Algebra and Graphs

The One Thing

Algebra and graphs provide complementary ways to represent and analyse quantities, relationships, patterns, equations, inequalities, sequences, and functions. Algebraic manipulation must preserve equality or the truth of an inequality, while graphs reveal features such as intercepts, gradients, turning points, areas, and intersections.

Definitions and Results

• Algebraic expression: A combination of numbers, variables, and operations without an equals sign, such as $3x^2-5x+2$.
• Term, coefficient, and constant: A term is part of an expression separated by addition or subtraction; its coefficient multiplies the variable, while a constant has no variable.
• Like terms: Terms with the same variables raised to the same powers. They can be combined by adding or subtracting their coefficients.
• Expansion: Removing brackets by multiplying every term, using the distributive law or special identities.
• Factorisation: Writing an expression as a product of factors, often by taking out a common factor or using the difference of two squares.
• Index laws: For powers with the same base,

$$a^m\times a^n=a^{m+n},a^m÷a^n=a^{m-n},(a^m{)}^n=a^{mn}.$$

• Surd: An irrational root that cannot be written exactly as a terminating or recurring decimal, such as $\sqrt{2}$. Surds may be simplified and rationalised.
• Linear equation: An equation in which the variable has a maximum power of one, usually producing a straight-line relationship.
• Quadratic equation: An equation of the form

$$a{x}^2+bx+c=0,$$ where $a\ne 0$.

• Quadratic formula: The roots of $a{x}^2+bx+c=0$ are

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Discriminant: The expression $b^2-4ac$ in a quadratic equation. It determines the nature of the roots:

- $b^2-4ac>0$: two distinct real roots; - $b^2-4ac=0$: one repeated real root; - $b^2-4ac<0$: no real roots.

• Inequality: A statement comparing quantities using $<$, $>$, $\le$, or $\ge$.
• Simultaneous equations: Two or more equations solved together because their variables must satisfy all equations at the same time.
• Sequence: An ordered list of numbers following a rule or pattern.
• Arithmetic sequence: A sequence with a constant difference $d$ between consecutive terms. Its $n$th term is

$$a+(n-1)d.$$

• Geometric sequence: A sequence with a constant multiplier or common ratio $r$. Its $n$th term is

$$a{r}^{n-1}.$$

• Function: A rule assigning exactly one output to each allowed input.
• Domain and range: The domain is the set of allowed input values; the range is the set of corresponding output values.
• Composite function: A function formed by applying one function after another, written $f(g(x))$.
• Inverse function: A function that reverses the input-output process of another function, where it exists.
• Gradient: The steepness of a straight line, calculated as change in $y$ divided by change in $x$.
• Y-intercept and x-intercept: The $y$-intercept is where a graph crosses the $y$-axis; the $x$-intercept is where it crosses the $x$-axis.
• Gradient-intercept form: A straight-line equation written as

$$y=mx+c,$$ where $m$ is the gradient and $c$ is the $y$-intercept.

• Transformation of graphs: A change such as a translation, reflection, stretch, or compression that alters a graph’s position or shape.
• Graph interpretation: Using coordinates, intercepts, gradients, turning points, intersections, and other features to understand mathematical or real-world relationships.

Worked Methods

1. Algebraic manipulation

• Apply the order of operations: brackets, powers, multiplication or division, then addition or subtraction.
• Identify terms, coefficients, constants, and like terms.
• When simplifying, combine only like terms.
• When expanding, multiply every term inside a bracket by the term outside it.
• Use the identities

$$(a+b{)}^2=a^2+2ab+b^2,$$ $$(a-b{)}^2=a^2-2ab+b^2,$$ $$(a+b)(a-b)=a^2-b^2.$$

• When factorising, take out a common factor or use an appropriate pattern, such as the difference of two squares.
• Apply the index laws when multiplying, dividing, or raising powers.

Different forms communicate different information: expanded form helps identify coefficients, factorised form helps find roots, and completed-square form helps identify a quadratic’s turning point.

2. Solving equations and inequalities

• Perform the same valid operation on both sides of an equation.
• Continue until the unknown is isolated.
• Check the solution in the original equation.
• For inequalities, maintain the direction of the inequality under positive operations.
• When multiplying or dividing by a negative number, reverse the inequality sign.
• Check restrictions, including zero denominators, square-root conditions, domain limits, and the original equation.

Algebraic manipulation must preserve equality or the truth of an inequality; every operation should be applied consistently.

3. Solving quadratic equations

For $$a{x}^2+bx+c=0,$$ use $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Identify $a$, $b$, and $c$.
• Calculate the discriminant $b^2-4ac$.
• Use the discriminant to determine whether there are two distinct real roots, one repeated real root, or no real roots.
• Substitute into the quadratic formula.
• Simplify both values arising from the $\pm$ sign.
• Check the solutions in the original equation where appropriate.

