Maths IA · Algebra and functions · topic 2 of 97
Drawing a shape with functions: how close can you get? · IB Maths IA idea: guide, sample graph and step-by-step plan
a picture made of mathsA picture is just a set of curves, if you are patient enough to write them down. You're going to rebuild your own drawing from functions, and then find out how close you really got.
- ✓ you like drawing, design or building things in Desmos
- ✓ you want to see functions turn into a picture
- ✓ you enjoy measuring how good your own work is
The big idea
Place axes over a drawing you made, read points off its outline and write each piece as a function with a limited domain: a line, a quadratic through three points, a sine curve stretched to fit. Three points give a 3 × 3 system, and transformations let one sine or one parabola be moved, stretched and flipped into place. Then comes the part many drawings skip. Measure the gap between your picture and the original, piece by piece. A root-mean-square error turns a feeling that it looks right into a number. Ask how many pieces are enough: where do extra pieces stop helping, and what do they cost? You can also find areas by integration.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Read three points off the outline and solve for the quadratic that passes through them.
→ A 3 × 3 system you solve by hand and then check with technology. Say why three points are exactly enough.Fit a sine curve to a wavy edge using amplitude, period, shift and vertical shift.
→ Each parameter is a transformation you can read off the image and justify.Compare each piece with points read from the original and find the root-mean-square error.
→ A number to judge success. Compare the pieces and say which is the weakest link.Redraw one edge with 1, 2, 3 and more pieces and plot the error against the number of pieces.
→ A graph with a reason: where does the error stop falling, and what is limited by how well you can read the image?Find the area of a region of your drawing by integrating the pieces and compare it with the area from a pixel count.
→ Definite integrals between curves, with a check from a different method.Rotate one piece through an angle and write its new equation.
→ A harder transformation: rotated curves are not always functions. Say what breaks and how you handled it.Your Maths IA research question: run the check
which image? which error?Write your own question first. Then make sure it ticks every box:
- Names the image, which you drew or whose creator allows your use, and how you will put axes over it.
- Says which function types you will use and why each suits its part of the picture.
- States the error measure and the points it will be calculated from.
- Says how you will compare designs: the number of pieces, the error and the effort.
- Is narrow enough for 12 to 20 pages: one drawing, fully analysed, rather than many drawings briefly.
The set-up
An outline with axes over it, points read off, three pieces and the gaps between them. Draw your own outline and mark where each piece starts and stops.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your drawing, why you chose it and what you predict about how many pieces it needs.
How you placed axes and read points, the scale and how precise your readings are.
Each piece as a function with a domain: the working for a line, a quadratic from a system and a sine curve from transformations.
The error measure defined, a table of errors by piece and one graph.
The error against the number of pieces for one edge, and where it stops falling.
A region of the drawing integrated piece by piece and checked another way.
Back to your aim: how close you got and how few pieces were enough, with numbers.
Reading error from the image, pieces that join with a corner, what a designer would use instead. Say which limit matters most.
Tools and sources used, and where the image came from.
What your Maths IA graph could look like
mark the floor!A made-up data set to show the form of a strong graph, not the answer: the error of a drawing against the number of pieces used. Yours will look different. example only
- Error against pieces shows where more work stops paying. Plot it for at least two ways of drawing.
- Axis labels say what the error is measured in, even if the unit is only the unit of your axes.
- Mark the floor set by how precisely you read the image.
- Plot each trial as a dot. A model line through them is a second step.
- Another graph to consider: the error of each piece along the edge, to find the weakest piece.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictLook at your drawing and guess which edges are lines, which are curves and which will be hardest.
- Draw your own imageUse a simple original shape you drew. Do not trace a copyrighted character, logo or artwork.
- Set the axesLay axes over the image, choose a scale and read at least a dozen points off the outline. Say how precise your readings are.
- Trial run (1 lesson)Build one edge with one or two pieces in Desmos. Check it against the image before drawing the rest.
- Fit the piecesFit each piece by hand: a line by two points, a quadratic by a 3 × 3 system, a sine curve by transformations. State each domain.
- Measure the errorDefine your error measure, then calculate it for each piece and for the whole edge.
- Vary the piecesRedraw one edge with more and fewer pieces and plot the error against the number used.
- Go further, with a reasonFind an area by integration and check it another way, or rotate a piece. HL: justify each step and compare two methods.
- Reflect as you goAfter every result, write one line on what it means for how good your drawing is and what you assumed. These lines become your evaluation.
- Cut, check, then sendCheck every function has a stated domain, every graph has labelled axes and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Quadratics from three points – Solve the system for , , so that passes through three points read from the outline.
- Transformations – Fit with stretches and shifts read from the image, and justify each parameter.
- Piecewise functions – Write each piece with its domain, check the pieces meet at the joins and decide whether a corner is acceptable.
- Error measure – Calculate for each piece and for the whole edge.
- Definite integrals – Find the area between an upper and a lower curve by integrating, and compare with the area from counting squares.
Key equations
Syllabus links
- 2.1Straight lines
- 2.6Quadratic functions
- 2.11Transformations of graphs
- 3.7Circular functions and their graphs
- 5.11Definite integrals and areas between curves
- 1.16Systems of up to three linear equationsHL
- 2.1Straight lines
- 2.5Linear, quadratic, exponential, cubic and sinusoidal models
- 2.8Transformations of graphsHL
- 2.9Logarithmic, logistic, piecewise and half-life modelsHL
- 1.8Solving systems and polynomial equations with technology
- 5.8Trapezoidal rule
Read up first
Data and sources to start from
Points you read from your own drawing, using Desmos or GeoGebra with the image placed behind the axes.
References to cite
- Desmos Studio. (2026). Desmos graphing calculator. https://www.desmos.com/calculator
- Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
give every piece a domain- Tracing a copyrighted character or logo. Use a shape you drew or one that is free to use.
- Writing a piece's equation with no domain, so the pieces overlap.
- Judging success by eye with no error measure.
- Adding pieces until the picture looks right without asking what each piece costs.
- Ignoring the reading error: you cannot be more accurate than your points.
- Showing screenshots of Desmos but no working for how each function was found.
More Maths IA ideas like this
Fonts and logos: how do Bézier curves work?●●○
Bridge arch: is a cubic or quartic a real shape?●○○
Area under a curve: trapezium or Simpson?●○○
Vase or bowl: how close can integration get?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.