B: Algebra and functions — AQA A-Level Mathematics
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B: Algebra and functions explained
This topic extends the familiar rules for integer powers to any rational exponent.
Read the full explanation
You need to know that a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a raised to the power m, or equivalently (a^(1/n))^m. The three core laws are: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), and (a^m)^n = a^(mn). These hold for all rational m and n, provided the base a is positive when roots are involved. For example, 8^(2/3) = (8^(1/3))^2 = 2^2 = 4. You also need to handle negative and zero exponents: a^0 = 1 (a ≠ 0) and a^(-n) = 1/a^n. When simplifying expressions, convert roots to fractional powers, combine using the laws, and leave answers in the form requested. This skill underpins later work on exponential functions, logarithms, and calculus.
Use and manipulate surds, including rationalising the denominator.
A surd is an irrational root such as √2 or √5. Simplify by extracting the largest square factor: √ab = √a × √b, so √50 = √(25×2) = 5√2. Combine like surds: 3√2 + 5√2 = 8√2. Multiply using √a × √b = √(ab), and expand brackets normally, e.g. (2+√3)(2−√3) = 4 − 3 = 1. Rationalising the denominator rewrites a fraction so the denominator is rational. For a single surd, multiply numerator and denominator by that surd: 1/√2 = √2/2. For a denominator a ± √b, multiply numerator and denominator by the conjugate a ∓ √b, using the difference of two squares. This skill gives exact answers in geometry, trigonometry and calculus.
Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown.
This topic covers quadratic functions from multiple angles. You must sketch and interpret their parabolic graphs, identifying the vertex, axis of symmetry, intercepts and whether the curve opens upwards or downwards. Completing the square rewrites ax²+bx+c as a(x+p)²+q, revealing the vertex and enabling solution. The discriminant b²−4ac determines the number of real roots: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots. You solve quadratic equations by factorising, completing the square or using the formula. You also solve equations that are quadratic in a function of the unknown, such as x⁴−5x²+4=0, by substituting u=x². For example, 2x²−8x+6=0 completes to 2(x−2)²−2, has discriminant 16, and roots x=1 and x=3.
Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.
This topic requires you to solve pairs of equations in two variables. For two linear equations, elimination is often efficient: multiply one or both equations to make the coefficients of one variable equal, then add or subtract to eliminate it. Substitution is also valid: rearrange one equation to make a variable the subject and substitute into the other. When one equation is linear and the other is quadratic, substitution is usually the best method. Rearrange the linear equation to express one variable in terms of the other, substitute into the quadratic equation, and solve the resulting quadratic. Each solution pair must satisfy both original equations. For example, solving y = x + 1 and x² + y² = 25 gives x² + (x+1)² = 25, leading to 2x² + 2x − 24 = 0, so x = 3 or x = −4, with corresponding y values 4 and −3.
Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions. Express solutions through correct use of ‘and’ and ‘or’, or through set notation. Represent linear and quadratic inequalities such as y > x + 1 and y > ax² + bx + c graphically.
Solve linear inequalities by expanding brackets and collecting terms; reverse the sign if multiplying or dividing by a negative. For quadratics, rearrange to make one side zero. Find critical values and sketch the parabola to identify the correct regions. If the x² coefficient is positive, < 0 lies between the roots and > 0 outside. If there are no real roots, the expression is always positive or always negative. Here 'fractions' means numerical fractions, e.g. (x+1)/2 > (2x−3)/3; clear these by multiplying by the positive common denominator, not by algebraic denominators. Express solutions using 'and', 'or', or set notation like {x : x < −1} ∪ {x : x > 3}. Graphically, represent strict inequalities with dashed lines/curves and shade the required region.
Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only).