A quadratic graph has the form $$y=a{x}^2+bx+c.$$ If $a>0$, it opens upward; if $a<0$, it opens downward. Its axis of symmetry is $$x=-\frac{b}{2a}.$$

4. Solving simultaneous equations

• Use the equations together because the variables must satisfy both at the same time.
• Eliminate one variable by substitution or by combining equations.
• Solve for the remaining variable.
• Substitute back to find the other variable.
• Check the ordered pair in both original equations.

The intersection of the graphs of the equations represents the simultaneous solution.

5. Finding sequence rules

Arithmetic sequence

• Identify the first term $a$.
• Find the constant difference $d$.
• Use

$$\text{nth term}=a+(n-1)d.$$

• Substitute the required value of $n$.

Geometric sequence

• Identify the first term $a$.
• Find the common ratio $r$.
• Use

$$\text{nth term}=a{r}^{n-1}.$$

• Substitute the required value of $n$.

These rules allow unknown terms and long-term behaviour to be found efficiently.

6. Working with functions

• Represent the function using a formula, mapping diagram, table, set of ordered pairs, or graph.
• Identify the allowed inputs, which form the domain.
• Identify the corresponding outputs, which form the range.
• For a composite function, apply the inner function first and then the outer function:

$$f(g(x)).$$

• To find the inverse of a one-to-one function:

- write $y=f(x)$; - interchange $x$ and $y$; - solve for $y$.

The inverse reverses the input-output process and exists in this form only where the function is one-to-one.

7. Finding and using straight-line features

For a line through $(x_1,y_1)$ and $(x_2,y_2)$, calculate the gradient using $$m=\frac{y_2-y_1}{x_2-x_1}.$$

Then write the equation in gradient-intercept form: $$y=mx+c.$$

• Find the gradient $m$.
• Identify or calculate the $y$-intercept $c$.
• Substitute into $y=mx+c$.
• Find the $x$-intercept by setting $y=0$, and the $y$-intercept by setting $x=0$.

Parallel lines have equal gradients. Perpendicular non-vertical lines have gradients whose product is $-1$.

8. Transforming graphs

Apply the specified transformation to the original function:

• $y=f(x)+a$: vertical movement by $a$ units.
• $y=f(x-a)$: horizontal movement by $a$ units to the right.
• $y=-f(x)$: reflection in the $x$-axis.
• $y=f(-x)$: reflection in the $y$-axis.

Transformations may also include stretches and compressions, which alter the graph’s shape.

9. Interpreting graphs

• Label the axes and use suitable scales.
• Plot coordinates accurately and include units where appropriate.
• Identify intercepts, gradients, turning points, and intersections.
• Interpret each feature in context.
• For a velocity-time graph, calculate displacement from the area under the graph.
• For a distance-time graph, interpret speed from its gradient.
• Interpret intersections as simultaneous solutions to the equations represented by the graphs.

The meaning of a graph depends on its context; gradients, intercepts, areas, and turning points must therefore be assigned appropriate real-world meanings and units.

Where It Goes Wrong

• Combining unlike terms instead of only combining like terms with the same variables raised to the same powers.
• Failing to multiply every term inside a bracket when expanding.
• Applying an operation to only one side of an equation, rather than applying the same valid operation to both sides.
• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
• Using quadratic results without considering the discriminant, restrictions such as zero denominators and square-root conditions, domain limits, or checking in the original equation.
• Misreading graph features by ignoring axis labels, scales, units, or the contextual meaning of gradients, intercepts, areas, turning points, and intersections.

What Gets Asked

This material supports questions requiring students to:

• Simplify algebraic expressions using order of operations, like terms, expansion, factorisation, identities, and index laws.
• Identify terms, coefficients, constants, domains, ranges, intercepts, gradients, and other structural features.
• Solve linear equations, inequalities, quadratic equations, and simultaneous equations.
• Use the quadratic formula and discriminant to find and classify roots.
• Determine the $n$th term of arithmetic and geometric sequences.
• Evaluate composite functions and find inverse functions where they exist.
• Find the equation of a straight line, its gradient, $x$-intercept, and $y$-intercept.
• Identify whether lines are parallel or perpendicular from their gradients.
• Describe and apply graph transformations, including translations and reflections.
• Interpret coordinates, gradients, turning points, intersections, areas under velocity-time graphs, and gradients of distance-time graphs.
• Construct accurate graphs using labelled axes, appropriate scales, accurate plotting, and suitable units.