Mathematics · IGCSE 0580 · §E2.9–E2.13

Graphs, Calculus & Functions

Where algebra becomes shape — reading a graph’s meaning, measuring the rate at which a curve changes, and treating a function as a machine you can run forwards and backwards.

Mathematics · 0580 Extended Topic 5 of 12
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Graphs in practical situations

P tangent linear quadratic cubic reciprocal exponential
FIG 2.0 A curve and the tangent whose gradient the derivative measures, above the five function families this chapter learns to sketch.

Many questions hand you a graph drawn from a real situation — a journey, a currency conversion, a change in speed — and ask you to read meaning from its shape. Two features carry most of the marks: the gradient of a line, and the area beneath it.

Definition
Gradient
The steepness of a line: change in the vertical quantity divided by change in the horizontal quantity.

Travel and conversion graphs

On a distance–time graph the gradient is speed, and a horizontal segment means the object is stationary. A conversion graph is a single straight line, usually through the origin; to convert between the two quantities, read across to the line and then up or down to the other axis.

Speed–time graphs

On a speed–time graph the two readings change meaning. The gradient of the line is acceleration — how quickly the speed itself is changing — and the area between the line and the time axis is the distance travelled. For straight-line segments that area is a triangle or a trapezium, found by ordinary mensuration.

Definition
Acceleration
The gradient of a speed–time graph — the rate at which speed changes, in m/s².
area = distance gradient = acceleration time speed
FIG 2.1 On a speed–time graph, the gradient gives acceleration and the shaded area gives the distance travelled.
Examiner note
Read the axis labels before the shape. The same slope means speed on a distance–time graph but acceleration on a speed–time graph — confusing them loses the interpretation mark.
Why this matters
The gradient as a rate you read here is exactly what differentiation calculates for curves, a few pages on.

Graphs of functions

Before a graph can be sketched or read, it often has to be plotted from a table of values. The skill is mechanical, but marks are lost through arithmetic slips and mis-plotted points.

Definition
Function family
A group of functions sharing the same algebraic form and characteristic graph shape.
Definition
Reciprocal function
y = a/x: undefined at x = 0, with the two axes as asymptotes.

Building a table of values

Take the x-values given in the question, substitute each into the function, and record y to the accuracy asked for. Plot each point to within half a small square, then join a non-linear graph with a single smooth curve — never a chain of straight segments.

x−10123
y30−103
A table of values for y = x² − 2x, ready to plot.

The function families

IGCSE graphs come from a small set of families. Power functions y = axⁿ cover linear (n = 1), quadratic (n = 2) and cubic (n = 3), together with the reciprocal (n = −1). Exponential functions y = abˣ + c model growth when b > 1 and decay when 0 < b < 1, levelling off towards the value c.

Examiner note
Plot every given x-value, including negatives and zero. A curve drawn through too few points, or forced straight between them, is not credited as a smooth graph.

Sketching curves

A sketch is not a plot. It need not be to scale, but it must show the features that identify the curve: where it crosses the axes, its turning points, its symmetry, and how it behaves far from the origin.

Definition
Root
A value of x where the curve meets the x-axis (y = 0).

Label the roots, any turning point, the y-intercept, and any asymptote the curve approaches but never meets. Getting the long-run behaviour right — which way each end of the curve heads — is often worth a mark on its own.

Definition
Asymptote
A line the curve approaches ever more closely but never reaches.
linear quadratic cubic reciprocal exponential
FIG 2.2 The five function families every Extended candidate should be able to sketch from memory.

Exponentials deserve special care: growth and decay are mirror images about the y-axis, both flattening towards a horizontal asymptote.

y = c growth (b > 1) decay (0 < b < 1)
FIG 2.3 Exponential growth and decay share a starting value and a horizontal asymptote y = c.
Examiner note
Plot means mark given points accurately on a grid; Sketch means a freehand curve showing key features, not to scale; Draw means an accurate line or curve. The command word sets the accuracy expected.

Differentiation

Differentiation turns the informal idea of a gradient into an exact calculation. For the polynomials on this syllabus, one rule does all the work: d/dx(axⁿ) = anxⁿ⁻¹ — multiply by the power, then reduce the power by one.

Definition
Derivative (gradient function)
dy/dx: a new function giving the gradient of y = f(x) at any value of x.
Definition
Stationary point
A point where dy/dx = 0 and the tangent is horizontal — a maximum or a minimum.

