Sketching curves

Sketching standard curves from their key features — intercepts, repeated-root behaviour, and the asymptotes of reciprocals — and finding where curves meet.

Pure · Algebra & functions

Sketching curves

Sketching standard curves from their key features — intercepts, repeated-root behaviour, and the asymptotes of reciprocals — and finding where curves meet.

Why it works

Sketching isn't plotting dozens of points — it's capturing the shape and the few features that pin a curve down.

Intercepts. The $y$ -intercept is the value at $x = 0$ ; the $x$ -intercepts

(roots) are where

y = 0

. For a factorised curve

y = (x+2)(x-1)(x-3)

the roots are read straight off —

x = -2, 1, 3

— and

x = 0

gives

y = (2)(-1)(-3) = 6

Shape from the leading term. A positive cubic ( $+x^3$ ) rises overall from

bottom-left to top-right; a negative one falls. A positive parabola is a U, a negative one an ∩. A positive quartic (

+x^4

) has both ends pointing up (a negative quartic, both down) — with up to four roots it often takes a "W" shape.

Touch vs cross. A repeated factor changes the behaviour at that root:

(x-2)^2

makes the curve touch the axis at

x = 2

(bouncing back like a parabola), while a single factor

(x-2)

crosses it.

Reciprocals. $y = \dfrac1x$ can never equal $0$ and is undefined at $x = 0$ ,

so it approaches but never reaches the axes — they are asymptotes. As

x \to \pm\infty

y \to 0

; as

x \to 0

y \to \pm\infty

Intersections. Where two curves meet, their $y$ -values are equal — so you

solve them simultaneously. The number of solutions is the number of crossing points.