Cubic, reciprocal and exponential graphs

The shape vocabulary of the Higher course — S-curve cubics, two-branch reciprocals with asymptotes, ever-positive exponentials — and why each equation forces its shape.

Algebra · Graphs of functions

Cubic, reciprocal and exponential graphs

The shape vocabulary of the Higher course — S-curve cubics, two-branch reciprocals with asymptotes, ever-positive exponentials — and why each equation forces its shape.

Why it works

Exams show a curve and ask "which equation?" — or name an equation and expect its shape on sight. Each family's shape follows from how its formula treats

x

:

Cubic, $y = x^3$ : an S through the origin. Cubing keeps the sign — negative in, negative out — so the curve comes from far BELOW on the left, flattens through

(0, 0)

, and climbs away to the right. No mirror symmetry (that's the giveaway against a parabola: a parabola U-turns; a cubic never does). With a negative coefficient,

y = -x^3

, the S runs the other way. A factorised cubic like

y = x^3 - 4x = x(x-2)(x+2)

crosses the

x

-axis at every root:

-2, 0, 2

.

Reciprocal, $y = \frac{1}{x}$ : two branches and a forbidden line. Dividing by zero is impossible, so the graph simply HAS NO POINT at

x = 0

— the curve splits into two branches (positive quadrant and negative quadrant for

y = \frac{1}{x}

) that hug the axes ever closer without touching. The axes are asymptotes: approached forever, reached never. Small inputs give huge outputs (

x = \frac{1}{2}

gives

y = 2

), huge inputs give tiny ones.

Exponential, $y = k^x$ (with $k > 1$ ): growth that never lands. Every power of a positive number is positive, so

y = 2^x

sits entirely ABOVE the

x

-axis: it crosses the

y

-axis at

(0, 1)

— anything to the power 0 is 1 — sweeps up steeply to the right, and to the left decays toward the axis without ever reaching it (

2^{-3} = \frac{1}{8}

: tiny, never zero, never negative). "Why does the graph never touch the

x

-axis?" has a one-line answer:

2^x > 0

for every

x

.

Telling them apart in a line-up: parabola = symmetric U; cubic = unsymmetric S; reciprocal = two separate branches; exponential = one sweeping curve trapped above the axis. Check a single easy point (

x = 0

, or

x = 1

) if two candidates look alike.