GCSEMathsEdexcelHigherRevision Notes2. AlgebraGraphs of FunctionsTypes of Graphs

Types of Graphs (Edexcel GCSE Maths: Higher): Revision Note

Exam code: 1MA1

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 20 January 2026

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Understand this topic with deeper explanations and understanding checks.

Types of graphs

What types of graphs do I need to know?

You need to be able to recognise, sketch, and interpret the following types of graph:
- Linear ( $y = \pm x$ )
  - $y = m x + c$ or $a x + b y = c$
- Quadratic ( $y = \pm x^{2}$ )
  - $y = a x^{2} + b x + c$ where $a \neq 0$
- Cubic ( $y = \pm x^{3}$ )
  - $y = a x^{3} + b$ or $y = a x^{3} + b x^{2} + c x$ where $a \neq 0$
- Reciprocal ( $y = \pm \frac{1}{x}$ )
  - $y = \frac{a}{x} + b$ where $a \neq 0$
- Exponential ( $y = k^{\pm x}$ ) where $k > 1$
  - $y = a k^{x} + b$ where $a \neq 0$

Examples of linear, quadratic, cubic, reciprocal, and exponential graphs

You must also be able to recognise the three basic trigonometric graphs, covered in the Trigonometry section

Where are the asymptotes on reciprocal graphs?

An asymptote is a line on a graph that a curve becomes closer to but never touches
- These may be horizontal or vertical
The reciprocal graph, $y = \frac{a}{x}$ (where $a$ is a constant)
- does not have a y-intercept
- and does not have any roots
This graph has two asymptotes
- A horizontal asymptote at the x-axis: $y = 0$
  - This is the limiting value when the value of x gets very large (or very negative)
- A vertical asymptote at the y-axis: $x = 0$
  - This is the value that causes the denominator to be zero

The reciprocal graph, $y = \frac{a}{x} + b$ (where $a$ and $b$ are both constants)
- is the same shape as $y = \frac{a}{x}$
- but is shifted upwards by $b$ units
  - $y = \frac{a}{x} - 3$ would be $y = \frac{a}{x}$ shifted down by 3 units
- This means the horizontal asymptote also shifts up by $b$ units
  - The vertical asymptote remains on the y-axis

How do I draw exponential growth and decay?

The equation $y = k^{x}$ represents exponential growth when $k > 1$
- $y = k^{x}$ represents exponential decay when $0 < k < 1$
  - $k$ is positive but less than 1
Both of these graphs:
- have a horizontal asymptote at $y = 0$
- do not have a vertical asymptote
- have a $y$ -intercept of $(0, 1)$
The graph of $y = a k^{x} + b$ is a similar shape to $y = k^{x}$ , but there are some differences
- It is first stretched vertically by $a$
- It is then shifted $b$ units upwards
  - Therefore it has a horizontal asymptote at $y = b$
  - and a $y$ -intercept of $(0, a + b)$
For example, a population may be modelled as $y = 400 \times {(\frac{1}{2})}^{x} + 100$ , where $y$ is the population and $x$ represents time
- This is an exponential decay as $0 < k < 1$
- The initial population (when $x = 0$ ) will be 400 + 100 = 500
  - The $y$ -intercept is (0, 500)
- Over a long period of time (large $x$ -value) the population will settle to 100
  - The asymptote is at $y = 100$
Exponential decay can also be identified by a negative power using index laws
- ${(\frac{1}{2})}^{x} = {(2^{- 1})}^{x} = 2^{- x}$ so $y = 400 \times 2^{- x} + 100$ is the model above
- This has the form $y = k^{- x}$ where $k > 1$

Worked Example

Match the graphs to the equations.

5 different shapes of graph; exponential, reciprocal, negative quadratic, linear, and negative cubic

(1) $y = 0.6 x + 2$ , (2) $y = 3^{x}$ , (3) $y = - 0.7 x^{3}$ , (4) $y = \frac{4}{x}$ , (5) $y = - x^{2} + 3 x + 2$

Answer:

Starting with the equations,

(1) is a linear equation (y = mx + c) so matches the only straight line, graph D

(2) is an exponential equation with a positive coefficient so matches graph A

(3) is a cubic equation with a negative coefficient so matches graph E

(4) is a reciprocal equation with a positive coefficient so matches graph B

(5) is a quadratic equation with a negative coefficient so matches graph C

Graph A → Equation 2

Graph B → Equation 4

Graph C → Equation 5

Graph D → Equation 1

Graph E → Equation 3

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Questions students ask about Types of Graphs

Real questions from students revising this topic on Save My Exams

What does an exponential graph look like?

An exponential growth graph, y = kˣ with k > 1, is a smooth curve that starts almost flat just above the x-axis on the left, passes through (0, 1), then rises more and more steeply.

Key features of y = kˣ:

y-intercept (0, 1), because k⁰ = 1
Horizontal asymptote y = 0 — the curve gets closer to the x-axis but never touches it
No vertical asymptote, and the curve always stays above the x-axis

Exponential decay, y = kˣ with 0 < k < 1 (the same as y = k⁻ˣ with k > 1), is the mirror image: high on the left, through (0, 1), flattening towards the x-axis on the right.

For y = akˣ + b the curve is stretched vertically by a then shifted up by b, so the asymptote becomes y = b and the y-intercept (0, a + b).

Example from this page: y = 3ˣ is graph A. At x = 0, y = 1; at x = 1, y = 3; at x = 2, y = 9; at x = −1, y = ⅓.

What type of graph is y = kˣ?

y = kˣ is an exponential graph — the variable x sits in the power (the exponent), which is where the name comes from.

If k > 1 it is exponential growth: the curve rises more and more steeply
If 0 < k < 1 it is exponential decay: the curve falls and levels off towards the x-axis
Decay can also be written with a negative power, y = k⁻ˣ where k > 1
It models anything changing by a constant multiplier each step

How to tell it apart from a cubic: y = x³ has x as the base with a fixed power, while y = 3ˣ has a fixed base with x as the power.

Features of y = kˣ: y-intercept (0, 1), horizontal asymptote y = 0, no vertical asymptote.

Example from this page: y = 400 × (½)ˣ + 100 is decay, since (½)ˣ = (2⁻¹)ˣ = 2⁻ˣ. At x = 0 the population is 400 + 100 = 500, and over a long time it settles at the asymptote y = 100.

What are the shapes of the different types of graphs in GCSE Maths?

Each shape matches one type of equation: a straight line is linear, a parabola is quadratic, an S-shape is cubic, a hyperbola is reciprocal, and an exponential curve comes from y = kˣ.

Linear: y = mx + c — a straight line
Quadratic: y = ax² + bx + c — a parabola, U-shaped if a > 0, ∩-shaped if a < 0
Cubic: y = ax³ + bx² + cx — an S-shape, running bottom-left to top-right if a > 0 and the other way if a < 0
Reciprocal: y = a/x — two separate branches, with asymptotes x = 0 and y = 0
Exponential: y = kˣ — passes through (0, 1), asymptote y = 0
You also need the three trig graphs, which repeat at regular intervals

Example from this page: matching the equations to the graphs gives y = 0.6x + 2 → D, y = −x² + 3x + 2 → C, y = −0.7x³ → E, y = 4/x → B and y = 3ˣ → A.