2 Algebra and functions — 9MA0 spec coverage

2.1

Understand and use the laws of indices for all rational exponents.

For a non-zero base, $a^m a^n=a^{m+n}$ , $a^m/a^n=a^{m-n}$ and $(a^m)^n=a^{mn}$ .
Interpret rational powers through roots: $a^{p/q}=\sqrt[q]{a^p}=(\sqrt[q]{a})^p$ whenever the real-valued expression is defined.
For example, $16^{3/4}=(\sqrt[4]{16})^3=2^3=8$ .
A negative exponent means reciprocal, not a negative value: $a^{-r}=1/a^r$ ; also avoid applying index laws across addition.

Tier 1 · Easy

2 marks

ORIGINAL

Evaluate $81^{3/4}$ without using a calculator.

Tier 2 · Standard

2 marks

ORIGINAL

Given $x>0$ , simplify $\dfrac{x^{5/2}}{x^{-1/2}}$ to a single power of $x$ .

Tier 3 · Hard

4 marks

ORIGINAL

Solve $8^{2x-1}=4^{x+2}$ , giving the exact value of $x$ .

2.2

Use and manipulate surds, including rationalising the denominator.

A surd is an exact irrational root; simplify it by extracting the largest square factor, as in $\sqrt{50}=5\sqrt2$ .
Combine only like surds, and rationalise a binomial denominator by multiplying numerator and denominator by its conjugate.
For example, $1/(3+\sqrt2)=(3-\sqrt2)/(9-2)=(3-\sqrt2)/7$ .
Do not split roots across addition: $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ ; after squaring an equation, check every candidate in the original.

Tier 1 · Easy

2 marks

ORIGINAL

Write $\sqrt{72}$ in the form $a\sqrt2$ , where $a$ is an integer.

Tier 2 · Standard

3 marks

ORIGINAL

Rationalise and simplify $\dfrac{5}{2+\sqrt3}$ .

Tier 3 · Hard

5 marks

ORIGINAL

Solve $\sqrt{x+6}-\sqrt{x-3}=1$ .

2.3

Work with quadratic functions and their graphs; the discriminant, including conditions for real and repeated roots; completing the square; solution of quadratic equations, including solving quadratic equations in a function of the unknown.

For $ax^2+bx+c$ , the discriminant $b^2-4ac$ is positive for two distinct real roots, zero for a repeated root and negative for no real roots.
Complete the square to expose the turning point and range, or use factorisation and the quadratic formula to solve equations.
For example, $x^2-6x+11=(x-3)^2+2$ , so its minimum point is $(3,2)$ and it has no real roots.
When the quadratic is in a function of the unknown, substitute a new variable, solve the quadratic in that variable, then solve every resulting equation and reject invalid roots.

Tier 1 · Easy

2 marks

ORIGINAL

Write $x^2-8x+5$ in completed-square form.

Tier 2 · Standard

3 marks

ORIGINAL

Find the two values of $k$ for which $2x^2+(k-1)x+8=0$ has a repeated root.

Tier 3 · Hard

6 marks

ORIGINAL

Solve $(x^2-5x)^2-5(x^2-5x)-14=0$ , giving exact answers.

2.4

Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

A simultaneous solution is an ordered pair satisfying every equation; graphically, solutions are intersection points.
Use elimination when coefficients align conveniently, or substitute an expression from the linear equation into the nonlinear equation.
For example, substituting $y=5-x$ into $xy=6$ gives $x(5-x)=6$ , whose roots generate the two intersection pairs.
After finding one coordinate, substitute each possible value back separately; losing the second quadratic root or mismatching coordinates are common errors.

Tier 1 · Easy

2 marks

ORIGINAL

Solve $x+y=9$ and $x-y=1$ simultaneously.

Tier 2 · Standard

4 marks

ORIGINAL

Find every solution of $y=x+1$ and $x^2+y^2=25$ .

Tier 3 · Hard

5 marks

ORIGINAL

Solve the system $x+2y=7$ and $x^2+y^2=13$ .