Polynomial manipulation underpins much of algebra. Expanding brackets means multiplying every term in one bracket by every term in the other, then collecting like terms; for example (2x + 3)(x² − 4x + 1) gives 2x³ − 8x² + 2x + 3x² − 12x + 3 = 2x³ − 5x² − 10x + 3. Factorisation reverses this, often by extracting a common factor or using known identities. Simple algebraic division by a linear expression can be done by long division or by comparing coefficients. The factor theorem states that (x − a) is a factor of f(x) exactly when f(a) = 0, which allows you to find unknown constants and factorise cubics. Rational expressions are simplified by factorising numerator and denominator fully, then cancelling common factors, remembering that the cancelled factor must not be zero. Division by a linear expression may leave a remainder, which can be written as a fraction.
Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs.
This topic builds graphical fluency across key function families. You must recognise and sketch polynomials, the modulus of a linear function, and reciprocal curves y = a/x and y = a/x². Mark axial intercepts, asymptotes and end behaviour. For y = a/x and y = a/x², the axes act as vertical and horizontal asymptotes. Graphically, solving f(x) = g(x) means finding the x-coordinates of intersection points. Proportional relationships include direct proportion (e.g. y = kx, y = kx²) and inverse proportion (e.g. y = k/x). Practise plotting key points and using symmetry, especially for modulus graphs y = |ax + b| where the section below the x-axis is reflected in the x-axis.
Understand and use composite functions; inverse functions and their graphs.
This topic covers combining functions and reversing them. A composite function applies one function to the output of another: (f ∘ g)(x) = f(g(x)), read as 'f of g of x'. Order matters: f(g(x)) is generally not the same as g(f(x)). The domain of f(g(x)) consists of x in the domain of g for which g(x) lies in the domain of f. An inverse function f⁻¹ reverses f: if f(a) = b then f⁻¹(b) = a. To find f⁻¹, write y = f(x), swap x and y, and rearrange for y. The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x. For f⁻¹ to exist as a function, f must be one-to-one on its domain; restricting the domain can achieve this. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
Understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax), and combinations of these transformations.
A transformation changes a graph's position, size or orientation. For y = af(x), every y-coordinate is multiplied by a, so the curve stretches vertically by factor a (a reflection in the x-axis if a < 0); points where f(x) = 0 stay fixed. For y = f(x) + a, the whole graph translates by a units parallel to the y-axis. For y = f(x + a), the graph translates by −a units parallel to the x-axis, so f(x + 2) moves left. For y = f(ax), x-coordinates are divided by a, giving a horizontal stretch of factor 1/a (a reflection in the y-axis if a < 0). Combinations apply these in turn, and sketching means marking key features such as intercepts, turning points and asymptotes after transformation.
Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear).
Partial fractions reverse adding algebraic fractions: a rational function with a factorised denominator is rewritten as a sum of simpler fractions whose denominators are those factors. For distinct linear factors use one constant per factor, e.g. (2x+1)/((x−1)(x+2)) = A/(x−1) + B/(x+2). For a repeated linear factor include a term for each power, e.g. A/(x−1) + B/(x−1)^2. If the numerator is linear and the denominator has three linear factors, use three constants. Multiply through by the full denominator to form an identity valid for all x, then substitute convenient values (roots of factors) or compare coefficients. The method is assessed through correct form, accurate constants and valid algebraic manipulation.
Use of functions in modelling, including consideration of limitations and refinements of the models.
Modelling uses functions to represent real-world situations, enabling predictions and explanations. You must evaluate how well a model fits reality. For example, a cooling cup of tea modelled by $T = 20 + 75e^{-0.05t}$ assumes the ambient temperature remains exactly 20°C. While it correctly models the cooling rate decreasing as temperature drops, a limitation is that it ignores factors like evaporation or draughts. Refinements could involve adjusting parameters, adding terms for fluctuating room temperatures, or restricting the valid domain (e.g., $t \ge 0$). You must state assumptions, interpret parameters in context, evaluate the model's limitations, and suggest sensible improvements.
Your focus
- Evaluate numerical expressions with rational exponents without a calculator, such as 25^(3/2) or 8^(-2/3).
- Simplify algebraic expressions involving rational exponents using the laws of indices, including negative and fractional powers.
- Convert between radical and exponent notation and use this to solve equations or simplify expressions.
