This SL mini-lesson is about modelling with functions: reading and using linear, quadratic, cubic, exponential, logarithmic and trigonometric models, plus direct/inverse variation, and — the heart of AI — fitting models to data with regression and judging fit with the correlation coefficient r.
AI flavour: you rarely derive functions from scratch; you choose the right model, fit it on your GDC and interpret parameters in context.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.
Function toolkit
The AI model families
Each real situation suggests a shape:
Linear f(x)=mx+c — constant rate of change (m = gradient).
Quadratic f(x)=ax²+bx+c — a single turning point (projectiles, profit).
Cubic f(x)=ax³+bx²+cx+d — up to two turning points.
?A cup of coffee cools quickly at first, then ever more slowly toward room temperature. Which model fits best?
Linear models
Gradient, intercept & interpretation
For f(x)=mx+c, the gradient m is the rate of change and c the initial value. In context, units matter: a taxi at $2.50 flat + $1.20/km is C = 1.20d + 2.50.
gradient m = Δy ÷ Δxc = value of y when x = 0
Worked example
Points (1, 52) and (5, 80).
m = (80 − 52) ÷ (5 − 1) = 28 ÷ 4 = 7
c: 52 = 7(1) + c ⇒ c = 45, so f(x) = 7x + 45
Calculate
Your turn — gradient
1A line passes through (2, 10) and (6, 30). Find its gradient m.
Hint: m = (30 − 10) ÷ (6 − 2).
Regression
Line of best fit & correlation
Least-squares regression finds the line y = a + bx that best fits data. The correlation coefficient r (−1 ≤ r ≤ 1) measures linear strength; r² is the fraction of variation explained.
Worked example — study hours vs score
Data x = 1,2,3,4,5 with y = 52,58,67,71,80.
GDC gives y = 44.9 + 6.9x, with r = 0.995 and r² = 0.989.
Predict x = 6: y = 44.9 + 6.9×6 = 86.3
Interpret, do not over-claim: r near ±1 means strong linear association — not causation. Only interpolate within the data range.
Calculate
Your turn — use the regression line
2A regression line is y = 44.9 + 6.9x. Predict y when x = 6.
Hint: substitute x = 6 into 44.9 + 6.9x.
Quick check
Read the correlation
?A data set gives r = 0.995. What does this tell you?
Exponential models
Exponential growth & decay
f(x)=k·ax grows (a>1) or decays (0<a<1) by a constant percentage. A population of 100 growing 8% per year is P = 100 × 1.08t.
f(x) = k · ax% rate: a = 1 + r (growth) or a = 1 − r (decay)
Horizontal asymptote: f(x)=k·ax+c approaches y = c but never reaches it — the model's long-run level.
Calculate
Your turn — exponential model
3A colony follows P = 100 × 1.08t (t in years). Find the population at t = 10 years, to the nearest whole number.
Hint: 100 × 1.0810.
Quadratic models
Quadratics & optimisation
A quadratic f(x)=ax²+bx+c has a vertex (turning point) at x = −b⁄(2a). If a < 0 the vertex is a maximum (e.g. maximum height or profit).
Worked example
Height h = −5t² + 20t (metres).
Vertex at t = −20 ÷ (2×−5) = 2 s
Max height = −5(2²) + 20(2) = −20 + 40 = 20 m
Calculate
Your turn — vertex
4A ball's height is h = −5t² + 20t. At what time t (seconds) does it reach maximum height? Use t = −b⁄(2a).
s
Hint: a = −5, b = 20, so t = −20 ÷ (2×−5).
Sort it
Which model?
Tap a real situation, then the model family that fits it best.
📈 Linear
🚀 Quadratic
🦠 Exponential
Calculate
Your turn — model value
5For the linear taxi fare C = 1.20d + 2.50 (d in km), find the cost of a 15 km trip in dollars.
$
Hint: 1.20 × 15 + 2.50.
Quick check
r² meaning
?A model has r² = 0.989. The best interpretation is:
Match it
Match each situation to its model
Tap an item on the left, then its match on the right.
Item
Match
Direct & inverse variation
Variation models
Two more AI staples: direct variation y = kx (double x → double y) and inverse variation y = k⁄x (double x → halve y).
Inverse-square laws (light, gravity) use y = k⁄x²; recognise the shape and fit k from one data point.
Quick check
Inverse variation
?The intensity of light follows I = k⁄d². If the distance d doubles, the intensity becomes:
Recap
The big ideas to take away
Model families: linear, quadratic, cubic, exponential, logarithmic, sinusoidal
Linear: f(x)=mx+c; m is the rate of change
Regression: GDC gives y = a + bx; r measures linear strength, r² the variation explained