For $y=ax^2+bx+c$, the sign of $a$ tells you the shape immediately: $a>0$ gives a U-shape (minimum point); $a<0$ gives an N-shape (maximum point). Checking this before plotting gives you a built-in error check — if your calculated points look like the wrong shape, you've made a substitution mistake.

Substitute each required $x$-value into the equation. The single biggest source of error is substituting negative $x$-values without brackets.

Example: For $y=-x^2+8x$ at $x=-3$: correctly, $y=-(-3)^2+8(-3) = -(9)-24=-33$. Without brackets, $-3^2$ is often miscalculated as $-9$ instead of the correct $(-3)^2=9$ — always bracket the substituted value before applying the power.

• $y$-intercept: set $x=0$; for $y=ax^2+bx+c$ this is always $(0,c)$.
• Roots ($x$-intercepts): solve $ax^2+bx+c=0$ — up to two solutions, one (repeated), or none.
• Axis of symmetry: $x=\dfrac{-b}{2a}$, or exactly midway between two known roots.
• Turning point: substitute the axis-of-symmetry $x$-value back into the equation to get its $y$-value.

Plot each coordinate as a small, precise cross — examiners require accuracy to within half a small grid square. Then join the points with a single smooth freehand curve, never a ruler and never short straight segments. The curve must flow through the turning point without a sharp corner.

To solve $2x^2-4x=2.5$ using the plotted graph of $y=2x^2-4x$: draw the horizontal line $y=2.5$ on the same axes, and read off the $x$-coordinates where it crosses the curve — those are your solutions.

Worked Example

Complete a table of values for $y=2x^2-4x$ for $x=-2$ to $3$, then state the coordinates of the turning point.

The values are symmetric about $x=1$ (e.g. $y(0)=y(2)=0$), confirming the axis of symmetry is $x=1$. Substituting back, $y(1)=2(1)^2-4(1)=-2$.

Turning point: $(1,-2)$, a minimum (since $a=2>0$).

Quadratic graph questions are typically split into parts — (a) complete the table, (b) plot the graph, (c) use it to solve something — and each part builds directly on the previous one. If part (c) asks you to "use your graph" to solve an equation, the expectation is that you draw an extra line (often a simple horizontal one) on the same axes and read off an intersection, rather than solving the equation fresh with algebra. Treating each part as dependent on the one before it, rather than restarting from scratch, is what the word "hence" or "using your graph" is signalling.