Show all 33 objectives
- Simplify surd expressions by extracting square factors and combining like terms.
- Rationalise denominators involving single surds and binomial surds using conjugates.
- Apply surd manipulation to solve problems requiring exact answers, such as in geometry or algebra.
- Complete the square for a quadratic expression and use it to find the vertex and solve equations.
- Calculate and interpret the discriminant to determine the number of real roots of a quadratic equation.
- Solve quadratic equations in a function of the unknown by substitution and back-substitution.
- Solve two linear simultaneous equations using elimination and substitution.
- Solve simultaneous equations with one linear and one quadratic equation by substitution.
- Interpret the solutions as points of intersection of the corresponding graphs.
- Solve linear inequalities with brackets and numerical fractions, reversing the sign when required.
- Solve quadratic inequalities by finding critical values and interpreting a sketch, including cases with negative x² coefficients or no real roots.
- Express solution sets correctly using ‘and’, ‘or’ or set notation, and represent linear and quadratic inequalities graphically.
- Expand and collect like terms in polynomial products accurately.
- Factorise polynomials and use the factor theorem to identify linear factors.
- Divide polynomials by linear expressions and simplify rational expressions by cancelling common factors.
- Sketch polynomials, modulus of a linear function, and reciprocal curves y = a/x and y = a/x², showing intercepts and asymptotes.
- Solve equations graphically by identifying intersection points of curves and lines.
- Recognise and use direct and inverse proportional relationships, including those involving powers, and their graphs.
- Form and evaluate composite functions, including stating their domains.
- Find inverse functions algebraically and sketch their graphs as reflections in y = x.
- Determine when an inverse function exists and use domain restrictions appropriately.
- Describe the geometric effect of each of y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax) on the graph of y = f(x).
- Apply a combination of transformations to a given curve and determine the resulting equation or graph.
- Sketch a transformed graph showing correct intercepts, turning points and asymptotes.
- Select the correct partial fraction form for denominators with distinct or repeated linear factors.
- Determine unknown constants by substitution and coefficient comparison.
- Verify a partial fraction decomposition by recombination or substitution.
- Construct and interpret a function to model a real-world situation, stating assumptions and defining variables.
- Use the model to make predictions and solve problems within its domain.
- Critically evaluate the model's limitations and suggest specific refinements.
B: Algebra and functions exam tips
Marking Points
- Correctly interpret a^(m/n) as the nth root of a raised to the power m, or as (a^(1/n))^m, and evaluate numerical examples such as 27^(2/3) = 9.
- Apply the multiplication law a^m × a^n = a^(m+n) to simplify expressions with rational exponents, including cases with different bases that cannot be combined.
- Apply the division law a^m ÷ a^n = a^(m-n) and the power law (a^m)^n = a^(mn) correctly, including negative and fractional exponents.
- Use the zero and negative exponent rules: a^0 = 1 and a^(-n) = 1/a^n, and combine these with other laws to simplify expressions.
- Convert between radical and exponent notation, e.g. √x = x^(1/2) and 1/∛x = x^(-1/3), to simplify or evaluate expressions.
- Simplify surds by extracting the largest square factor, e.g. √72 = 6√2, and combine like surds through addition and subtraction.
- Multiply and divide surds, including expanding brackets such as (3+√2)(3−√2) = 7, and simplify the result.
- Rationalise a denominator containing a single surd by multiplying numerator and denominator by that surd, e.g. 5/√3 = 5√3/3.
- Rationalise a denominator of the form a ± √b by multiplying numerator and denominator by the conjugate, and simplify the resulting expression.
- Use surds in contextual problems, such as finding exact lengths in right-angled triangles or simplifying expressions in coordinate geometry.
- Sketch or describe a quadratic graph, showing correct shape, y-intercept and any x-intercepts or vertex.
- Complete the square for a quadratic expression, including when the coefficient of x² is not 1, and use the completed-square form to identify the vertex or solve the equation.
- Calculate the discriminant b²−4ac and interpret its sign to state the number and nature of real roots.
