Algebra and graphs

Terms are like terms only when their variable parts, including indices, match. Coefficients of like terms may be added or subtracted.

Expansion removes brackets by multiplying every term. Factorisation reverses expansion. Always look for a common factor before trying identities or quadratic methods.

Treat an algebraic fraction like a numerical fraction: factor before cancelling, use a common denominator for addition, and multiply by the reciprocal when dividing.

A factor may be cancelled only when it multiplies the whole numerator and denominator. State excluded values when a denominator could be zero.

Keep an equation balanced by performing the same valid operation on both sides. Clear fractions early when this makes the structure simpler.

Simultaneous equations share the same solution. Use elimination or substitution for two linear equations; substitute the linear equation into the non-linear one when a curve is involved.

Quadratics may be solved by factorisation, completing the square or the quadratic formula. A negative discriminant means there are no real roots.

Strict inequalities use open circles or broken boundary lines. Inclusive inequalities use closed circles or solid boundary lines. On region diagrams, follow the question's shading convention carefully.

A sequence rule can connect each term to the next or connect the position n directly to the term. Constant first differences indicate a linear sequence; constant second differences indicate a quadratic sequence; constant third differences indicate a cubic sequence.

An exponential sequence has a constant multiplier between consecutive terms. Mixed sequences may combine linear, quadratic, cubic or exponential patterns, so compare both differences and ratios.

In direct proportion, both variables change in the same scale direction. In inverse proportion, multiplying one variable by a factor divides the other by that factor.

On a distance–time graph, gradient is speed. A horizontal section means stationary. On a speed–time graph, gradient is acceleration and area under the graph is distance travelled.

Linear, quadratic, cubic, reciprocal and exponential families have distinctive shapes. Roots are x-intercepts; turning points are local maxima or minima; reciprocal graphs have asymptotes that the curve approaches.

Common function forms include powers axⁿ, combinations of up to three power terms and exponentials abˣ + c. A table of values should include enough points to show roots, turning points and asymptotic behaviour.

Equations can be solved graphically by reading x-intercepts or intersections. Graphical answers are estimates and should match the graph's scale.

Differentiation gives a formula for gradient. For each term axⁿ, multiply by n and reduce the power by one. A stationary point occurs where dy/dx = 0.

The second derivative can classify a stationary point: d²y/dx² > 0 indicates a local minimum and d²y/dx² < 0 indicates a local maximum. A sketch or the gradient on either side may also be used.

A function maps each allowed input to one output. The domain is the set of allowed inputs and the range is the resulting set of outputs.

An inverse function reverses the original mapping. A composite function such as gf(x) means apply f first, then g. Mapping diagrams can show inputs, outputs and whether a relation is a function.