2.5

Solve linear and quadratic inequalities in one variable and interpret them graphically, including with brackets and fractions; express solutions using 'and'/'or' or set notation; represent linear and quadratic inequalities graphically.

An inequality solution is a set of values; endpoints are included for $\leq$ or $\geq$ and excluded for $<$ or $>$ .
For a quadratic or rational inequality, locate every zero and undefined value, then use a sign diagram or graph on the resulting intervals.
For example, $(x-2)(x+3)<0$ between its roots, so $-3<x<2$ .
Multiplying by an expression of unknown sign can reverse the inequality unpredictably; bring a fractional inequality to one side and analyse signs instead.

Tier 1 · Easy

2 marks

ORIGINAL

Solve $5-2x<11$ .

Tier 2 · Standard

3 marks

ORIGINAL

Solve $(x-4)(x+1)\leq0$ .

Tier 3 · Hard

4 marks

ORIGINAL

Solve $\dfrac{x+2}{x-3}>2$ and give the result in set notation.

2.6

Manipulate polynomials algebraically, including expanding brackets, collecting like terms, factorisation and simple algebraic division; use the factor theorem; simplify rational expressions by factorising, cancelling and algebraic division.

Polynomial manipulation uses expansion, collection and factorisation while preserving equality; choose the form that exposes the required structure.
The factor theorem states that $x-a$ is a factor of $f(x)$ exactly when $f(a)=0$ ; after finding a factor, divide to obtain the remaining polynomial.
For example, $f(2)=0$ for $f(x)=x^3-3x^2-4x+12$ , and division by $x-2$ gives $x^2-x-6$ .
Cancel factors, not separate terms, in rational expressions, and retain exclusions from the original denominator even when a factor cancels.

Tier 1 · Easy

2 marks

ORIGINAL

Factorise $x^2-9x+20$ completely.

Tier 2 · Standard

4 marks

ORIGINAL

Use the factor theorem to factorise $x^3-4x^2+x+6$ fully.

Tier 3 · Hard

5 marks

ORIGINAL

Simplify $\dfrac{x^3-4x^2+x+6}{x^2-4x+4}$ as far as possible, then use algebraic division. State any excluded value.

2.7

Understand and use graphs of functions; sketch curves including polynomials, the modulus of a linear function, y = a/x and y = a/x² with asymptotes; use graph intersections to solve equations; understand proportional relationships.

A useful sketch records intercepts, turning points, end behaviour, symmetry and asymptotes rather than relying on a table of isolated points.
For $y=a/x$ the coordinate axes are asymptotes and the relation is inverse proportionality; $y=a/x^2$ is inverse-square, symmetric about the $y$ -axis and has the same sign as $a$ .
An equation $f(x)=g(x)$ is solved graphically at the $x$ -coordinates where the two graphs intersect; modulus reflects negative outputs above the $x$ -axis.
Do not draw a reciprocal curve crossing an asymptote, and distinguish direct proportionality $y=kx$ from a relation that merely has positive correlation.

Tier 1 · Easy

2 marks

ORIGINAL

For the curve $y=-\dfrac{6}{x}$ , state both asymptotes and the two quadrants containing its branches.

Tier 2 · Standard

4 marks

ORIGINAL

Find the intersection values of $x$ for the graphs $y=|2x-3|$ and $y=x+4$ .

Tier 3 · Hard

4 marks

ORIGINAL

The curves $y=\dfrac4x$ and $y=x+3$ meet twice. Determine the exact coordinates of both intersections.

2.8

Understand and use composite functions; inverse functions and their graphs.

The composite $fg(x)$ means $f(g(x))$ : apply the right-hand function first and keep its whole output inside the outer function.
To find an inverse, write $y=f(x)$ , rearrange for $x$ , then exchange the variable labels; a restricted domain may be needed to make $f$ one-to-one.
A function and its inverse have graphs reflected in $y=x$ , with domain and range interchanged.
In general $fg\ne gf$ ; for inverse quadratics, use the stated domain to choose the correct square-root branch.

