A sketch is a quick, reasonably accurate diagram showing only the key features of a graph — its general shape, where it crosses the axes, and any symmetry. It is not built from a table of plotted values, and at Core level, turning points never need to be calculated exactly — only their approximate horizontal position, found from symmetry.

Mark just two points, then rule a straight line through both, extending it in both directions:

• the $y$-intercept, $(0,c)$;
• the $x$-intercept, found by setting $y=0$.

Sketch $y=2x-4$: setting $x=0$ gives $y=-4$, so it passes through $(0,-4)$; setting $y=0$ gives $x=2$, so it passes through $(2,0)$. Since the gradient is positive, the line slopes upward left to right.

A quadratic graph $y=ax^2+bx+c$ is a parabola, and the sign of $a$ (the coefficient of $x^2$) decides its direction:

• $a>0$: opens upwards (∪ shape, minimum point).
• $a<0$: opens downwards (∩ shape, maximum point).

   a > 0 (∪ shape)        a < 0 (∩ shape)
   \    |    /                 ___
    \   |   /                 /   \
     \__|__/                 /_____\

Checking this sign should always be the very first step of any quadratic sketch — getting the direction wrong makes every other correct feature (roots, symmetry) land on an upside-down curve.

The roots (where the curve crosses the $x$-axis) come from solving $ax^2+bx+c=0$, and the curve's line of symmetry passes exactly halfway between them.

Sketch $y=(x-1)(x-5)$, showing the $x$-intercepts and line of symmetry:

1. The expanded coefficient of $x^2$ is positive, so the parabola opens upwards.
2. Roots: $x-1=0 \implies x=1$, and $x-5=0 \implies x=5$.
3. Line of symmetry, halfway between the roots:

1. Sketch a U-shaped curve through $(1,0)$ and $(5,0)$, with its lowest point directly above $x=3$.

If the two roots of a quadratic are equal (a repeated root), the curve doesn't cross the $x$-axis at two separate points — it just touches it once, right at its turning point, before curving back away.

Sketch $y=(x-4)^2$: this is $(x-4)(x-4)$, so both roots equal $4$. The curve opens upwards (coefficient of $x^2$ is positive) and touches the $x$-axis at exactly one point, $(4,0)$, which is also its minimum point and its own line of symmetry.

A quadratic graph has roots at $x=-2$ and $x=6$, and opens downwards. State the $x$-coordinate of its line of symmetry, and sketch its general shape.

Solution

Line of symmetry: $\dfrac{-2+6}{2}=2$. Since the parabola opens downwards, sketch an n-shaped curve crossing the $x$-axis at $(-2,0)$ and $(6,0)$, with its highest point directly above $x=2$ — no coordinates for that maximum point are required at Core level, only its horizontal position.