- Solve quadratic equations by an appropriate method, including factorising, completing the square or the quadratic formula, and give exact or decimal answers as required.
- Recognise and solve an equation that is quadratic in a function of the unknown, such as a polynomial in x² or a trigonometric equation, by using a substitution and then solving the resulting quadratic.
- Link the discriminant to graph sketches, for example showing a curve that does not meet the x-axis when the discriminant is negative.
- Solve two linear simultaneous equations by elimination, correctly scaling equations and adding or subtracting to eliminate one variable.
- Solve two linear simultaneous equations by substitution, rearranging one equation and substituting into the other.
- Solve a pair of simultaneous equations where one is linear and one is quadratic by substituting the linear expression into the quadratic equation.
- Form and solve the resulting quadratic equation correctly, including expanding brackets and collecting like terms.
- Find both variables for each solution and present the solutions as coordinate pairs, checking that each pair satisfies both original equations.
- Interpret the number of solutions geometrically, for example a line intersecting a circle or parabola at two points, one point or no points.
- Expands brackets and collects like terms to solve linear inequalities, reversing the inequality sign when dividing by a negative.
- Solves linear inequalities containing numerical fractions by multiplying by a positive common denominator, e.g. (x+1)/2 > (2x−3)/3 gives 3(x+1) > 2(2x−3).
- Finds critical values for quadratic inequalities and uses a sketch to select the correct region, accounting for the sign of the x² coefficient.
- Identifies that a quadratic inequality with no real roots is either satisfied by all real values or no real values.
- Expresses solution sets accurately using 'and' for bounded intervals, 'or' for disjoint intervals, or formal set notation.
- Draws a dashed boundary for strict inequalities (<, >) or a solid boundary for inclusive inequalities (≤, ≥) and shades the correct region.
- Expands products of polynomials correctly, including brackets with more than two terms.
- Collects like terms accurately and writes the polynomial in descending powers.
- Factorises by taking out common factors and by using the factor theorem to find linear factors.
- Performs algebraic division by a linear expression, showing the quotient and any remainder.
- Uses the factor theorem to test whether a given linear expression is a factor.
- Simplifies rational expressions by factorising and cancelling common factors.
- States any restrictions on the variable where a cancelled factor could be zero.
- Handles division by a linear expression only, as required by the specification.
- Sketching a polynomial: identify degree, leading coefficient and roots; mark axial intercepts and show correct end behaviour.
- Modulus of a linear function: sketch y = |ax + b| by drawing y = ax + b and reflecting the part below the x-axis; the vertex occurs where ax + b = 0.
- Reciprocal curves: for y = a/x, draw branches in quadrants determined by the sign of a; for y = a/x², draw two branches above the x-axis (if a > 0). Both have the axes as asymptotes.
- Graphical solution of equations: rearrange to f(x) = g(x), sketch both graphs, and read off x-coordinates of intersection points.
- Proportional relationships: recognise direct proportion (e.g. y = kx, y = kx²) and inverse proportion (e.g. y = k/x, y = k/x²). Use given points to find the constant of proportionality k.
- Forming composite functions: substitute the whole expression for the inner function into the outer function and simplify correctly.
- Evaluating composite functions at a given value: work from the inside out, applying the inner function first.
- Finding inverse functions algebraically: write y = f(x), interchange x and y, then rearrange to make y the subject; denote the result f⁻¹(x).
- Graphs of inverse functions: sketch y = f⁻¹(x) as the reflection of y = f(x) in y = x, and identify that points (a, b) on f correspond to (b, a) on f⁻¹.
- Domain and range: state the domain and range of composite and inverse functions, using the original function's domain and range appropriately.
- Existence of inverses: recognise that a one-to-one function has an inverse; if not one-to-one, restrict the domain to create an inverse.
- Identifies y = af(x) as a vertical stretch of factor a, or a reflection in the x-axis when a is negative, leaving x-intercepts unchanged.
- States that y = f(x) + a translates the graph by a units parallel to the y-axis, moving every point including turning points and asymptotes.