Tier 1 · Easy

2 marks

ORIGINAL

Let $f(x)=3x-1$ and $g(x)=x^2+2$ . Calculate $fg(2)$ .

Tier 2 · Standard

4 marks

ORIGINAL

For $f(x)=\dfrac{3x+2}{x-1}$ , find $f^{-1}(x)$ and state its domain.

Tier 3 · Hard

6 marks

ORIGINAL

The function $f(x)=(x-3)^2+1$ has domain $x\geq3$ . Find $f^{-1}(x)$ , state its domain, and solve $f^{-1}(2x+1)=x+3$ .

2.9

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.

Transformations outside $f$ act on output coordinates: $y=af(x)$ scales vertically by factor $|a|$ and reflects in the $x$ -axis when $a<0$ .
Transformations inside $f$ act oppositely on inputs: $f(x+a)$ moves left by $a$ , while $f(ax)$ scales horizontally by factor $1/|a|$ and may reflect in the $y$ -axis.
For example, a point $(u,v)$ on $y=f(x)$ maps to $(u/2,v+3)$ on $y=f(2x)+3$ .
Apply combined transformations to coordinates or rewrite the complete formula; describing $f(x+a)$ as a shift right is the standard sign error.

Tier 1 · Easy

2 marks

ORIGINAL

Describe fully the transformation from $y=f(x)$ to $y=f(x-4)+2$ .

Tier 2 · Standard

3 marks

ORIGINAL

The point $(6,-1)$ lies on $y=f(x)$ . Find the corresponding point on $y=-2f(3x)+1$ .

Tier 3 · Hard

5 marks

ORIGINAL

Let $f(x)=x^2-4x+1$ . For $g(x)=-f(2x+2)+4$ , find the turning point and both $x$ -intercepts, then state whether it is a maximum or minimum.

2.10

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).

Factor the denominator first and assign one partial-fraction term to each linear factor.
A repeated factor needs a term for every power: $(x-a)^2$ requires $A/(x-a)+B/(x-a)^2$ .
After multiplying through by the common denominator, substitute convenient roots or compare coefficients to determine the constants.
If the numerator degree is at least the denominator degree, divide first; omitting a repeated-factor term produces an identity that cannot hold.

Tier 1 · Easy

3 marks

ORIGINAL

Express $\dfrac7{(x+1)(x+2)}$ as partial fractions.

Tier 2 · Standard

4 marks

ORIGINAL

Decompose $\dfrac{5x+1}{(x-1)(x+2)}$ into partial fractions.

Tier 3 · Hard

5 marks

ORIGINAL

Express $\dfrac{5x+1}{(x-1)^2(x+2)}$ as three partial-fraction terms.

2.11

Use of functions in modelling, including consideration of limitations and refinements of the models.

A function model links clearly defined variables over a stated domain; parameters should have units or contextual meaning where possible.
Use the given data to determine parameters, then calculate with the model and interpret the result in context with appropriate rounding.
For example, in $P(t)=P_0r^t$ , $P_0$ is the value at $t=0$ and $r$ is the multiplier per time interval.
A model is an approximation: check whether extrapolation, a fixed growth rate, ignored constraints or measurement variation makes a prediction unreliable, and suggest a refinement tied to that weakness.

Tier 1 · Easy

2 marks

ORIGINAL

A colony is modelled by $P(t)=120(1.08)^t$ , where $t$ is measured in days. Estimate the number of organisms after $3$ days.

Tier 2 · Standard

5 marks

ORIGINAL

A cooling model is $T(t)=18+62e^{-0.15t}$ , where $T$ is in degrees Celsius and $t$ is in minutes. Find the first whole minute for which the model gives $T<30$ , and state one limitation.

Tier 3 · Hard

7 marks

ORIGINAL

The cross-section of a greenhouse roof is modelled by $h(x)=kx(30-x)$ for $0\leq x\leq30$ , with distances in metres. The point $(5,10)$ lies on the model. Find $k$ , the maximum modelled height, and the horizontal width over which $h(x)\geq12$ . Give one limitation of the model.