The gradient function

Differentiating y with respect to x gives dy/dx, the gradient function: substitute any x-value into it to get the gradient of the curve there. Differentiate term by term; the derivative of a constant is zero.

run rise P tangent
FIG 2.4 The derivative gives the gradient of the tangent to the curve at each point: rise ÷ run.

Stationary points

Where dy/dx = 0 the tangent is horizontal — a stationary, or turning, point. Solving dy/dx = 0 gives the x-coordinates; substituting each back into y gives the coordinates in full.

Worked example: the curve y = x³ − 3x² has gradient function dy/dx = 3x² − 6x. Setting the gradient function to zero: 3x² − 6x = 0. Factorise: 3x(x − 2) = 0, so x = 0 or x = 2. Substitute back: x = 0 gives y = 0; x = 2 gives y = 8 − 12 = −4. The curve is momentarily flat at both — turning points (0, 0) and (2, −4).

Examiner note
Setting dy/dx = 0 earns the method mark; the accuracy marks need both coordinates. Points of inflection are not required — only maxima and minima.
Why this matters
The rate of change met informally on the speed–time graph is exactly what dy/dx measures — now made exact for any curve.

Functions

A function is a rule that assigns exactly one output to each input. Function questions test three skills: evaluating, inverting, and composing. The composite is written fg(x) = f(g(x)); if y = f(x) then f⁻¹(y) = x.

Definition
Domain and range
Domain: the set of inputs. Range: the set of outputs the function can produce.
Definition
Function
A rule assigning exactly one output to each input value.

Notation and evaluation

Writing f(x) = 2x + 5 names a rule; f(3) means substitute x = 3, giving 11. The domain is the set of allowed inputs; the range is the set of outputs it produces.

Inverse and composite functions

The inverse f⁻¹ reverses the rule: set y = f(x), rearrange to make x the subject, then rename. The composite fg(x) means apply g first, then f — read the notation right to left.

x g(x) f(g(x)) g f fg(x) = f(g(x))
FIG 2.5 The composite fg applies g first, then f: the notation is read right to left.

Worked example: for f(x) = 2x + 5 and g(x) = x − 3, find f⁻¹(x) and fg(x). Inverse: write y = 2x + 5, so x = (y − 5)/2; hence f⁻¹(x) = (x − 5)/2. Composite: fg(x) = f(x − 3) = 2(x − 3) + 5 = 2x − 1. The inverse undoes f; the composite is a brand-new rule.

Examiner note
f⁻¹(x) is the inverse function, not the reciprocal 1/f(x) — a frequent and costly confusion. The −1 is notation, not a power.

Exam advice

Common mistakes

Classifying a turning point without testing it
Finding a stationary point but declaring it a maximum or minimum without checking the gradient on either side — the nature of the point needs justifying.
Reading f⁻¹(x) as 1/f(x)
The inverse function and the reciprocal are different operations; treating the −1 as a power gives the wrong expression entirely.
A reciprocal sketch with no asymptotes
Drawing y = a/x without the excluded value at x = 0, or the axes as asymptotes, loses the long-run-behaviour mark.
Misreading a speed–time graph
Taking the gradient as speed rather than acceleration, or mishandling the sign of an area that falls below the axis.

Model answer

A curve has equation y = xⁿ + qx² + 9x and gradient function dy/dx = 3x² − 12x + 9. (a) Find n and q. (b) Find the coordinates of the turning points.
[6 marks]
B1 B1
Differentiate term by term and match coefficients
nxⁿ⁻¹ = 3x² gives n = 3; 2q = −12 gives q = −6.
M1
Set the gradient function to zero for the turning points
3x² − 12x + 9 = 0.
A1
Solve the quadratic — factorising, completing the square or the formula are all credited
3(x − 1)(x − 3) = 0, so x = 1 or x = 3.
A1 A1
Substitute both x-values into y = x³ − 6x² + 9x
x = 1 gives (1, 4); x = 3 gives (3, 0).

Recall checklist

  • Draw and read travel and conversion graphs, using gradient as a rate.
  • Construct a table of values and plot axⁿ and abˣ + c graphs.
  • Sketch the five families, labelling roots, symmetry and asymptotes.
  • Estimate a curve’s gradient at a point by drawing a tangent.
  • Differentiate simple polynomials with the power rule.
  • Locate turning points by solving dy/dx = 0.
  • Distinguish a maximum from a minimum by a valid test.
  • Evaluate, invert and compose functions.

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