What are non-linear functions in the context of IB MYP Mathematics?

In IB MYP Mathematics, non-linear functions are functions whose graphs do not form a straight line. These include quadratic functions, exponential functions, and logarithmic functions, among others. Unlike linear functions, non-linear functions can have curves, peaks, and troughs, making them essential for modeling more complex real-world scenarios.

How do you identify a non-linear function from its equation?

A non-linear function can be identified by the presence of variables raised to a power other than one, or variables in the denominator, or within a function like sine, cosine, or exponential. For example, the equation y = x^2 + 3 is non-linear because the variable x is squared. Similarly, y = 2^x is non-linear due to the exponential term.

What are some common examples of non-linear functions studied in the IB MYP curriculum?

Common examples of non-linear functions in the IB MYP curriculum include quadratic functions (e.g., y = ax^2 + bx + c), cubic functions (e.g., y = ax^3 + bx^2 + cx + d), exponential functions (e.g., y = a * b^x), and logarithmic functions (e.g., y = a * log_b(x)). Each of these functions has unique properties and applications.

How do non-linear functions differ from linear functions in terms of their graphs?

Non-linear functions differ from linear functions in that their graphs are not straight lines. While linear functions have a constant rate of change and are represented by straight lines, non-linear functions have varying rates of change and can be represented by curves, such as parabolas, hyperbolas, or exponential curves. This variation allows non-linear functions to model more complex relationships.

Why are non-linear functions important in real-world applications?

Non-linear functions are crucial in real-world applications because they can model complex phenomena that linear functions cannot. For instance, they are used in physics to describe motion under gravity, in biology to model population growth, and in economics to analyze market trends. Their ability to represent non-constant rates of change makes them invaluable in various scientific and engineering fields.

Why is "Forgetting the minus sign in $x = -\dfrac{b}{2a}$." a common mistake in Non Linear Functions?

Why it happens: Slight formula error. How to avoid it: Memorise WITH the minus. For y = x^2 + 6x, b = 6, so x = -62 = -3 at the vertex.

Why is "Confusing $a^x$ growth (constant ratio) with $x^a$ power (constant difference of logs)." a common mistake in Non Linear Functions?

Why it happens: Both involve exponents. How to avoid it: 2^x is EXPONENTIAL — base fixed, exponent is x. x^2 is a POWER FUNCTION — exponent fixed, base is x. These have very different shapes.

Why is "Saying the graph touches the asymptote at infinity." a common mistake in Non Linear Functions?

Why it happens: Imprecise. How to avoid it: Asymptote is APPROACHED but never reached. The graph gets arbitrarily close — but no actual touching point exists.

AlgebraIB MYP Maths Extended Level

Non Linear Functions

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Non-Linear Functions — IB MYP Mathematics (Extended): quadratic, cubic, reciprocal, exponential

Beyond straight lines, functions come in characteristic shapes. This MYP Mathematics Extended note covers the main non-linear function families and their key features.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Identify each function family from its formula or graph.
MYP Mathematics A — Find intercepts and key points of non-linear graphs.
MYP Mathematics A — Sketch the four key non-linear functions.
MYP Mathematics D — Apply to real situations (projectiles, growth, decay).

Quadratic — the parabola

$y = ax^2 + bx + c$ — U or ∩.

A quadratic $y = ax^2 + bx + c$ graphs as a PARABOLA.

Key features:

Opening direction: $a > 0$ → U-shape (opens UP). $a < 0$ → ∩-shape (opens DOWN).
Vertex (turning point): at $x = -\dfrac{b}{2a}$ . Substitute back to find $y$ .
Y-intercept: at $(0, c)$ .
X-intercepts (roots): solve $ax^2 + bx + c = 0$ .
Axis of symmetry: vertical line through the vertex.

Worked example. Sketch $y = x^2 - 4x + 3$ .

Opens UP ( $a = 1 > 0$ ).
Vertex: $x = -\dfrac{-4}{2} = 2$ . $y = 4 - 8 + 3 = -1$ . Vertex at $(2, -1)$ .
Y-intercept: $(0, 3)$ .
X-intercepts: $x^2 - 4x + 3 = (x-1)(x-3) = 0$ , so $x = 1, 3$ .
Sketch: U-shape, min at $(2, -1)$ , crossing x-axis at $1$ and $3$ .

Four standard non-linear shapes. Recognising these instantly is a key skill.

$a > 0$ : U. $a < 0$ : ∩.
Vertex at $x = -\dfrac{b}{2a}$ .
Roots from factorisation or formula.

Cubic, reciprocal, exponential

Three more families to recognise.

Cubic. $y = ax^3 + bx^2 + cx + d$ . The standard cubic $y = x^3$ goes from bottom-left to top-right (S-curve through origin).

Key features:

One inflection point (where curvature changes).
May have 0, 1 or 2 turning points.
ALWAYS spans all real values (range $\mathbb{R}$ ) unless restricted.

Reciprocal. $y = \dfrac{k}{x}$ . Two-branch HYPERBOLA.

