What is a graph of a function in the context of Edexcel IGCSE Mathematics?

In Edexcel IGCSE Mathematics, a graph of a function is a visual representation of the relationship between two variables, typically x and y. It shows how the value of y changes in response to changes in x. This is crucial for understanding how different types of functions behave, such as linear, quadratic, and exponential functions.

How do you plot the graph of a linear function?

To plot the graph of a linear function, you first need to identify the equation of the line, usually in the form y = mx + c, where m is the slope and c is the y-intercept. Start by plotting the y-intercept on the y-axis. Then, use the slope to determine another point on the line by rising and running from the y-intercept. Connect these points with a straight line.

What are the key features of a quadratic function's graph?

The graph of a quadratic function, typically in the form y = ax^2 + bx + c, is a parabola. Key features include the vertex, which is the highest or lowest point, the axis of symmetry, which is a vertical line through the vertex, and the direction of the parabola, which opens upwards if a > 0 and downwards if a < 0. The roots or x-intercepts, where the graph crosses the x-axis, are also significant.

How can you determine the intersection points of two functions' graphs?

To find the intersection points of two functions' graphs, you need to set their equations equal to each other and solve for x. This will give you the x-coordinates of the intersection points. Substitute these x-values back into either of the original equations to find the corresponding y-values. These (x, y) pairs are the intersection points.

What is the significance of the slope in the graph of a function?

The slope of a function's graph indicates the rate of change of the dependent variable (usually y) with respect to the independent variable (usually x). In a linear function, the slope is constant and represents the steepness and direction of the line. A positive slope means the line rises as x increases, while a negative slope means it falls. Understanding the slope is essential for interpreting and predicting the behavior of functions.

Why is "Treating a double root as crossing" a common mistake in Graph of Functions?

Why it happens: All roots look the same. How to avoid it: DOUBLE root: tangent (touches but doesn't cross). SINGLE root: crosses. Source: 4MA1/2H Jan 2024 — examiner report Q17

Why is "Drawing reciprocal graph crossing the axes" a common mistake in Graph of Functions?

Why it happens: Forgetting the asymptotes. How to avoid it: y = k/x: NEVER touches x = 0 or y = 0. Always two branches.

Why is "Wrong direction for quadratic" a common mistake in Graph of Functions?

Why it happens: Sign of x^2 coefficient. How to avoid it: +x^2: opens UP (smile). -x^2: opens DOWN (frown).

Sequences, Functions and GraphsEdexcel IGCSE Mathematics (4MA1)

Graph of Functions

Q: How can you determine the intersection points of two functions' graphs?

To find the intersection points of two functions' graphs, you need to set their equations equal to each other and solve for x. This will give you the x-coordinates of the intersection points. Substitute these x-values back into either of the original equations to find the corresponding y-values. These (x, y) pairs are the intersection points.

Q: What is the significance of the slope in the graph of a function?

The slope of a function's graph indicates the rate of change of the dependent variable (usually y) with respect to the independent variable (usually x). In a linear function, the slope is constant and represents the steepness and direction of the line. A positive slope means the line rises as x increases, while a negative slope means it falls. Understanding the slope is essential for interpreting and predicting the behavior of functions.

Q: Why is "Treating a double root as crossing" a common mistake in Graph of Functions?

Why it happens: All roots look the same. How to avoid it: DOUBLE root: tangent (touches but doesn't cross). SINGLE root: crosses. Source: 4MA1/2H Jan 2024 — examiner report Q17

Q: Why is "Drawing reciprocal graph crossing the axes" a common mistake in Graph of Functions?

Why it happens: Forgetting the asymptotes. How to avoid it: y = k/x: NEVER touches x = 0 or y = 0. Always two branches.

Q: Why is "Wrong direction for quadratic" a common mistake in Graph of Functions?

Why it happens: Sign of x^2 coefficient. How to avoid it: +x^2: opens UP (smile). -x^2: opens DOWN (frown).

