Further Graphs & TangentsCambridge IGCSE Maths: Revision notes

Section 1

Graphs of $ax^n$ and $ab^x+c$

Beyond linear and quadratic graphs, curves can be built from terms of the form $ax^n$ (up to three added terms), where $n \in \{-2, -1, -\frac{1}{2}, 0, \frac{1}{2}, 1, 2, 3\}$ , and exponential-type expressions $ab^x + c$ .

Negative powers of $n$ (e.g. $n=-1, -2$ ) create reciprocal graphs
Fractional powers (e.g. $n=\frac{1}{2}$ ) create root-type graphs
$ab^x + c$ creates exponential curves

Every point on such a curve can still be calculated algebraically by substituting an $x$ -value — no diagram is needed to find a value on the curve.

Key termsreciprocal graphexponential curve

Section 2

Recognising graph shapes

Being able to identify a function type from its equation (or vice versa) is a key exam skill:

Function type	General form	Key feature
Linear	$y=mx+c$	Straight line
Quadratic	$y=ax^2+bx+c$	One turning point, symmetric U or ∩ shape
Cubic	$y=ax^3+bx^2+cx+d$	Up to two turning points, S-shaped
Reciprocal	$y=\frac{a}{x}$	Two separate branches, never touches the axes
Exponential	$y=ab^x+c$	Constant ratio growth/decay, flattens towards an asymptote

Each type has a distinctive algebraic signature you can spot directly from the equation.

Key termscubic function

Section 3

Asymptotes

An asymptote is a line that a curve approaches but never touches or crosses.

Reciprocal graphs (e.g. $y=\frac{a}{x}$ ) have a vertical asymptote at $x=0$ (the function is undefined there) and a horizontal asymptote at $y=0$
Exponential graphs $y=ab^x+c$ have a horizontal asymptote at $y=c$ , since as $x \to \pm\infty$ , $ab^x \to 0$ and $y \to c$

To find an asymptote algebraically, consider what value the function approaches as $x$ becomes very large (positive or negative) or as it approaches an excluded value.

Example: For $y = \frac{2}{x} + 5$ , the horizontal asymptote is $y=5$ (as $x \to \pm\infty$ , $\frac{2}{x}\to0$ ) and the vertical asymptote is $x=0$ .

Key termsasymptotevertical asymptotehorizontal asymptote

Common mistake

Don't confuse an asymptote with a root — the curve never touches an asymptote, but it can cross the $x$ -axis at a root.

Section 4

Gradients of curves and tangents

The gradient of a curve at a point equals the gradient of the tangent to the curve at that point.

A tangent touches the curve at exactly one point locally and matches the curve's steepness there
Since curve steepness constantly changes, the gradient must be estimated (or calculated, for polynomial curves, using differentiation) at each specific point of interest
The gradient is interpreted as a rate of change: e.g. on a distance–time type curve, the gradient at a point gives the instantaneous speed at that moment

Exponential growth/decay problems (e.g. population, depreciation) are interpreted the same way: the steeper the tangent, the faster the quantity is changing at that instant.

Key termstangentrate of change

Think of it like this

Think of the gradient of a tangent like a speedometer reading at one exact instant, rather than the average speed over a whole journey.

Must Know

$ax^n$ graphs (with $n$ including negative and fractional values) and $ab^x+c$ graphs extend beyond linear and quadratic shapes
Reciprocal graphs have a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$ (or wherever the function is shifted to)
Exponential graphs $y=ab^x+c$ have a horizontal asymptote at $y=c$
An asymptote is approached but never touched; a root is where the curve crosses the axis
The gradient of a curve at a point equals the gradient of its tangent there, interpreted as a rate of change
Every graph feature (asymptote, gradient, intersection) must be found by calculation, not by reading a picture

That's the notes covered.

Carry on to the next subtopic.