Functions, Relations and Graphs: Modelling Change
Core concepts
You must recognise and work with the standard function families: linear, quadratic, power, exponential, logarithmic, and piecewise/hybrid functions. Know how to find domain and range from a rule and from a graph, identify intercepts, turning points and asymptotes, and use transformations - translations, dilations and reflections - to move between a base graph and its image. Be able to sketch graphs from a rule, read a graph to solve equations and inequalities, and interpret gradient as a rate of change in context. Understand inverse functions and the relationship between a function and its inverse graph (reflection in y = x). For exponential models, interpret the base as a growth or decay factor and the coefficient as the initial value; for logarithmic scales, recognise that equal multiplicative changes appear as equal steps.
Traps to watch
- Confusing the domain (permitted x-values) with the range (resulting y-values).
- Forgetting that the graph of y = a(x − h)² + k has its turning point at (h, k), not (−h, k).
- Treating an asymptote as a value the function reaches.
- Ignoring practical restrictions, such as a length or time that cannot be negative.
- Reading a gradient without units when the axes carry real-world quantities.
Worked example
For y = 3(2)^(x−1) + 4: the horizontal asymptote is y = 4, the y-intercept occurs at x = 0 giving y = 3(2)^(−1) + 4 = 5.5, and the function is increasing because the base 2 exceeds 1. The range is y > 4, and the domain is all real x.
Rule of thumb
Sketch the base graph first, then apply transformations in order - dilations, reflections, then translations - and check with two or three plotted points.
Space and Measurement: Length, Area, Volume and Similarity
Core concepts
This domain tests measurement with meaning. Know the perimeter and area formulas for triangles, quadrilaterals, circles and composite shapes, and the surface area and volume formulas for prisms, cylinders, pyramids, cones and spheres. Be able to convert units of length, area and volume correctly, including between metric and derived units such as litres and cubic centimetres. Understand similarity and scale factors: if lengths scale by k, areas scale by k² and volumes by k³. Apply Pythagoras' theorem and right-triangle trigonometry (sine, cosine, tangent) to find unknown sides and angles, and use the sine and cosine rules in non-right triangles. Interpret contour maps and bearings, and estimate error in measurement.
Traps to watch
- Converting area or volume with a single factor of 10 or 100 instead of squaring or cubing it.
- Using the diameter as the radius in circle and cylinder formulas.
- Applying the length scale factor to area or volume without squaring or cubing.
- Forgetting to include all faces when calculating surface area of a composite solid.
- Using the sine rule when the cosine rule is required, or ignoring the ambiguous case.
Worked example
A cylindrical tank has radius 1.5 m and height 4 m. Volume = πr²h = π × 2.25 × 4 ≈ 28.27 m³, which is 28 270 litres since 1 m³ = 1000 L. If a similar tank has all lengths doubled, its volume is multiplied by 2³ = 8, giving about 226.2 m³.
Rule of thumb
Write the formula, substitute with units attached, then check the magnitude of your answer against the physical situation - a sensible estimate catches most unit errors.