2.1 Functions - SL content

A straight-line model can be written y=mx+c, where m is the gradient and c is the y-intercept. The gradient measures change in y per unit change in x.

Use the form that matches the data: two points give m=(y₂−y₁)/(x₂−x₁); a point and a gradient give y−y₁=m(x−x₁). Check the scale and units before interpreting m.

Through (2,5) and (6,13), m=8/4=2, so y−5=2(x−2), or y=2x+1. At x=0 the model predicts 1, not 0.

A negative gradient means y falls as x rises; it does not mean the intercept is negative. Do not swap coordinates or treat correlation as proof that the line is causal.

The inverse function f⁻¹ reverses the mapping: f⁻¹(f(x))=x and f(f⁻¹(x))=x on their appropriate domains. A function needs to be one-to-one for an inverse function to exist without restricting the domain.

To find an inverse, write y=f(x), interchange x and y, then solve for y. Graphically, f and f⁻¹ reflect in y=x, so the domain and range swap.

For f(x)=2x+3, y=2x+3 gives x=(y−3)/2, so f⁻¹(x)=(x−3)/2. A quadratic needs a domain restriction before its inverse is a function.

f⁻¹(x) is not 1/f(x). Always check composition and state the domain restriction when the original graph fails the horizontal-line test.

A graph shows how outputs depend on inputs. A useful sketch preserves key behaviour—intercepts, turning points, asymptotes, end behaviour and domain—not every plotted pixel.

Start with the domain and scale, then find features that determine the shape. Use a calculator or points to check the sketch, but interpret the graph rather than copying an unlabelled screen.

For y=(x−2)²−1, the vertex is (2,−1), the x-intercepts are 1 and 3, and the parabola opens upward. Those three facts are more informative than a dense table of values.

A sketch need not be to scale, but it must not invent features. Distinguish an intercept from a turning point and do not ignore a restricted domain.

Key graph features translate a picture into statements: intercepts show zeros or initial values, a gradient shows rate of change, a turning point marks a local extreme, and an asymptote describes a value the graph approaches.

Read the axes and units first. Then connect the feature to the context: a maximum may be a capacity or peak, while a negative gradient may represent decline. A graph can show association without proving why it occurs.

If a concentration curve levels off at 8 mg L⁻¹, the horizontal tendency suggests a limiting value; it does not prove the process has stopped or that 8 is a universal constant.

Do not confuse a horizontal tangent with a horizontal asymptote. Check whether the statement concerns one point, a local interval or long-run behaviour.

A mathematical model uses a function to represent a real relationship. The model is useful only over a stated context and with assumptions about variables, units and measurement.

Choose a form from behaviour: constant change suggests linear, repeated proportional change suggests exponential, and a single peak may suggest a quadratic. Fit parameters from data, then test residuals or predictions and state the domain where the model is credible.

A population rising by roughly 4% per year may be modelled by P(t)=P₀(1.04)^t. The model predicts a value, not a biological law; resource limits can make the extrapolation fail.

A close fit does not validate an unlimited extrapolation. Explain why the chosen form matches the observed mechanism and where the assumption breaks.

To model a situation, define variables and units, choose a functional form, estimate parameters, compare predictions with evidence and revise or qualify the model.

The context determines what counts as a reasonable assumption. Keep a train/test or observed/predicted distinction when possible, and inspect whether errors are random or systematic. A model that fits one interval may fail outside it.

A delivery-time model based on distance may work for local routes but overpredict long routes if motorway travel changes the relationship. Refit or split the model rather than hiding the systematic error.

The calculator selecting a curve is not the modelling process. The final claim must include assumptions, units, validation evidence and a domain of use.