📈 Curve-Craft: Mastering Linear & Quadratic Sketches!

Curves are everywhere in O-Level Maths exam papers! From straight-line graphs that predict costs to parabolas describing projectile motion, being able to **sketch** and **interpret** linear and quadratic graphs quickly can grab you easy marks and boost your problem-solving confidence.

Spotting the Shape: Lines vs Parabolas

A linear graph takes the form

y=mx+c

and always appears as a **straight line**. A quadratic graph,

y=ax^{2}+bx+c

, draws a **parabola** that opens upward if

a>0

and downward if

a<0

. When sketching, focus on three pieces of information: **intercepts**, **turning point**, and **overall shape (slope or concavity)**.

For $y=mx+c$ : • y-intercept $(0,c)$ • gradient $m$ tells you the steepness & direction.
For $y=ax^{2}+bx+c$ : • x-intercepts (roots) from factorising or quadratic formula • y-intercept $(0,c)$ • turning point at $x=-\dfrac{b}{2a}$ with $y$ found by substitution.

Worked Example: Sketching a Quadratic

Problem: Sketch the curve

y=x^{2}-4x-5

, clearly showing intercepts and the turning point.

Step 1: Find $x$-intercepts by factorising –
$x^{2}-4x-5=(x-5)(x+1)=0 \Rightarrow x=-1,\;5$
so roots are $(-1,0)$ and $(5,0)$ .
Step 2: Find the turning point. The axis of symmetry is
$x=\frac{-b}{2a}=\frac{4}{2}=2$
. Substitute $x=2$ :
$y=(2)^{2}-4(2)-5=4-8-5=-9$
giving vertex $(2,-9)$ .
Step 3: Identify the $y$-intercept by setting $x=0$ :
$y=-5$
, so point $(0,-5)$ .
Step 4: Draw a smooth U-shaped curve opening upward (since $a=1>0$ ) through $(-1,0)$ , $(5,0)$ , dipping to $(2,-9)$ , and crossing the y-axis at $(0,-5)$ .

Practice sketching without a calculator: pick simple values, plot key points, and trust the symmetry or gradient. The more you sketch, the faster you’ll see patterns—and in exams, that speed translates into extra checking time and higher marks. Keep curving with confidence!