代做Polynomial Designs 2025代做留学生SQL 程序
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Investigation: Polynomial Designs
Due: Friday Week 5 (30th May) at the start of the lesson via Canvas and printed.
PURPOSE
To demonstrate your ability to:
• Use mathematical modelling and problem-solving strategies as well as your knowledge, skills and understanding of mathematical ideas and processes.
• Effectively and appropriately communicate relevant mathematical information with your solutions.
DESCRIPTION OF ASSESSMENT
Students are to investigate sequences of numbers that can be represented by polynomials and develop conjectures to find these polynomial functions. These skills will then be applied to replicate an image or logo made up of various polynomial functions. All solutions are to be supported with calculations and appropriate representation and notation.
ASSESSMENT CONDITIONS
Use of technology is required. The report should include
• an introduction that demonstrates your understanding of the problem to be explored.
• the method required to find a solution, in terms of the mathematical model or strategy used
• the application of the mathematical model or strategy, including:
- relevant data and/or information
- mathematical calculations and results, using appropriate representations
- the analysis and interpretation of results, including consideration of the reasonableness and limitations of the results
• the results and conclusions in the context of the problem.
• a bibliography and appendices, if required.
Each investigation report, excluding bibliography and appendices, must be a maximum of eight A4 pages. The maximum page limit is for single-sided A4 pages with minimum font size 10.
Appendices are used only to support the report, and do not form. part of the assessment decision.
For this assessment type, students provide evidence of their learning in relation to the following assessment design criteria:
• concepts and techniques
• reasoning and communication.
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Concepts and Techniques |
Reasoning and Communication |
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A |
Comprehensive knowledge and understanding of concepts and relationships. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. |
Comprehensive interpretation of mathematical results in the context of the problem. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Effective development and testing of valid conjectures. |
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B |
Some depth of knowledge and understanding of concepts and relationships. Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Some development and successful application of mathematical models to find mostly accurate solutions. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. |
Mostly appropriate interpretation of mathematical results in the context of the problem. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly effective development and testing of valid conjectures. |
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C |
Generally competent knowledge and understanding of concepts and relationships. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts. Successful application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. |
Generally appropriate interpretation of mathematical results in the context of the problem. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Development and testing of generally valid conjectures. |
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D |
Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems. |
Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted development or testing of a reasonable conjecture. |
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E |
Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems. |
Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, their reasonableness or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to develop or test a conjecture. |
INTRODUCTION:
The equation of a quadratic function can be found using the x-intercepts and one other point on the quadratic or using the coordinates of the vertex and one other point. These methods are limited as this information may not be known. The aim of this investigation is to develop another method for finding the equation of a quadratic function from points without relying on the x-intercepts or the vertex and then extend this method to finding the equation of other polynomials from points. These findings will then be applied to recreating a logo or image made up of polynomial functions.
PART A: Finding Equations of given Quadratic sequences
Consider the quadratic: y = 2xz + 3x 一 7 and a table of some of its points.
Consider adding two further rows to the table:
Δ1 which gives the differences between successive y values and
Δ2 which gives the differences between successive Δ1 values.
The difference table below illustrates this.
A sequence of numbers with a constant second difference is a feature of a quadratic pattern.
Complete a difference table, as shown above, to show that the following sequences of y values have a second difference which is constant.
Using technology (see instructions on page 5) to verify that the y values in the sequences above follow a quadratic pattern. State the quadratic function that generate each of these values and use them to find three other terms in each of the sequences.
PART B: Finding patterns connecting the coefficients of all Quadratic Sequences
Consider the general case of a quadratic sequence with equation:
T(x) = ax2 + bx + c
Find the terms T(0), T(1), T(2), T(3), T(4) and T(5)
Set up difference table for T(x) in terms of a, b and c.
Use this table to make a conjecture in terms of a, b and/or c for the:
• y 一 intercept of T(x).
• second differences.
• nth term of the first differences.
Test the conjectures by using them to verify the equations of the quadratic sequences found in Part A.
PART C: Finding Equations of all Cubic Sequences
Extend the investigation to form. conjectures to find the equation connecting numbers in all cubic sequences.
Find an example of a cubic sequence and use it to verify these conjectures.
PART D: Using polynomials to recreate an image or logo of your own choice
Find a logo/image that already exists. Provide a reference of the image.
https://au.pinterest.com/logolearn/logo-design-sketches/might provide some inspiration.
The logo/image must be able to be recreated using polynomials. Transfer your image onto DESMOS. Position the image and select suitable points on each section of the design.
Investigate possible models that best fit the data on each section of the design.
The design must include sections that can be modelled by:
• at least 1 quadratic function that can be found using factorised form.
• at least 1 quadratic function that can be found using vertex form.
• at least 1 quadratic function that can be found using your findings from PART B.
• at least 1 cubic function that can be found using your findings from PART C.
Other polynomials can be included such as lines and quartics. Circles can be included.
Clearly explain and show the development of each function algebraically.
State all equations and their relevant domains. Graph the completed image.
Analyse how well your chosen models fit the data by comparing and appraising the reasonableness of each model visually and using technology.
CONCLUSION:
Summarise the findings. Discuss the reasonableness and limitations of the findings.
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Graphics Calculator Instructions Use STATISTICS mode • Type x data in LIST 1 and y data into LIST 2 • Select GRAPH(F1) • Select SET (F6) Graph Type: Scatter 1Var X List: List 1 1Var Y List: List 2 Frequency: 1 • EXE • Select GRAPH 1 (F1) • Select CALC (F1) • Select X2 (F4) • The coefficients of the quadratic, ( a, b, and C) are shown on the screen. |