Different forms of the equation of a straight line. Gradient; intercepts. Lines with gradients m1 and m2 Parallel lines m1 = m2. Perpendicular lines m1 × m2 = … 1.

Concept of a function, domain, range and graph. Function notation, for example f (x) , v(t) , C(n). The concept of a function as a mathematical model. A Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y = x, and the notation f …1(x).

The graph of a function; its equation y = f (x). Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.

Determine key features of graphs. Finding the point of intersection of two curves or lines using technology.

Modelling with the following functions: Linear models. f x = mx + c. Quadratic models. f x = ax + bx + c ; a ≠ 0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis. Exponential growth and decay models. x f (x) = ka + c f (x) = ka…x + c (for a > 0) f (x) = kerx + c Equation of a horizontal asymptote. Direct/inverse variation: f x = axn, n ∈ ℤ The y-axis as a vertical asymptote when n < 0. Cubic models: f x = ax3 + bx2 + cx + d. Sinusoidal models: f x = asin(bx) + d,

Modelling skills: Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs. Develop and fit the model: Given a context recognize and choose an appropriate model and possible parameters. Determine a reasonable domain for a model. Find the parameters of a model. Test and reflect upon the model: Comment on the appropriateness and reasonableness of a model. Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation. Use the model: Reading, interpreting and making predictions based on the model.

Composite functions in context. The notation ( f ∘ g)(x) = f (g(x)). …1 Inverse function f Finding an inverse function.

Transformations of graphs. Translations: y = f (x) + b ;y = f (x … a). Reflections: in the x axis y = … f (x), and in the y axis y = f ( … x). Vertical stretch with scale factor p: y = p f (x). Horizontal stretch with scale factor : y = f (qx) Composite transformations.

In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions: Exponential models to calculate half-life. Natural logarithmic models: f (x) = a + blnx Sinusoidal models: f x = asin b x … c + d Logistic models: L f x = ; L, C, k > 0 1 + Ce…kx Piecewise models.

Scaling very large or small numbers using logarithms. Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters. Interpretation of log-log and semi-log graphs.