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Mathematical Methods
Quadratic functions
Chapter 3
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Recognise whether an equation is linear from the power on its unknowns
CH 1
Solve a one-step linear equation
FOUNDATIONS
Multiplication and division
CH 2
Decimal place value
FOUNDATIONS
Tabulate a sequence by term number and term value
FOUNDATIONS
Build and read a total formula from two unit rates
FOUNDATIONS
Solve equations with a scaled bracket by expanding or dividing first
CH 1
Translate between a variation statement, its ∝ form and its equation
FOUNDATIONS
Quadratic forms, roots and graphs
Completing the square and finding rules
Quadratic formula and surd roots
Quadratic equations, inequalities and intersections
Quadratic models
Analysing quadratic models
Recognise a quadratic expression or equation by a nonzero coefficient of the squared term and no higher power after simplification
Expand two linear factors and factorise a simple quadratic with rational coefficients, preserving the leading coefficient and middle term
Connect roots of an equation, zeros of its polynomial, linear factors and horizontal-axis intercepts, including a repeated root
Rearrange to zero, factorise, apply the null factor law and solve the resulting linear equations
Identify a, b, c in zero form and calculate b squared minus 4ac with signs intact
Use the sign and square status of the discriminant to decide the number and rational or irrational nature of real roots, factors and horizontal-axis intercepts
Read reflection, dilation parallel to the vertical axis, and horizontal or vertical translation from a(x minus h) squared plus k, including the sign inside the bracket
Obtain concavity, axis of symmetry, turning point, maximum or minimum value, vertical intercept and zero, one or two horizontal intercepts from an exact quadratic rule
Connect general, completed-square and factorised forms of one quadratic and select the form that exposes the required feature
Solve two-step linear equations by inverse operations in reverse order
Simplify a fraction
Substitute known values into a formula and evaluate it correctly
Signed numbers
Build a table of values from a formula by independent substitution
Express a quadratic in completed-square form by extracting the leading coefficient and adding and subtracting the square of half the linear coefficient
Determine a quadratic rule from a turning point and another point, or from two horizontal intercepts and another point, choosing the efficient form first
Determine the three general-form coefficients from three points by forming and solving three simultaneous linear equations
Multiply and divide fractions
Translate a two-condition description into a pair of equations in two unknowns
Solve a pair of simultaneous linear equations for the shared (x, y)
Substitute exact coefficients into the quadratic formula, keep the whole denominator 2a, take both sign branches and simplify
Represent irrational real roots in a canonical exact surd form rather than prematurely converting them to decimals
For a polynomial with rational coefficients, recover the conjugate of a given quadratic surd root and use the pair to construct rational coefficients
Rounding
Identify a repeated subexpression, replace it with a temporary variable, solve the resulting quadratic and substitute back
Reject values excluded by the substitution domain or introduced by a non-equivalent operation such as squaring, by checking the original equation exactly
Determine the sign of a quadratic on intervals separated by its roots, accounting for concavity and even or odd root multiplicity
Solve a quadratic inequality and report the correct between/outside/all/none interval pattern with endpoints included only for a non-strict sign
Substitute a linear rule into a quadratic rule, use the resulting discriminant to count intersections and solve for any intersection coordinates
Express a discriminant in a parameter, impose the sign condition matching the required number of roots or intersections, and solve the resulting parameter equation or inequality
Check a claimed solution or point by substituting into the original equation
Solve equations with unknowns on both sides by collecting them onto one side
Define a variable, translate a constraint or stated quadratic relationship into an exact quadratic equation or rule, and retain its meaningful domain
Select feasible quadratic solutions using sign, discreteness, units and domain constraints, then state the requested quantity
Find the constant of variation for a stated direct relationship
Use a completed direct rule in both directions
Translate a described situation into a linear equation in one unknown
Turn an equation's exact solution into the answer the situation demands
Use the turning point and modelling domain to find and interpret an algebraic maximum or minimum, including threshold and crossing questions