- Recognises y = f(x + a) as a horizontal translation by −a units, so f(x + 3) shifts the graph three units left.
- Describes y = f(ax) as a horizontal stretch of factor 1/a, or a reflection in the y-axis when a is negative, leaving y-intercepts unchanged.
- Applies combinations in a consistent order, for example y = 2f(x − 1) as a translation right by 1 then a vertical stretch of factor 2.
- Sketches the transformed curve with correct key features: intercepts, stationary points, asymptotes and overall shape.
- Writes the correct partial fraction form, using one constant for each distinct linear factor and separate terms for repeated factors up to their power.
- Multiplies both sides by the full denominator to obtain an identity valid for all x, then solves for the unknown constants.
- Uses substitution of convenient x values, such as roots of the factors, to find constants efficiently; choosing a root makes all other terms vanish, which is the standard efficient method.
- Compares coefficients of powers of x when substitution alone is insufficient, for example with repeated factors.
- Handles numerators that are constant or linear correctly, including cases where the numerator has the same degree as a factor.
- Checks the decomposition by recombining the partial fractions or substituting a test value.
- State clearly what each variable and parameter represents in the real context, including units.
- Use the function to calculate values or solve equations within the model's valid domain.
- Identify at least one limitation, such as the model ignoring external environmental factors or failing for large values of the independent variable.
- Suggest a specific refinement, such as adding a term to account for a neglected factor or restricting the domain.
- Comment on the reasonableness of predictions by comparing them with given data or expected real-world behaviour.
- Explain how changing a parameter affects the model's output in context.
Examiner Tips
- 💡Always show intermediate steps when simplifying expressions with rational exponents, especially when converting between radical and exponent form, as method marks may be available.
- 💡Check whether the question asks for an exact value or a simplified expression; leave answers in index form if instructed, or evaluate fully if a numerical answer is required.
- 💡When evaluating expressions like 16^(3/4), take the root first (fourth root of 16 is 2) then raise to the power 3 to get 8, as this reduces arithmetic errors.
- 💡Remember that the laws of indices apply only when the bases are the same; if bases differ, look for ways to express them as powers of a common base, e.g. 4^x = 2^(2x).
- 💡Always simplify surds fully before combining; look for the largest square factor to avoid multiple steps.
- 💡When rationalising a denominator with two terms, multiply by the conjugate and show the expansion clearly, as method marks are often awarded for the correct conjugate.
- 💡Check whether the question requires an exact answer in surd form; do not convert to a decimal unless asked, as this may lose accuracy marks.
- 💡For expressions like (a+√b)(a−√b), recognise the difference of two squares to simplify quickly to a² − b.
- 💡When asked to sketch a quadratic graph, mark the coordinates of the vertex and any intercepts clearly; a rough sketch with key points is usually sufficient.
- 💡If a question asks for the number of real roots, calculate the discriminant and state its value and sign before concluding; this shows your reasoning.
- 💡For equations quadratic in a function of the unknown, always define your substitution clearly and check that your final answers satisfy the original equation.
- 💡Use completing the square to find the vertex or solve equations when factorising is not obvious; it is a reliable method that also gives exact answers.
- 💡For one linear and one quadratic equation, always use substitution; elimination is usually not suitable.
- 💡Show your substitution step clearly, including the expansion and simplification, to gain method marks even if the final answer is wrong.
- 💡Check each solution pair in both original equations; this helps you spot arithmetic errors and confirms the number of solutions.
- 💡If the resulting quadratic has no real roots, state that there are no real solutions and relate this to the graphs not intersecting.
- 💡Always sketch the quadratic graph to decide between 'and' and 'or' before writing the final answer, especially if the x² coefficient is negative.
- 💡When solving inequalities with numerical fractions, multiply by the positive common denominator and show each step clearly.
- 💡For graphical inequalities, clearly label the boundary as dashed or solid and use a test point like (0, 0) to confirm which side to shade.
- 💡Lay out algebraic division clearly with aligned powers and bring down each term carefully.
- 💡When using the factor theorem, show the substitution and the result f(a) = 0 explicitly.