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Graphs of Functions Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Sketch and recognise graphs: linear, quadratic, cubic, reciprocal, exponential. Identify roots, vertices, asymptotes.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

3.4 — Recognise and sketch graphs of common functions.
3.4 — Identify intercepts, roots, vertices, asymptotes.
3.4 — Use graphs to solve equations and inequalities.

Linear and quadratic — review

Linear: straight. Quadratic: parabola.

Linear $y = mx + c$ .

Straight line. Gradient $m$ , y-intercept $c$ .

Quadratic $y = ax^{2} + bx + c$ .

Parabola.

$a > 0$ : opens UP. Minimum vertex.
$a < 0$ : opens DOWN. Maximum vertex.

Roots = where parabola crosses $x$ -axis.

Discriminant	Roots	Graph
$b^{2} - 4ac > 0$	2 distinct	Crosses $x$ -axis twice
$b^{2} - 4ac = 0$	1 repeated	Touches (tangent)
$b^{2} - 4ac < 0$	no real	Doesn't cross

Sketching:

Find $y$ -intercept: at $x = 0$ .
Find roots: solve $ax^{2} + bx + c = 0$ .
Find vertex: $x_{v} = -b/(2a)$ , then compute $y_{v}$ .
Sketch.

Example. $y = x^{2} - 4x - 5$ .

Roots: $(x-5)(x+1) = 0$ → $-1, 5$ .
y-intercept: $-5$ .
Vertex: $x = 2$ , $y = -9$ .

Edexcel tip. Always show 3-4 key points: roots, y-intercept, vertex. Mark schemes credit each.

Linear: straight, gradient + intercept.
Quadratic: parabola, $a > 0$ up, $a < 0$ down.
Roots from factoring or formula.
Vertex at $x = -b/2a$ .

Cubic graphs

S-shape. Up to 3 roots. Double roots are tangent.

Cubic $y = ax^{3} + bx^{2} + cx + d$ .

S-shape (or 'wave' shape).

$a > 0$ : bottom-left to top-right.
$a < 0$ : top-left to bottom-right.

Roots: up to $3$ .

Single root: graph CROSSES $x$ -axis.
Double root: graph is TANGENT (touches but doesn't cross).
Triple root: graph has 'inflection' at the axis.

Example. $y = x^{3} - 3x^{2}$ .

Factor: $x^{2}(x - 3)$ .
Roots: $x = 0$ (double), $x = 3$ (single).
At $x = 0$ : tangent to axis.
At $x = 3$ : crosses axis.

Other cubics:

Function	Shape
$y = x^{3}$	Through origin, S-shape, 1 root
$y = x^{3} - x$	$x(x^{2}-1)$ , three roots $-1, 0, 1$
$y = -x^{3} + 4x$	$x(4 - x^{2})$ , three roots, opens opposite

Worked qualitative. What's different about a double root?

The graph TOUCHES the $x$ -axis without crossing.
Both 'sides' of the root are on the same side of the axis.
E.g. $y = x^{2}(x - 3)$ at $x = 0$ : $y$ goes from positive (left) to positive (right) of $0$ , touching at $0$ .

Edexcel tip. When a question gives a factored form like $y = x(x - 1)(x + 2)$ , identify the roots immediately: $0, 1, -2$ . Sketch using these.

Cubic: S-shape.
Up to 3 roots.
Double root: tangent.
Single root: crosses.

Reciprocal graphs

$y = k/x$ . Two branches. Asymptotes at axes.

Reciprocal $y = \dfrac{k}{x}$ ( $k \ne 0$ ).

Two-branch hyperbola.

$k > 0$ : branches in upper-right (positive $x, y$ ) and lower-left (negative $x, y$ ).
$k < 0$ : opposite (upper-left and lower-right).

Asymptotes:

VERTICAL: $x = 0$ (the $y$ -axis). Graph never touches.
HORIZONTAL: $y = 0$ (the $x$ -axis).

