Functions and graphs
Chapter 20 of 65 strong
RESTS ONSigned numbersCH 3Substitute known values into a formula and evaluate it correctlyCH 4Solve two-step linear equations by inverse operations in reverse orderCH 1Read coordinates off a graphed line respecting its axis scalesFOUNDATIONSForm or recognise the union, intersection, difference and complement of finite or interval-defined sets relative to an explicitly stated universeFOUNDATIONSCompute a slope from a graph as rise over run between two pointsFOUNDATIONSWrite a line's equation from its y-intercept and slopeCH 5Solve equations with unknowns on both sides by collecting them onto one sideCH 1Check a claimed solution or point by substituting into the original equationCH 1Write a linear model from a description with signed intercept and rateFOUNDATIONSConnect general, completed-square and factorised forms of one quadratic and select the form that exposes the required featureFOUNDATIONSUse the sign and square status of the discriminant to decide the number and rational or irrational nature of real roots, factors and horizontal-axis interceptsFOUNDATIONSRecognise a quadratic expression or equation by a nonzero coefficient of the squared term and no higher power after simplificationCH 1Translate a two-condition description into a pair of equations in two unknownsFOUNDATIONSConnect roots of an equation, zeros of its polynomial, linear factors and horizontal-axis intercepts, including a repeated rootCH 1Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variableCH 1Connect a root of P(x)=0, its linear factor and the corresponding horizontal-axis intercept of y=P(x)FOUNDATIONSInfer the two end directions of a cubic graph from the sign of its leading coefficientFOUNDATIONSState a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending powerCH 6Use the parity and size of a root's multiplicity to determine whether the graph crosses or touches the horizontal axis, distinguish an ordinary crossing from a stationary crossing, and decide whether the sign changesFOUNDATIONSConstruct a polynomial from specified roots and multiplicities, including a stated leading coefficient, and convert between factor and expanded coefficient formsFOUNDATIONSFind intersections between a cubic and a line or quadratic by equating their outputs, solving the resulting polynomial equation, and substituting each horizontal coordinate to obtain the corresponding vertical coordinateFOUNDATIONSIdentify the vertex, horizontal axis, opening direction, intercepts, domain and range of a translated and dilated sideways parabola relation, and identify that the full relation is not a functionFOUNDATIONSSimplify products, quotients and powers with a common positive base by adding, subtracting or multiplying exponents as appropriate, without applying an index law across addition or subtractionCH 7Highest common factorFOUNDATIONSMultiplication and divisionCH 9Reduce an angle to its first-quadrant reference angle measured from the horizontal axisCH 1Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundaryCH 1Apply reflection, parity and periodicity relations to rewrite a circular-function value in terms of a base angle and the correct signCH 1Rewrite a rule in canonical a·f(n(x−h))+k form and connect a, n, h and k with vertical dilation or reflection, horizontal dilation, phase shift and vertical translationFOUNDATIONSConstruct and extend the quarter-period key-point table for a basic sine or cosine graph over a stated interval and count complete or partial cycles without double-counting shared endpointsFOUNDATIONSState and apply the maximal domains and ranges of the sine, cosine and tangent functions, including tangent's excluded odd half-turn boundariesCH 1Transform the stated domain with an affine circular-function argument, solve for the inner argument, and map every solution back to the original variable without losing endpoints or periodsCH 1Interpret amplitude, period, equilibrium, phase anchor, maximum and minimum in the units and meaning of the contextFOUNDATIONSBuild a table of values from a formula by independent substitutionFOUNDATIONSDecide whether a given transformation linearises a data setFOUNDATIONSPercentage change as a multiplierFOUNDATIONSSolve an exponential equation that cannot be reduced to a common base by taking logarithms consistently, isolate the unknown and distinguish the exact logarithmic form from a guarded decimal approximationCH 4Turn an equation's exact solution into the answer the situation demandsCH 4Use a completed inverse rule in both directionsFOUNDATIONSTranslate a described situation into a linear equation in one unknownCH 6
Restricted domains and graph reading
Linear and polynomial graphs
Maximal domains and power functions
Combined and piecewise functions
Circular and exponential graphs
Model selection and interpretation