Mathematical Investigation: Print design investigation

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Mathematical Investigation: Print design investigation

For the following Mathematical Investigation task, students use Mathematics creatively to design an aesthetically pleasing pattern that can be used in real life by textile industry to produce fabric prints.

This activity is not covered in class creating a greater challenge for students, as one must utilize appropriately, concepts covered in calculus, including differentiation and integration techniques of various functions studied in our course, then apply their independent learning to completing the task.

Note: Use of Desmos, GeoGebra or any other maths software programs, including the graphic calculator,  is indicated as an exploration tool in order to develop deeper and more meaningful connections between each question, sections of the investigation and consequently the entire Mathematical Investigation.

Mathematical Investigation Task must include:

•    An introduction that demonstrates an understanding of the features of the problem or situation being investigated

•    Appropriate presentation of information used, calculations and results

•    Analysis and Conclusion

For this investigation there will be minimal teacher direction (initial guideline questions are provided with no

background and students are expected to research relevant material)

Students must extend the investigation in an open-ended context.

Stage 2 Mathematical Methods Subject Outline, 2021, SACE board of SA,https://www.sace.sa.edu.au/ “Students complete a report for the mathematical investigation.

In the report, students interpret and justify results, and draw conclusions. They give appropriate explanations and arguments. The mathematical investigation provides an opportunity to develop, test, and prove conjectures.

The investigation report, excluding bibliography and appendices if used, must be a maximum of 15 A4 pages if written minimum font size 10, or the equivalent in multimodal form.

For this assessment type, students provide evidence of their learning in relation to the following assessment design criteria:

•  concepts and techniques

•  reasoning and communication.”

This investigation exploreshow complex functions, differentiation and integration can be used to create a graphic design for a fabric.

Introductory Task:

Part 1

First pattern has been designed for you. You are required to find the functions and the circle used for this design, based on the description below.

The following functions/relations have been used to design this pattern:

-      one logarithmic function and its inverse

-      two cubic functions

-      one circle

Recommended function types: y1  = f(x) = a ln(bx + c)

y2  =  f −1 (x)

g(x) = ax [(x − ℎ)2  + k] ℎ(x) = bx [(x − ℎ)2  + k]

Mathematical rules used for this design:

(i)   the four functions start at the origin.

(ii)  logarithmic function and its exponential inverse end at (4,4) (iii) logarithmic function has  a  = 0.5

(iv) the point of inflection of the top cubic (green function) is located atx  =  1.6

(v)  the maximum value of the top cubic (green function) aligns vertically with the stationary point of inflection of the bottom cubic (blue function)

(vi) the minimum point of the top cubic (green function) is located at (2, 2)

(vii)  the area of the circle is one third of the area between the two cubic functions and its centre aligns with the minimum point of the top cubic.

(viii) the two cubic functions always meet on the logarithmic function.

Based on the above information find:

1.    The four functions and the equation of the circle.

2.    Calculate the areas between each curve combination (use of technology indicated).

Refer to the cubic functions found from above rules. Assume that theirrespective found hand k values do not change.