Graphs of Functions & Real-Life Graphs

Mathematics· 2.9, 2.10, 2.11· 25 min read

1. 1. Real-Life Graphs (All Tiers, Core Content)★★☆☆☆⏱ 8 min

📘 Definition

Real-life graph

A graph representing real-world quantities, with axes labelled with units, used to interpret relationships between two variables.

The two most frequently examined real-life graphs at Core tier are distance-time graphs and conversion graphs. The gradient of any real-life graph equals the rate of change of the y-axis variable with respect to the x-axis variable.

Distance-time graph: y-axis = total distance travelled, x-axis = time, gradient = speed
Conversion graph: converts between two units (e.g. km to miles), gradient = conversion factor

📐 Worked Example

A distance-time graph shows a cyclist travelling 40 km in 2 hours, then stopping for 1.5 hours. Calculate the cyclist's speed while moving, and state what the flat section of the graph represents.

1
Speed equals the gradient of the distance-time graph, calculated as change in distance divided by change in time:
$gradient = \frac{\Delta distance}{\Delta time} = \frac{40}{2} = 20$
2
The flat section has a gradient of 0, so speed = 0: this represents the cyclist being stationary.

✓ Quick check

What is the gradient of a distance-time graph for a stationary object?
Reveal answer
0 —
A stationary object has no change in distance over time, so the gradient equals zero.

2. 2. Core Function Graphs (All Tiers)★★★☆☆⏱ 7 min

At Core tier, you will be asked to construct tables of values, plot, read, and identify three core function types: linear, quadratic, and simple reciprocal.

📘 Definition

Root of a function

The x-coordinate(s) where a function graph crosses the x-axis, i.e. where $y = 0$ . Quadratic functions can have 0, 1, or 2 roots.

Linear: $y = a x + b$ : straight line, constant gradient, 1 root (unless horizontal)
Quadratic: $y = \pm x^{2} + a x + b$ : U or ∩ shaped parabola, symmetric about its turning point
Reciprocal: $y = \frac{a}{x} (x \neq = 0)$ : two separate curve branches, no roots, no y-intercept

📐 Worked Example

Complete the table of values for $y = x^{2} - 2 x - 3$ for $x = - 1, 0, 1, 2, 3$ , then state the roots of the function.

1
Substitute each x value into the function to calculate y:
$x=-1: (-1)^2 -2(-1) -3 = 0; x=0: 0 - 0 -3 = -3; x=1: 1 -2 -3 = -4; x=2: 4 -4 -3 = -3; x=3: 9 -6 -3 = 0$
2
Roots are where $y = 0$ , so roots are $x = - 1$ and $x = 3$ .

3. 3. Extended Additional Content (Extended Tier Only)★★★★☆Extended only⏱ 10 min

✓ Calculator OK

Extended tier adds three key areas of content: estimating curve gradients via tangents, kinematics with speed-time graphs, and sketching higher-order function graphs with key features.

📐 Worked Example

A tangent is drawn to a quadratic curve at $x = 2$ . The tangent passes through points $(1, - 1)$ and $(3, 7)$ . Estimate the gradient of the curve at $x = 2$ .

1
The gradient of the tangent equals the gradient of the curve at the point of contact:
$gradient = \frac{7 - (-1)}{3 - 1} = \frac{8}{2} = 4$
2
The estimated gradient of the curve at $x = 2$ is 4.

For linear speed-time graphs, the area under the graph equals the total distance travelled. You can calculate this area using standard shape formulas (triangles, rectangles, trapezia).

📐 Worked Example

A speed-time graph shows a car accelerating from 0 to 10 m/s in 5 seconds, then travelling at constant speed for 10 seconds. Calculate the total distance travelled.

1
Split the area under the graph into a triangle (acceleration phase) and rectangle (constant speed phase):
$Area_{triangle} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 5 \times 10 = 25$
2
$Area_{rectangle} = length \times width = 10 \times 10 = 100$
3
Total distance = $25 + 100 = 125$ m.

Cubic: $y = a x^{3}$ : S-shaped curve, 1 or 3 roots, one turning point
Reciprocal: $y = \frac{a}{x}$ : asymptotes at $x = 0$ and $y = 0$ , no turning points
Exponential: $y = a b^{x} + c$ : horizontal asymptote at $y = c$ , no turning points

4. Common Pitfalls

Wrong move:

Confusing distance-time and speed-time graph gradients

Why:

Distance-time gradient = speed, speed-time gradient = acceleration, they measure different physical quantities

Correct move:

Always read axis labels and units first before calculating gradient for kinematic graphs

Wrong move:

Joining reciprocal graph points with a straight line through x=0

Why:

Reciprocal functions are undefined at x=0, with two separate curve branches

Correct move:

Draw two smooth separate curves, never crossing the x or y axes for $y = a / x$

Wrong move:

Drawing a chord instead of a tangent to estimate curve gradient

Why:

A chord connects two points on a curve, giving average gradient, not instantaneous gradient at a point

Correct move:

Draw a straight line touching the curve at exactly one point, extending equally on both sides of the point

Wrong move:

Forgetting units when calculating rate of change from real-life graphs

Why:

Examiners award marks for correct units as part of rate answers

Correct move:

Always include units derived from axis labels e.g. km/h for speed, £/kg for cost rate

Wrong move:

Attempting to calculate area under non-linear speed-time graphs

Why:

This syllabus only requires area calculations for linear speed-time graph segments

Correct move:

Only calculate area for straight line segments using triangle, rectangle, or trapezium formulas

5. Quick Reference Cheatsheet

Graph Type	Core Feature	Extended Only Feature
Distance-Time	Gradient = Speed	N/A
Speed-Time	N/A (Extended only)	Gradient = Acceleration; area under linear segments = Distance travelled
Linear $y = a x + b$	Constant gradient, 1 root	N/A
Quadratic $y = \pm x^{2} + a x + b$	U/∩ shape, symmetric	Estimate gradient via tangent at any point
Reciprocal $y = a / x$	2 separate branches	Asymptotes at $x = 0$ , $y = 0$
Exponential $y = a b^{x} + c$	Not assessed at Core	Horizontal asymptote at $y = c$

6. Frequently Asked

What is the difference between a distance-time and speed-time graph?

A distance-time graph plots total distance travelled against time, so its gradient equals speed. A speed-time graph plots instantaneous speed against time, so its gradient equals acceleration, and the area under linear segments equals total distance travelled (Extended only).

Do I need to use calculus to find the gradient of a curve for this topic?

No, calculus is not required for this sub-topic. For Extended tier, you only need to draw a tangent to the curve at the required point and calculate the gradient of that tangent line.

Going deeper

past_paperCIE IGCSE 0580 Past Papers 2025-2027Official past papers to practice this topic

What's Next

Now that you have mastered graphs of functions and real-life graphs for CIE IGCSE Mathematics 0580, you are ready to move to more advanced algebra and graph topics. For Core tier students, next you will learn to solve simultaneous equations using graphs, a common exam question that builds directly on your ability to read and interpret linear and quadratic function plots. For Extended tier students, you can progress to differentiation, where you will learn to calculate exact gradients of curves and turning points without drawing tangents, as well as graph transformation topics that build on your knowledge of cubic, reciprocal and exponential graph shapes. Make sure you practice past paper questions on this topic to reinforce your understanding, as it appears frequently across all papers.