Algebra and graphs

A term is separated by + or −. Its numerical factor is the coefficient. Substitute by replacing each letter with its value, using brackets around negative values before applying powers.

Construct algebra in the order described. For example, the product of two consecutive even numbers can be written 2n(2n + 2).

First extract a common factor. Then check for grouping, a difference of two squares, a perfect square or a quadratic. Cubic expressions in this syllabus can be factorised by taking out a common factor before handling the remaining quadratic.

Completing the square rewrites ax² + bx + c as a(x − p)² + q. The form reveals the axis of symmetry x = p and turning point (p, q).

For addition or subtraction, use the lowest common denominator and combine the resulting numerator. For division, multiply by the reciprocal of the second fraction.

Index laws apply to matching bases. Fractional indices represent roots and negative indices represent reciprocals. Simple exponential equations can be solved by writing both sides with the same base.

Two linear simultaneous equations may be solved by elimination or substitution. For a linear and non-linear pair, substitute the linear expression into the non-linear equation, solve the resulting quadratic, then find both matching values.

Solve quadratics by factorisation, completing the square or the quadratic formula. The discriminant b² − 4ac indicates two, one or no real roots.

If the subject appears in several terms, move all of those terms to one side, factor out the subject and divide by its remaining factor.

On a number line, strict boundaries use open circles and inclusive boundaries use closed circles. On a coordinate grid, strict boundaries are broken and inclusive boundaries are solid.

For a two-variable inequality, test a point such as (0, 0) when it is not on the boundary. Cambridge questions normally shade the unwanted side unless told otherwise.

Constant first, second and third differences indicate linear, quadratic and cubic rules. For a quadratic sequence, half the constant second difference is the coefficient of n².

An exponential sequence has a constant multiplier. Combined sequences may need a familiar polynomial pattern plus an alternating or exponential part.

Replace ∝ with = k, use one known pair to calculate k, then use the equation for new values. Keep the same units throughout.

A negative speed–time gradient represents deceleration. A tangent estimates the instantaneous gradient of a curve; choose two well-separated points on the tangent, not on the curve.

Conversion graphs translate between units or currencies. Read from one axis to the line and across to the other axis using the stated scale.

Linear, quadratic, cubic, reciprocal and exponential functions have characteristic shapes. Roots are x-intercepts. Reciprocal graphs approach vertical and horizontal asymptotes; exponential graphs may approach a horizontal asymptote.

Solve f(x) = 0 at the x-intercepts and f(x) = g(x) at intersections. Graphical answers are estimates and must be reported to an accuracy supported by the scale.

A stationary point occurs where dy/dx = 0. Classify it using a sketch, the sign of the gradient on each side, or the second derivative: positive gives a minimum and negative gives a maximum.

The syllabus uses polynomials containing no more than three terms and non-negative integer powers. Points of inflection are not required.

An inverse function reverses a one-to-one function. Write y = f(x), rearrange for x, then interchange x and y. A composite such as gf(x) means apply f first and then g.

A mapping diagram represents each input with an arrow to its output. If one input has two outputs, the relation is not a function.