Recognising and sketching quadratic, cubic, reciprocal and exponential graphs, reading roots and turning points, and using function notation at Higher tier.

What this dot point is asking

AQA wants you to recognise the standard shapes of quadratic, cubic, reciprocal and exponential graphs, sketch them, read roots and turning points, and at Higher tier use function notation including composite and inverse functions. Recognising a graph from its equation (and the reverse) is a recurring multiple-choice and short-answer skill, and function notation is a reliable Higher topic.

Recognising the standard graph shapes

Each family of equations has a fingerprint shape you should know on sight.

• Quadratic $y = ax^2 + bx + c$: a parabola with one turning point and a vertical line of symmetry. Positive $a$ opens upward (minimum), negative $a$ opens downward (maximum).
• Cubic $y = ax^3 + bx^2 + cx + d$: typically an S-shape. A simple $y = x^3$ rises steeply, flattens near the origin, then rises steeply again, passing through $(0,0)$.
• Reciprocal $y = \dfrac{k}{x}$: two branches in opposite quadrants, never crossing the axes. The $x$-axis and $y$-axis are asymptotes (the curve approaches but never reaches them).
• Exponential $y = k^x$ with $k > 0$: passes through $(0, 1)$, rises rapidly if $k > 1$ (growth) or falls toward zero if $0 < k < 1$ (decay), with the $x$-axis as an asymptote.

Sketching a quadratic with key features

To sketch $y = ax^2 + bx + c$, mark the $y$-intercept at $(0, c)$, find the roots by factorising or using the formula, and locate the turning point on the line of symmetry $x = -\tfrac{b}{2a}$. The turning point sits midway between the two roots.

Reading roots and turning points

The roots are where $y = 0$ (the $x$-intercepts), and they tell you the solutions of the corresponding equation. The turning point gives the maximum or minimum value of $y$: for a profit or height model this is often the key answer. Completing the square is the quickest algebraic route to the turning point because $y = (x - p)^2 + q$ has its vertex at $(p, q)$.

Function notation at Higher tier

A function $f(x)$ is a rule that turns an input into an output. Evaluating means substituting: if $f(x) = 2x + 1$ then $f(3) = 7$. A composite function applies one function then another: $fg(x)$ means apply $g$ first, then $f$, so if $g(x) = x^2$ then $fg(x) = 2x^2 + 1$. The inverse $f^{-1}(x)$ reverses $f$: to find it, write $y = 2x + 1$, swap to $x = 2y + 1$, and rearrange to $y = \tfrac{x - 1}{2}$, so $f^{-1}(x) = \tfrac{x - 1}{2}$.

Using graphs to solve equations

A powerful idea is that the solutions of an equation are where two graphs cross. To solve $x^2 - 3 = x + 1$ graphically, plot $y = x^2 - 3$ and $y = x + 1$ and read off the $x$-coordinates of the intersection points. The roots of $f(x) = 0$ are simply where the graph of $y = f(x)$ crosses the $x$-axis. This means a sketch can confirm how many solutions an equation has: a line crossing a parabola twice gives two solutions, touching it gives one, and missing it gives none. Exam questions often supply a curve and ask you to draw a straight line whose intersections solve a given equation.

Real-world graph shapes

The standard curves model real situations. Exponential growth ($y = k^x$ with $k > 1$) describes compound interest and population growth, where the increase accelerates over time, while exponential decay ($0 < k < 1$) describes depreciation and radioactive decay, falling fast then levelling toward zero. A reciprocal graph models inverse proportion, such as the time for a journey against speed. Recognising which shape fits a described scenario, and reading meaning from its features (the intercept as a starting value, the steepness as a rate), turns the abstract families into useful tools.