Functions: models, graphs and interpretation
Functions in AI SL is about reading and building models rather than algebraic manipulation. You are expected to use function notation, state domain and range in context, sketch and describe graphs, and fit linear, quadratic and exponential models to real data, using the GDC to find roots, vertices and intersections.
Core rules
- Linear model f(x) = mx + c: m is the constant rate of change, c the initial value.
- Quadratic model f(x) = ax² + bx + c: the vertex is at x = −b/(2a); a > 0 gives a minimum, a < 0 a maximum.
- Exponential model f(x) = a·bˣ: a is the initial value; b > 1 gives growth, 0 < b < 1 decay, and the x-axis is an asymptote.
- Domain is the set of valid inputs; in context it is often restricted (x ≥ 0, integer values, a maximum capacity).
- Solve f(x) = k, or f(x) = g(x), graphically on the GDC and interpret the intersection in context.
Worked example
A company's profit, in dollars, is modelled by P(x) = −2x² + 40x − 150, where x is the number of items sold in hundreds. This is a downward parabola, so the maximum is at the vertex: x = −40/(2 × −2) = 10. Then P(10) = −2(100) + 400 − 150 = 50. Maximum profit is therefore $50, reached at 1000 items. The model's domain here is 0 ≤ x ≤ 20 in practice, because P(x) = 0 at the edges of that interval.
Traps that cost marks
Quoting a vertex or root without interpreting it in context; ignoring a restricted domain and reporting a negative or fractional answer; treating the growth factor b as a percentage (b = 1.05 means 5% growth, not 105%); confusing f(x) + 2 with f(x + 2); extrapolating a model far beyond the data it was fitted to.