Algebra · equations and reasoning

Explanation

• An equation is satisfied by particular values, whereas an identity states that two expressions are equivalent for every permitted value.
• To show expressions are equivalent, expand or factorise one side until it matches the other.
• Foundation tier: use algebra to support and construct an argument by defining quantities with variables, translating the claim and linking the resulting algebra back to it.
• Higher tier: extend this to a proof by using a general form, such as $2n$ for an even integer or $2n+1$ for an odd integer, and reasoning that covers every permitted case.
• Checking examples alone is not a proof.

Explanation

• A function maps each allowed input to one output. To evaluate $f(a)$, substitute the complete input $a$ into every occurrence of the variable and then simplify.
• A function can be represented by a rule, mapping diagram or input-output table.
• Higher tier: an inverse function reverses a one-to-one function, while a composite applies two functions successively; in $fg(x)=f(g(x))$, $g$ acts first.
• Formal function notation is expected for those extensions.
• Examiners require careful brackets when the input is an expression, because the whole input replaces $x$.

Explanation

• A coordinate $(x,y)$ gives horizontal position first and vertical position second. Positive $x$ is right, negative $x$ is left, positive $y$ is up and negative $y$ is down.
• The signs identify the quadrant, numbered anticlockwise from the top right.
• Find a displacement by subtracting starting coordinates from ending coordinates.
• Find a midpoint by averaging the two $x$-coordinates and separately averaging the two $y$-coordinates.
• Examiners expect coordinate order and negative signs to be preserved, with the two component calculations shown clearly.

Explanation

• A non-vertical straight line has equation $y=mx+c$, where $m$ is the gradient and $(0,c)$ is the $y$-intercept. Plot a line by calculating at least two accurate coordinate pairs and joining them carefully with a ruler.
• From two points, find $m=\dfrac{y_2-y_1}{x_2-x_1}$, then substitute either point to find $c$.
• Parallel lines have equal gradients.
• Higher tier: perpendicular non-vertical lines have gradients whose product is $-1$.
• Examiners expect the gradient calculation, substitution for the intercept and a final equation in a requested form.

Explanation

• The gradient of a linear function is the change in the vertical quantity per unit change in the horizontal quantity. Its sign shows whether the line rises or falls.
• The $y$-intercept is the output when $x=0$; the $x$-intercept is where the output is zero.
• Read axis scales and units before calculating or interpreting either feature.
• In a context, a gradient is a rate and an intercept is often an initial value or fixed charge.
• Examiners require a value, its unit and a sentence explaining what it represents.

Explanation

• A root of a quadratic is an $x$-coordinate where its graph meets the $x$-axis, so $y=0$. The $y$-intercept is found by setting $x=0$.
• A turning point is the maximum or minimum point, and its vertical line is the axis of symmetry.
• Factorising can reveal roots algebraically and the graph confirms their positions.
• Higher tier: completing the square into $a(x-h)^2+k$ reveals turning point $(h,k)$.
• Examiners expect coordinates for points, equations for axes, and algebraic working when roots are to be deduced.

Explanation

• Recognise graph families from their defining shapes and features: linear graphs have constant gradient, quadratics are symmetric parabolas, simple cubics have an S-shape, and $y=\dfrac{1}{x}$ has two reciprocal branches with asymptotes $x=0$ and $y=0$.
• Sketch by marking intercepts, roots, turning points, asymptotes and representative values.
• Higher tier: also recognise exponentials and sine, cosine and tangent graphs for angles of any size, using their periods and standard values.
• Examiners expect a sketch to show correct shape and position, not merely a collection of plotted points.

Explanation

• For $y=f(x)+a$, translate the graph vertically by vector $\begin{pmatrix}0\\a\end{pmatrix}$. For $y=f(x-a)$, translate it horizontally by $\begin{pmatrix}a\\0\end{pmatrix}$; the sign inside the function appears opposite to the movement.
• The graph $y=-f(x)$ is the reflection of $y=f(x)$ in the $x$-axis, while $y=f(-x)$ is its reflection in the $y$-axis.
• Track distinctive points, including intercepts and turning points, and preserve the graph's exact shape and scale.
• A point $(p,q)$ provides a reliable coordinate check after transforming.
• Examiners expect a fully described transformation, including the correct axis or translation vector.