- 💡After simplifying a rational expression, check whether any value of x must be excluded.
- 💡Label all intercepts and asymptotes on sketches; examiners reward clear communication of key features.
- 💡When using graphs to solve equations, state the x-coordinates of intersection points clearly.
- 💡Show the substitution step clearly when forming a composite function, especially if the inner function is a fraction or contains a modulus.
- 💡When finding an inverse, always swap x and y explicitly before rearranging; this makes your method clear.
- 💡For graph questions, draw the line y = x as a guide and reflect key points to sketch the inverse accurately.
- 💡Sketch the original curve lightly first, then transform a few labelled key points to anchor the new graph.
- 💡Use a numerical check such as comparing f(0) with the transformed value to confirm the direction of a translation.
- 💡When several transformations are combined, state the order you are using and apply each one to the key points rather than redrawing freehand.
- 💡Factorise the denominator fully before deciding the partial fraction form.
- 💡Substitute roots of each factor first, then use coefficient comparison for any remaining constants.
- 💡Show the identity clearly and substitute back one value to verify your constants.
- 💡Read the modelling question carefully and underline the context, variables, and any data given before calculating.
- 💡When asked for limitations, think about assumptions made in deriving the function (e.g., constant ambient temperature) and how they might fail in reality.
- 💡For refinements, suggest a concrete mathematical change to the function or its domain, rather than just stating 'make it more accurate'.
Common Mistakes
- Error: assuming (a + b)^n = a^n + b^n, e.g. (4 + 9)^(1/2) = 2 + 3 = 5 instead of √13. Correction: the laws of indices apply to products and quotients, not sums; evaluate the sum first or leave as a single term.
- Error: mishandling negative exponents in fractions, e.g. simplifying 2x^(-1) as 1/(2x) instead of 2/x. Correction: the negative exponent applies only to the variable, so 2x^(-1) = 2 × (1/x) = 2/x.
- Error: confusing the addition law with multiplication, e.g. writing 2^3 × 2^4 = 2^12 instead of 2^7. Correction: when multiplying powers with the same base, add the exponents; when raising a power to a power, multiply them.
- Error: evaluating a^(m/n) by taking the power first and then the root, which can create large intermediate numbers, e.g. 16^(3/4) as (16^3)^(1/4) = 4096^(1/4) = 8. Correction: this is not wrong, but taking the root first, (16^(1/4))^3 = 2^3 = 8, is usually safer; either order gives the same value.
- Error: writing √(a+b) = √a + √b, e.g. √(9+16) = 3+4 = 7 instead of 5. Correction: the square root of a sum is not the sum of the square roots; evaluate the sum first or leave as a single surd.
- Error: multiplying only the numerator by the conjugate when rationalising, e.g. writing 1/(2+√3) = 2−√3 without multiplying the denominator. Correction: multiply both numerator and denominator by the conjugate: 1/(2+√3) = (2−√3)/((2+√3)(2−√3)) = (2−√3)/1 = 2−√3. The result is correct here, but the method must apply to both parts to be valid in general.
- Error: simplifying √(a² + b²) as a + b, e.g. √(3²+4²) = 3+4 = 7 instead of 5. Correction: this is a common error in Pythagoras; evaluate the sum of squares first, then take the square root.
- Error: incorrectly expanding brackets with surds, e.g. (√2+√3)² = 2+3 = 5 instead of 5+2√6. Correction: use (a+b)² = a² + 2ab + b², remembering that √2×√3 = √6.
- Forgetting to divide by a when completing the square if the coefficient of x² is not 1. Correction: always factor out the coefficient of x² from the first two terms before completing the square.
- Misinterpreting the discriminant when it is negative as meaning there are no solutions at all. Correction: a negative discriminant means no real roots, but there may be complex roots; for A-Level Mathematics, state that there are no real roots.
- Solving an equation like x⁴−5x²+4=0 by treating it as a quadratic in x without substitution, leading to errors. Correction: substitute u=x², solve the quadratic in u, then solve u=x² for x, remembering both positive and negative roots.
