Quadratic & Other Graphs

Recognising a Quadratic Graph

A quadratic function has the form $y = ax^2 + bx + c$ where $a \neq 0$. Its graph is always a parabola — a smooth, symmetrical U-shape.

If $a > 0$ the parabola opens upwards (a valley, with a minimum point).
If $a < 0$ it opens downwards (a hill, with a maximum point).

The bigger $|a|$ is, the narrower the curve. The constant $c$ is the $y$-intercept, because when $x = 0$ we get $y = c$.

Key terms Parabola — the U-shaped curve of any quadratic.

Root — a value of $x$ where the curve crosses the $x$-axis, i.e. where $y = 0$.

Turning point — the single lowest (minimum) or highest (maximum) point.

Line of symmetry — the vertical line through the turning point; the curve is a mirror image across it.

Roots, Turning Point and Symmetry

To plot a quadratic, build a table of values, substituting each $x$ carefully (watch negative-squared signs), then join the points with a smooth curve — never straight line segments.

The Turning Point by Completing the Square

Completing the square rewrites the quadratic as $y = a(x - h)^2 + k$. The turning point is then read straight off as $(h,\,k)$, and the line of symmetry is $x = h$.

Worked example Find the turning point and line of symmetry of $y = x^2 - 6x + 11$.

Halve the $x$-coefficient: $-6 \div 2 = -3$.

Exam tip The line of symmetry always sits exactly halfway between the two roots. If you know the roots are $x = 1$ and $x = 5$, the symmetry line is $x = \tfrac{1+5}{2} = 3$ — no algebra needed.

Solving Quadratics Graphically

The roots of $ax^2 + bx + c = 0$ are simply where the graph $y = ax^2 + bx + c$ crosses the $x$-axis. Read those $x$-values off the plot.

To solve a different equation, say $x^2 - 3x - 1 = 0$ using a graph of $y = x^2 - 3x - 4$, rearrange so one side matches the drawn curve:

$$x^2 - 3x - 1 = 0 \;\Rightarrow\; x^2 - 3x - 4 = -3$$

So draw the straight line $y = -3$ and read the $x$-coordinates where it meets the curve. Those intersection points are the solutions.

Watch out A quadratic can have two, one, or zero real roots. One root means the curve just touches the axis (the turning point sits on it); zero roots means the whole curve floats above or below the axis.

Cubic Graphs

A cubic has the form $y = ax^3 + bx^2 + cx + d$. Its graph has a characteristic stretched-S shape with up to two turning points and can cross the $x$-axis up to three times.

If $a > 0$ the curve rises from bottom-left to top-right.
If $a < 0$ it falls from top-left to bottom-right.

Reciprocal Graphs

A reciprocal graph has the form $y = \dfrac{k}{x}$ (with $k \neq 0$). It forms two separate curves called a hyperbola.

The graph never touches the axes — they are asymptotes that the curve approaches but never reaches.
For $k > 0$ the branches sit in the top-right and bottom-left.
There is no value at $x = 0$, since dividing by zero is undefined.

Exponential Graphs

An exponential graph has the form $y = k\,a^x$ where $a > 0$. This models growth (when $a > 1$) and decay (when $0 < a < 1$).

The curve always passes through $(0, k)$, because $a^0 = 1$.
It rises (or falls) ever more steeply and has the $x$-axis as a horizontal asymptote — it gets very close to but never touches $y = 0$.

Key terms Asymptote — a line the curve approaches infinitely closely but never meets. Reciprocal and exponential graphs both have them.

The Graph of a Circle

The equation $x^2 + y^2 = r^2$ gives a circle centred on the origin with radius $r$. For example $x^2 + y^2 = 25$ is a circle of radius $5$, since $25 = 5^2$.

Exam tip Don't mistake $x^2 + y^2 = 25$ for a quadratic curve. The right-hand side is the radius squared, so the radius is $\sqrt{25} = 5$, not $25$.

Gradients and Areas from Graphs

For a curve, the gradient at a point is found by drawing a tangent — a straight line just touching the curve there — and calculating its gradient as $\dfrac{\text{rise}}{\text{run}}$ using two clear points on the tangent.

The area under a curve between two $x$-values can be estimated by splitting the region into strips (trapezia or counting squares) and adding them up.

Real-Life Graphs

These ideas have direct physical meaning in motion graphs.

On a distance–time graph, a steeper line is faster; a horizontal line is stationary; a curve means changing speed.
On a speed–time graph, a positive gradient is acceleration, a negative gradient is deceleration, and the area underneath gives total distance.

Worked example A car travels at a steady $20\ \text{m/s}$ for $30\ \text{s}$, shown as a horizontal line on a speed–time graph. The distance is the area of the rectangle beneath it:

$\text{distance} = 20 \times 30 = 600\ \text{m}$.

If the line then slopes down to $0$ over the next $10\ \text{s}$, that braking distance is a triangle: $\tfrac{1}{2} \times 10 \times 20 = 100\ \text{m}$, giving $700\ \text{m}$ in total.

Real world Engineers read speed–time graphs straight off a vehicle's data to find both how hard it accelerated (gradient) and how far it went (area) — the same maths you use in the exam runs inside a car's safety systems.