Behaviour:

As $x \to \infty$ , $y \to 0$ .
As $x \to 0^{+}$ , $y \to +\infty$ (or $-\infty$ for $k < 0$ ).
As $x \to 0^{-}$ , $y \to -\infty$ (or $+\infty$ ).

Example. $y = 6/x$ .

Test: $(1, 6), (2, 3), (3, 2), (6, 1)$ for positive branch.
Test: $(-1, -6), (-2, -3), (-3, -2)$ for negative branch.
Sketch curve approaching but never touching axes.

Worked qualitative. Why does $y = 1/x$ never cross the $y$ -axis?

$x = 0$ would mean dividing by zero — undefined.
Graph has a 'gap' at $x = 0$ .
The vertical asymptote.

Edexcel tip. Always draw asymptotes as DASHED lines. Mark schemes credit recognising them.

Two branches.
Asymptotes at $x = 0$ and $y = 0$ .
$k > 0$ : 1st & 3rd quadrants.
$k < 0$ : 2nd & 4th quadrants.

Exponential graphs

$y = a^{x}$ . Always positive. Through $(0, 1)$ . Asymptote $y = 0$ .

Exponential $y = a^{x}$ for $a > 0, a \ne 1$ .

Always positive (for real $x$ ): $a^{x} > 0$ .

Through $(0, 1)$ : $a^{0} = 1$ .

Behaviour:

$a > 1$ : GROWTH. As $x \to \infty$ , $y \to \infty$ . As $x \to -\infty$ , $y \to 0^{+}$ .
$0 < a < 1$ : DECAY. As $x \to \infty$ , $y \to 0^{+}$ . As $x \to -\infty$ , $y \to \infty$ .

Asymptote: $y = 0$ for all exponentials.

Common exponentials:

Function	Type	Through
$y = 2^{x}$	Growth	$(0,1), (1,2), (2,4), (-1, 0.5)$
$y = 3^{x}$	Growth	$(0,1), (1,3), (2,9)$
$y = (1/2)^{x}$	Decay	$(0,1), (1, 0.5), (2, 0.25)$

Real-world contexts:

Compound interest: exponential growth.
Radioactive decay.
Population growth (initially).
Cooling.

Worked qualitative. Why is $a^{x}$ always positive?

$a > 0$ : any positive number raised to any power is positive.
Even if $x$ is negative, $a^{-x} = 1/a^{x}$ which is still positive.
Graph never crosses the $x$ -axis.

Edexcel tip. Sketch with at least 3 points and the asymptote. Show the shape changes between $x < 0$ and $x > 0$ .

Always positive.
Through $(0, 1)$ .
Asymptote $y = 0$ .
$a > 1$ growth, $0 < a < 1$ decay.

How it’s examined

Function graphs every Higher Tier paper (4-7 marks). Sketching, intercepts, identifying types. Examiner reports flag (1) wrong direction for parabola, (2) double root crossing instead of tangent, (3) reciprocal crossing axes.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/2H Jan 2024. Last reviewed 2026-05-07.

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Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

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Step-by-step worked examples — Graph of Functions

Step-by-step solutions to past-paper-style questions on graph of functions, written exactly the way a tutor would explain them at the board.

