Modelling with linear, quadratic, exponential, variation, cubic and sinusoidal functions

Learning objectives

Model a situation with a linear function f(x) = mx + c and interpret the gradient and intercept in context, stating their units

Model with a quadratic function f(x) = ax² + bx + c, finding the vertex, the axis of symmetry, the zeros and the intercepts, and interpreting the vertex as a maximum or minimum in context

Model exponential growth and decay with f(x) = ka^x + c or f(x) = ke^(rx) + c, identify the horizontal asymptote y = c as a long-term limit, and find the doubling or half-life time

Model direct and inverse variation with f(x) = ax^n for integer n, recognising an inverse square relationship from data, and state the behaviour near x = 0 and for large x

Model with a cubic function f(x) = ax³ + bx² + cx + d and with a sinusoidal function f(x) = a sin(bx) + d or a cos(bx) + d, interpreting a as the amplitude, 360/b as the period since the course works in degrees, and d as the principal axis

Apply the modelling cycle by choosing a model, fitting it with technology from data or given conditions, deciding on a reasonable domain and testing the fit, then using it to predict, distinguishing interpolation within the data range from extrapolation beyond it and commenting on the reduced reliability of extrapolated predictions and on the limitations of the model