic Function’ displayed in Figure 7 had clearly displayed the greatest correlation to the coordinates
listed in Figure 1, modelling a curve which had best fit in relation to the structure of the original bridge. The
function displayed in Figure 7 also revealed a reasonable solution in relation to the purpose of modelling the
Mackinac bridge through its accuracy and feasibility in the results despite minor limitations and exceeded domain
and range. Furthermore, the solution of the investigation could be further extended by introducing the use of an
extra function of 𝑥 6 in the combination of the models, to provide a result of greater accuracy and feasibility. In
addition, the investigation could be further improved by recording the values of the bridge in intervals of 0.01
rather than 0.1 units of ‘𝑦’ as displayed in Figure 1. This refinement could result in a greater level of accuracy in
creating a model for the Mackinac Bridge.
6.0
o
o
References
Graphing calculator. (n.d.). Desmos. https://www.desmos.com/calculator
Mackinac bridge (Big Mac / Mighty Mac) - HistoricBridges.org. (n.d.). Historic Bridges .org.
https://historicbridges.org/bridges/browser/?bridgebrowser=other/mackinac/
10
7.0
Appendices
Figure 9: The Working for Each Formulated Equation
3 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 ℎ𝑎𝑑 𝑏𝑒𝑒𝑛 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑎𝑠 3 𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑠 𝑤𝑒𝑟𝑒 𝑡𝑜 𝑏𝑒 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑.
𝟑 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔:
𝑇ℎ𝑒 3 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑡𝑜 𝑏𝑒 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑜𝑟𝑑𝑒𝑟 𝑡𝑜 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑡𝑜 𝑏𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡
𝑦 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 0.56, 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡. 𝑇ℎ𝑒𝑠𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑤𝑒𝑟𝑒 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑎𝑠 𝑎 𝑟𝑒𝑠𝑢𝑙𝑡
𝑜𝑓 𝑡ℎ𝑒𝑖𝑟 𝑤𝑖𝑑𝑒𝑙𝑦 𝑠𝑝𝑟𝑒𝑎𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒.
(𝟒. 𝟗𝟒, 𝟎. 𝟎𝟗𝟑) = 𝒆𝒔𝒕𝒊𝒎𝒂𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒖𝒓𝒏𝒊𝒏𝒈 𝒑𝒐𝒊𝒏𝒕
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
0.093 = (4.94)2 𝑎 + 4.94𝑏 + 𝑐
0.093 = 24.4036𝑎 + 4.94𝑏 + 𝑐
(𝟐. 𝟒𝟒, 𝟎. 𝟓𝟔) = 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆 𝟏 𝒐𝒇 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒚 𝒗𝒂𝒍𝒖𝒆
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
0.56 = (2.44)2 𝑎 + 2.44𝑏 + 𝑐
0.56 = 5.9536𝑎 + 2.44𝑏 + 𝑐
(𝟕. 𝟓𝟒, 𝟎. 𝟓𝟔) = 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆 𝟐 𝒐𝒇 𝒉𝒊𝒈𝒉𝒆𝒔𝒕 𝒚 𝒗𝒂𝒍𝒖𝒆
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
0.56 = (7.54)2 𝑎 + 7.54𝑏 + 𝑐
0.56 = 56.8516𝑎 + 7.54𝑏 + 𝑐
𝑇𝑒𝑐ℎ𝑛𝑜𝑙𝑜𝑔𝑦 𝑤𝑎𝑠 𝑢𝑡𝑖𝑙𝑖𝑠𝑒𝑑 𝑡𝑜 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎, 𝑏 𝑎𝑛𝑑 𝑐.
