Linear and Quadratic Graphs

Recognising and Sketching Linear and Quadratic Graphs

Being able to recognise a function's graph type from its equation — and to produce an accurate sketch — is a key skill. A linear function produces a straight line; a quadratic produces a parabola. Sketching means showing the correct shape and marking key features (intercepts, turning point) without precise plotting of every point.

Linear y = mx + c Quadratic (a > 0) U-shape, minimum Quadratic (a < 0) ∩-shape, maximum

Sketching a Linear Graph

Plot the y-intercept and one or two more points (using the gradient or setting \(x = 0\) and \(y = 0\)), then draw a straight line through them. Label the key intercepts.

Sketching a Quadratic Graph

A sketch of \( y = ax^2 + bx + c \) should show:

  • The correct shape (U or ∩) from the sign of \(a\).
  • The y-intercept \((0, c)\).
  • The roots (if real), found by factorising or formula.
  • The turning point, found using the axis of symmetry \(x = -b/(2a)\) or completing the square.

A sketch does not need to be to scale, but key features must be clearly and correctly labelled.

Recognising the Graph Type

From the equation, identify the highest power of \(x\): degree 1 → linear; degree 2 → quadratic. For a quadratic, the sign of \(a\) determines whether the vertex is a minimum or maximum. A linear graph never curves; a quadratic never gives a straight line (unless it degenerates, which is not a standard case at GCSE).

Worked Examples

Classify each: \( y = 3x-7 \), \( y = -x^2+4 \), \( y = x^2-3x+2 \), \( y = 5 \).

\( y = 3x-7 \): linear (degree 1).

\( y = -x^2+4 \): quadratic (\(a=-1<0\), ∩-shape, maximum at \(x=0\)).

\( y = x^2-3x+2 \): quadratic (\(a=1>0\), U-shape).

\( y = 5 \): linear (horizontal line — a special case of \(y=mx+c\) with \(m=0\)).

Sketch \( y = 2x - 4 \), marking intercepts.

y-intercept: set \(x=0\): \(y=-4\). Point \((0,-4)\).

x-intercept: set \(y=0\): \(2x=4\), so \(x=2\). Point \((2,0)\).

Plot both intercepts, draw a straight line through them with a positive gradient.

Sketch \( y = x^2 - 9 \), marking all key features.

y-intercept: \((0,-9)\). Roots: \(x^2=9 \Rightarrow x = \pm 3\), so \((-3,0)\) and \((3,0)\).

Axis of symmetry: \(x = 0\) (by symmetry of the roots). Turning point: \((0,-9)\) — which is also the y-intercept. Minimum (since \(a=1>0\)).

Sketch: U-shape touching y-axis at its lowest point \((0,-9)\), crossing x-axis at \(x=-3\) and \(x=3\).

 Key Takeaways

  • The highest power of \(x\) determines graph type: degree 1 → straight line; degree 2 → parabola.
  • For quadratics: \(a>0\) → U-shape (minimum); \(a<0\) → ∩-shape (maximum).
  • A sketch must show: correct shape, y-intercept, any x-intercepts (roots), and the turning point.
  • A quick check: substitute any \(x\) value — if the answer is linear in \(x\), the graph is a straight line; if quadratic, a parabola.