- Incorrectly identifying the vertex from completed-square form, for example reading (x−2)²−3 as vertex (2, 3) instead of (2, −3). Correction: the completed-square form a(x−h)²+k has vertex (h, k).
- When eliminating, forgetting to multiply the constant term as well as the variable terms. Correction: multiply the entire equation by the same factor.
- Substituting into the same equation instead of the other equation. Correction: always substitute the rearranged expression into the other equation to avoid a trivial identity.
- Losing solutions when solving the resulting quadratic, for example by dividing by a variable that could be zero. Correction: factorise or use the quadratic formula, and consider all cases.
- Not pairing x and y values correctly after solving for one variable. Correction: substitute each value back into the linear equation to find the corresponding value of the other variable.
- Forgetting to reverse the inequality sign when dividing by a negative number; correct by checking the final range with a test value.
- Assuming < 0 always means 'between the roots' for quadratics without checking if the x² coefficient is positive; correct by always sketching the specific parabola.
- Multiplying an inequality by an algebraic denominator without squaring it, which loses valid negative regions; correct by multiplying by the denominator squared and noting the denominator ≠ 0. This technique is beyond this specification and is not required here.
- Writing disjoint quadratic solutions as a single continuous interval (e.g., −2 > x > 2); correct by separating them with 'or' or using union set notation.
- Cancelling terms rather than factors in a rational expression; correct by factorising first and cancelling whole brackets.
- Sign errors when substituting into the factor theorem, especially with negative values; correct by writing brackets carefully.
- Forgetting to include a zero coefficient for missing powers during algebraic division; correct by inserting 0x² or similar placeholders.
- Drawing y = a/x or y = a/x² crossing an axis: correct by showing the axes as asymptotes and ensuring the curve approaches but never touches them.
- Reflecting the wrong part of a linear graph for the modulus: correct by reflecting only the section below the x-axis, keeping the section above unchanged.
- Assuming all proportional relationships are linear: correct by checking if the relationship involves powers, such as y = kx² or y = k/x².
- Assuming f(g(x)) = g(f(x)): correct by always composing in the stated order and simplifying carefully.
- Writing f⁻¹(x) as 1/f(x): correct by finding the inverse by swapping x and y and rearranging, not by taking the reciprocal.
- Forgetting to restrict the domain when a function is not one-to-one: correct by stating a suitable restricted domain so that the inverse is a function.
- Mixing up domain and range of f and f⁻¹: correct by remembering that the domain of f⁻¹ is the range of f, and vice versa.
- Treating f(x + a) as a shift right by a; correct this by testing a point, since f(x + 2) at x = 0 gives f(2), so the graph moves left.
- Applying a vertical stretch to x-coordinates or a horizontal stretch to y-coordinates; correct by remembering that changes inside the bracket affect x and changes outside affect y.
- Assuming a horizontal stretch of factor a rather than 1/a; correct by noting f(2x) halves each x-coordinate, so the stretch factor is 1/2.
- Using only one term for a repeated factor such as (x − 1)^2; correct this by including both A/(x − 1) and B/(x − 1)^2.
- Forgetting to multiply every term by the full denominator when forming the identity; correct by writing the identity explicitly before substituting.
- Choosing an x value that is a root of a factor but then failing to substitute it into the full identity, or choosing a value that makes the algebra unnecessarily complicated; correct by substituting roots of the factors first, since this makes other terms vanish, and using coefficient comparison only for any remaining constants.
- Ignoring the real-world context and treating the function purely algebraically. Correction: always relate variables and parameters to the situation, explicitly stating units and physical meaning.
- Assuming the model is valid for all values of the variable, including negative time or extreme future values. Correction: identify the domain over which the model is intended and check whether predictions outside it are sensible.
- Misinterpreting exponential decay models by claiming they assume a 'constant cooling rate'. Correction: recognise that in models like $T = A + Be^{-kt}$, the rate of change is proportional to the difference from the ambient value; the rate itself decreases over time. State valid limitations like ignoring evaporation instead.