If the scatter bends, a straight line is the wrong tool: Plot the data. If it curves — bacteria multiplying, a hot drink cooling, light fading with distance — a straight line will miss it badly.
Instead pick a model whose shape matches the picture, and let the GDC find its constants. The common AI HL families are:
• Power y = a·xᵇ — growth/decline that starts steep then flattens (areas, biology scaling). • Exponential y = k·aˣ (or k·eʳˣ) — constant % change (growth or decay). • Quadratic / cubic — a rise-then-fall, or an S-shaped wiggle. • Sinusoidal y = a·sin(bx + c) + d — anything that repeats (tides, daylight).
The GDC does the arithmetic; your job is choosing the right family.
On the GDC (allowed on every AI paper): Enter the x-values in one list and the y-values in another, then choose the regression that matches the shape (ExpReg, PwrReg, QuadReg, CubicReg, SinReg…).
The GDC returns the constants (a, b, k, r…) and R² in one go. Write the model with those numbers substituted in.
IB-style question — fit an exponential decay model
A capacitor discharges. The voltage V (volts) is measured t seconds after the switch opens.
t: 0, 1, 2, 3, 4
V: 12.0, 7.8, 5.1, 3.3, 2.2
Fit a model of the form V = k·eʳᵗ and state R².
Step by step
- The values fall by roughly the same FACTOR each second (≈0.65×), so this is exponential decay — use ExpReg of the form k·eʳᵗ on the GDC.
- Entering the lists and running the regression gives the constants.
- Write the model with the numbers in, and read off the goodness-of-fit value.
Final answer
V = 12.0·e−0.425t, with R² = 0.9998 — an excellent fit. (r < 0 confirms decay.)
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R² scores the fit between 0 and 1: The coefficient of determination R² measures how much of the variation in the data the model explains.
• R² = 1 → the curve passes through every point (perfect fit). • R² close to 1 (say 0.95+) → a strong fit. • R² near 0 → the model barely beats just guessing the mean.
To choose between two candidate models, fit both on the GDC and compare their R² — the higher one usually wins. But also sanity-check the shape and whether it stays sensible in context (e.g. a population can't go negative).
IB-style question — compute a sum of squared residuals
A quadratic model gives predicted heights ŷ for three data points:
x: 1, 2, 3
y (data): 4.0, 9.0, 16.0
ŷ (model): 4.2, 8.5, 16.1
Find the sum of squared residuals, SSres.
Step by step
- A residual is data − model for each point.
- Square each residual and add.
- Total the squares.
Final answer
SSres = 0.30 volts² (smaller is better). Squaring stops positive and negative residuals from cancelling.
IB-style question — pick the better model
Cooling-coffee data is fitted two ways. The GDC reports R² = 0.982 for a power model and R² = 0.997 for an exponential model.
State, with a reason, which model fits better.
Step by step
- Compare the two R² values — the larger value explains more of the variation.
- Interpret: the exponential model leaves smaller residuals.
Final answer
The exponential model fits better because its R² (0.997) is closer to 1 than the power model's (0.982).