𝑎 = 467/6500
𝑏 = −0.7170246154
𝑐 = 1.8817968
𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝑼𝒏𝒓𝒐𝒖𝒏𝒅𝒆𝒅 𝑽𝒂𝒍𝒖𝒆𝒔
467 2
𝑦=
𝑥 − 0.7170246154𝑥 + 1.8817968
6500
𝑪𝒐𝒏𝒗𝒆𝒓𝒕 𝒕𝒐 𝑻𝒖𝒓𝒏𝒊𝒏𝒈 𝑷𝒐𝒊𝒏𝒕 𝑭𝒐𝒓𝒎
𝑳𝒆𝒕 𝒚(𝒙) = 𝒚
(𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑢𝑡)
(𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒)
(𝑈𝑠𝑒 𝑡ℎ𝑒 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑓𝑜𝑟𝑚𝑢𝑙𝑎)
(𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦)
(𝑒𝑥𝑝𝑎𝑛𝑑)
[commas (,) = decimal points(.)] {Program error}
𝑻𝒖𝒓𝒏𝒊𝒏𝒈 𝑷𝒐𝒊𝒏𝒕 𝑭𝒐𝒓𝒎 𝒇𝒐𝒓 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄
467
(𝑥 − 4.99)2 + 0.093
𝑦=
6500
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑡ℎ𝑒 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑖𝑠 (4.99, 0.093)
𝑻𝒖𝒓𝒏𝒊𝒏𝒈 𝑷𝒐𝒊𝒏𝒕 𝑭𝒐𝒓𝒎 𝒇𝒐𝒓 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄
𝑦 = 𝑏(𝑥 − 4.99)4 + 0.093
𝑆𝑢𝑏 𝑖𝑛 (7.54, 0.56)
0.56 = 𝑏(7.54 − 4.99)4 +0.093
0.467
𝑏=
(7.54 − 4.99)4
𝑏 = 0.011044757
𝑦 = 0.011044757(𝑥 − 4.99)4 + 0.093
11
𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒕𝒉𝒆 𝑪𝒐𝒎𝒃𝒊𝒏𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒂𝒏𝒅 𝑸𝒖𝒂𝒓𝒕𝒊𝒄 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏
𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑑(𝑥 − 𝑏)4 + 𝑐
467
(𝑥 − 4.99)2 ) + 0.093 + 𝐷(0.011044757(𝑥 − 4.99)4 ) + 0.093
𝑦 = 𝐴(
6500
467
(𝑥 − 4.99)2 ) + 𝐷(0.011044757(𝑥 − 4.99)4 ) + 0.093
𝑦 = 𝐴(
6500
𝑦 = 𝑎(𝑥 − 4.99)2 + 𝑑(𝑥 − 4.99)4 + 0.093
𝑠𝑢𝑏 𝑖𝑛 (2.44, 0.56)(𝑇ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑢𝑝𝑝𝑒𝑟𝑚𝑜𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑦)
0.56 = 𝑎(2.44 − 4.99)2 + 𝑑(2.44 − 4.99)4 + 0.093
0.467 − 42.28250625𝑑
= 𝑎
6.5025
2
𝑦 = 𝑎(𝑥 − 4.99) + 𝑑(𝑥 − 4.99)4 + 0.093
𝑆𝑢𝑏 𝑖𝑛 (4.94, 0.093) (𝐶𝑙𝑜𝑠𝑒𝑠𝑡 𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡)
0.093 = 𝑎(4.94 − 4.99)2 + 𝑑(4.94 − 4.99)4 + 0.093
0.467 − 42.28250625𝑑
Equation 1
= 𝑎
6.5025
Equation 2
0.093 = 𝑎(4.94 − 4.99)2 + 𝑑(4.94 − 4.99)4 + 0.093
0.467 − 42.28250625𝑑
0.093 = (
)(4.94 − 4.99)2 + 𝑑(4.94 − 4.99)4 + 0.093
6.5025
0.467 − 42.28250625𝑑
0=(
)(4.94 − 4.99)2 + 𝑑(4.94 − 4.99)4
6.5025
0.467 − 42.28250625𝑑
0=(
) (0.0025) + 0.00000625𝑑
6.5025
0.0011675 − 0.105706266𝑑 + 0.000040641𝑑
0=
6.5025
0 = 0.0011675 − 0.105706266𝑑 + 0.000040641𝑑
0 = 0.0011675 − 0.105665625𝑑
0.105665625𝑑 = 0.0011675
𝑑 = 0.0110490048
0.467 − 42.28250625𝑑
= 𝑎
6.5025
0.467 − 42.28250625(0.0110490048)
= 𝑎
6.5025
𝑎 = −0.000027624
𝑇𝑒𝑐ℎ𝑛𝑜𝑙𝑜𝑔𝑦 𝑤𝑎𝑠 𝑎𝑙𝑠𝑜 𝑢𝑡𝑖𝑙𝑖𝑠𝑒𝑑 𝑡𝑜 𝑠𝑖𝑚𝑢𝑙𝑡𝑎𝑛𝑒𝑜𝑢𝑠𝑙𝑦 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑎 𝑎𝑛𝑑 𝑑
𝑎 = −0.000027624
0.467
𝑑=
= 0.0110490048
42.26625
𝑦 = −0.000027624(𝑥 − 4.99)2 + 0.0110490048(𝑥 − 4.99)4 + 0.093
12
Solve for d