IB Maths IA idea · Art, music and design

Drawing a shape with functions: how close can you get?

A 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.

Art, music and design
SL ideal
HL ideal
Maths AA Maths AI Your own data
Syllabus: AA 2.1, AA 2.6, AA 2.11, AA 3.7, AA 5.11, AA 1.16

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

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.

The maths you will use

Quadratics from three points. Solve the $3 \times 3$ system for $a$ , $b$ , $c$ so that $y = ax^2 + bx + c$ passes through three points read from the outline.
Transformations. Fit $y = a\sin(b(x - c)) + d$ 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 $\mathrm{RMSE} = \sqrt{\frac{1}{n}\sum (y_i - f(x_i))^2}$ 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

\mathrm{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_i - f(x_i)\right)^2}

A = \int_{a}^{b}\left(f_{\text{upper}}(x) - f_{\text{lower}}(x)\right)dx

The maths in action

Left: a fish with a lens-shaped body, a sine-curve fin, a notched tail of straight lines and a circular eye, each labelled by its function type. Right: two falling error curves on a log scale, the quadratic one levelling off at a dashed noise line sooner than the straight-line one. — (a) An original fish built from labelled function pieces; integration gives the body area 22.9 and the tail area 5.5. (b) Error against the number of pieces for an edge traced from noisy points: quadratic pieces reach the noise level at about 7 pieces, lines need more.

Example research question

To adapt, never to copy

How closely can lines, quadratics and sine curves with restricted domains reproduce the outline of a shape I drew, measured by the root-mean-square error against points read from it, and how many pieces does each edge need?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

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.

Which course it fits

AA SL ideal
AA HL ideal
AI SL ideal
AI HL ideal

Syllabus links in Maths AA:

AA 2.1 Straight lines
AA 2.6 Quadratic functions
AA 2.11 Transformations of graphs
AA 3.7 Circular functions and their graphs
AA 5.11 Definite integrals and areas between curves
AA 1.16 Systems of up to three linear equations (HL only)

Steps

Read and predict. Look at your drawing and guess which edges are lines, which are curves and which will be hardest.
Draw your own image. Use a simple original shape you drew. Do not trace a copyrighted character, logo or artwork.
Set the axes. Lay 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 pieces. Fit 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 error. Define your error measure, then calculate it for each piece and for the whole edge.
Vary the pieces. Redraw one edge with more and fewer pieces and plot the error against the number used.
Go further, with a reason. Find an area by integration and check it another way, or rotate a piece. HL: justify each step and compare two methods.
Reflect as you go. After 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 send. Check every function has a stated domain, every graph has labelled axes and the conclusion answers your aim. Then send your teacher one draft.

Make it yours

Fit a piece through three points. Read three points off the outline and solve for the quadratic that passes through them.
Move a sine curve into place. Fit a sine curve to a wavy edge using amplitude, period, shift and vertical shift.
Measure your error. Compare each piece with points read from the original and find the root-mean-square error.
Pieces against accuracy. Redraw one edge with 1, 2, 3 and more pieces and plot the error against the number of pieces.
Find an area. Find the area of a region of your drawing by integrating the pieces and compare it with the area from a pixel count.
Rotate the picture. Rotate one piece through an angle and write its new equation.

HL extension

Set up and solve the 3 × 3 systems algebraically, justify the error measure, compare line and quadratic pieces in terms of how error scales with the number of pieces, and rotate a piece with a rotation of the axes.

The data you need

Points you read from your own drawing, using Desmos or GeoGebra with the image placed behind the axes.

Desmos graphing calculator

Traps

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.

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