1Sketch a quadratic
Higher• quadratic, sketch
Question
Sketch $y = x^{2} - 4x - 5$ . Show roots and y-intercept.
Step-by-step solution
1. Step 1
  Roots: solve $x^{2} - 4x - 5 = 0$ . Factor: $(x - 5)(x + 1) = 0$ . Roots: $x = 5, -1$ .
2. Step 2
  y-intercept: at $x = 0$ , $y = -5$ .
3. Step 3
  Coefficient of $x^{2}$ is positive → opens UP. Minimum between roots.
4. Step 4
  Vertex: midway between roots. $x = (5 + (-1))/2 = 2$ . $y = 4 - 8 - 5 = -9$ . Vertex $(2, -9)$ .
Answer
Parabola through $(-1, 0)$ , $(5, 0)$ , y-intercept $(0, -5)$ , vertex $(2, -9)$ , opens up.
2Sketch a cubic
Higher• Adapted from 4MA1/2H Jan 2024 Q17• cubic
Question
Sketch $y = x^{3} - 3x^{2}$ . Show key features.
Step-by-step solution
1. Step 1
  Factor: $y = x^{2}(x - 3)$ . Roots: $x = 0$ (double), $x = 3$ .
2. Step 2
  $x = 0$ is a TANGENT (double root) — graph touches axis at origin.
3. Step 3
  Cubic with positive leading coefficient: bottom-left to top-right.
4. Step 4
  Test values: at $x = 2$ , $y = 8 - 12 = -4$ . At $x = 4$ , $y = 64 - 48 = 16$ .
Answer
Cubic, touches $x$ -axis at origin (tangent), crosses at $(3, 0)$ , increasing for large $x$ .
Examiner tip
Double root: graph TOUCHES (tangent). Single root: graph CROSSES. Memorise the difference.
3Reciprocal graph
Higher• reciprocal
Question
Sketch $y = \dfrac{6}{x}$ . Show asymptotes.
Step-by-step solution
1. Step 1
  Two branches: one in upper-right (positive $x, y$ ), one in lower-left (negative $x, y$ ).
2. Step 2
  Asymptotes: $x = 0$ (vertical) and $y = 0$ (horizontal).
3. Step 3
  Test points: $(1, 6), (2, 3), (3, 2), (6, 1)$ and $(-1, -6), (-2, -3)$ , etc.
Answer
Hyperbola in 1st and 3rd quadrants, asymptotes at axes.
4Exponential graph
Higher• exponential
Question
Sketch $y = 2^{x}$ .
Step-by-step solution
1. Step 1
  Always POSITIVE. Passes through $(0, 1)$ since $2^{0} = 1$ .
2. Step 2
  As $x \to \infty$ , $y \to \infty$ (rapid growth).
3. Step 3
  As $x \to -\infty$ , $y \to 0^{+}$ (asymptote $y = 0$ ).
4. Step 4
  Test points: $(0, 1), (1, 2), (2, 4), (3, 8), (-1, 0.5), (-2, 0.25)$ .
Answer
Exponential growth curve, through $(0, 1)$ , asymptote $y = 0$ for negative $x$ .

Key Formulae — Graph of Functions

The formulae you need to memorise for graph of functions on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Quadratic vertex
$x_{\text{vertex}} = -\dfrac{b}{2a}\ \text{for}\ y = ax^{2} + bx + c$
$a, b$
coefficients
When to use
Quick way to find $x$ -coordinate of vertex.
Example
$y = x^{2} - 4x - 5$ : $x_{v} = 4/2 = 2$ . $y_{v} = -9$ .

Key Definitions and Keywords — Graph of Functions

Definitions to memorise and the exact keywords mark schemes credit for graph of functions answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Asymptote
Examiner keyword
A line the graph approaches but never touches. Vertical, horizontal, or diagonal.
Example
$y = 1/x$ has asymptotes $x = 0$ and $y = 0$ .
Root (graph)
$x$ -value where the graph crosses (or touches) the $x$ -axis. Where $y = 0$ .
Example
$y = x^{2} - 4$ has roots $x = \pm 2$ .
Vertex (parabola)
The minimum or maximum point of a parabola.

Common Mistakes and Misconceptions — Graph of Functions

The traps other students keep falling into on graph of functions questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Treating a double root as crossing
4MA1/2H Jan 2024 — examiner report Q17
Why it happens
All roots look the same.
How to avoid it
DOUBLE root: tangent (touches but doesn't cross). SINGLE root: crosses.
✕Drawing reciprocal graph crossing the axes
Why it happens
Forgetting the asymptotes.
How to avoid it
$y = k/x$ : NEVER touches $x = 0$ or $y = 0$ . Always two branches.
✕Wrong direction for quadratic
Why it happens
Sign of $x^{2}$ coefficient.
How to avoid it
$+x^{2}$ : opens UP (smile). $-x^{2}$ : opens DOWN (frown).

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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