Key features:

Asymptotes at $x = 0$ (vertical) and $y = 0$ (horizontal).
For $k > 0$ : branches in Q1 and Q3. For $k < 0$ : Q2 and Q4.
Domain: $x \ne 0$ . Range: $y \ne 0$ .

Asymptote = a line the graph approaches but NEVER touches.

Exponential. $y = a \cdot b^x$ where $b > 0, b \ne 1$ .

If $b > 1$ : GROWTH. Curve rises rapidly. Real-life: compound interest, bacterial growth.
If $0 < b < 1$ : DECAY. Curve falls rapidly. Real-life: radioactive decay, cooling.
Y-intercept: $(0, a)$ (since $b^0 = 1$ ).
Horizontal asymptote at $y = 0$ .

Worked examples:

$y = 2^x$ : grows from $(0, 1)$ .
$y = (1/2)^x$ : decays from $(0, 1)$ .
$y = 3 \cdot 4^x$ : starts at $(0, 3)$ , grows fast.

Cubic: S-shape; range $\mathbb{R}$ .
Reciprocal: hyperbola with asymptotes at $x = 0$ and $y = 0$ .
Exponential: growth ( $b > 1$ ) or decay ( $0 < b < 1$ ).
Exponential y-intercept at $(0, a)$ .

Real-life applications

Each family models particular real-world phenomena.

Quadratic models:

Projectile motion (height as function of time): $h = -5t^2 + ut + h_0$ .
Optimisation: max area for a given perimeter, max profit for a given price.

Cubic models:

Volume of a cube as function of side: $V = s^3$ .
Some growth processes with thresholds.

Reciprocal models (inverse proportion):

Speed vs time at fixed distance: $s = \dfrac{d}{t}$ .
Pressure vs volume in gases (Boyle's law).
Brightness of a light source vs distance (more precisely $\dfrac{1}{d^2}$ ).

Exponential models:

Compound interest: $A = P (1 + r)^t$ .
Population growth (when unconstrained).
Radioactive decay.
Cooling of a hot object (Newton's law of cooling).

MYP criterion D: identifying which model fits a given context is a high-value skill. Look at the SHAPE of the data, the units, the underlying mechanism. A growth that doubles every fixed period is exponential. A drop with bounded rebound is a parabola.

Projectile → quadratic.
Compound interest → exponential.
Inverse proportion → reciprocal.
Pick the model whose SHAPE matches your data.

How it’s examined

Criterion A: identify and sketch each family. Criterion C: clear sketch with key points labelled. Criterion D: choose appropriate model for real data.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Non Linear Functions, ready to print or save as PDF.

Step-by-step worked examples — Non Linear Functions

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

1Find the vertex of a quadratic
Getting started• vertex
Question
Find the vertex of $y = x^2 - 6x + 5$ .
Step-by-step solution
1. Step 1
  Vertex at $x = -\dfrac{b}{2a} = -\dfrac{-6}{2} = 3$ .
2. Step 2
  $y$ -value: $y = 9 - 18 + 5 = -4$ .
3. Step 3
  Vertex at $(3, -4)$ .
Answer
Vertex $(3, -4)$ .
2Asymptotes of a reciprocal
Building confidence• asymptotes
Question
State the asymptotes of $y = \dfrac{3}{x - 2} + 1$ .
Step-by-step solution
1. Step 1
  Vertical asymptote: where denominator is zero. $x - 2 = 0 \Rightarrow x = 2$ .
2. Step 2
  Horizontal asymptote: as $x \to \pm\infty$ , $\dfrac{3}{x-2} \to 0$ , so $y \to 1$ .
3. Step 3
  Asymptotes: $x = 2$ (vertical) and $y = 1$ (horizontal).
Answer
$x = 2$ and $y = 1$ .
3Compound interest (exponential)
Building confidence• exponential
Question
$2000 invested at 4% per year. Find the amount after 10 years.
Step-by-step solution
1. Step 1
  Exponential model: $A = P (1 + r)^t = 2000 \times 1.04^{10}$ .
2. Step 2
  $1.04^{10} \approx 1.4802$ .
3. Step 3
  $A \approx 2000 \times 1.4802 = 2960.49$ .
Answer
$\approx \$ 2960.49$
4Identify the function family
Stretch• classification
Question
The graph passes through $(0, 1)$ , $(1, 3)$ , $(2, 9)$ , $(3, 27)$ . Which family does it belong to and what's the equation?
Step-by-step solution
1. Step 1
  Check ratio: $3/1 = 3$ , $9/3 = 3$ , $27/9 = 3$ . Constant ratio → EXPONENTIAL.
2. Step 2
  $y$ -intercept is 1 at $x = 0$ , so $a = 1$ and $b = 3$ .
3. Step 3
  Equation: $y = 3^x$ .
Answer
Exponential: $y = 3^x$ .

Key Definitions and Keywords — Non Linear Functions

Definitions to memorise and the exact keywords mark schemes credit for non linear functions answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Vertex (of a parabola)
Examiner keyword
The turning point of a parabola — minimum if opens up, maximum if opens down.
Asymptote
Examiner keyword
A line that a graph approaches but never touches.
Exponential function
Examiner keyword
A function of the form $f(x) = a \cdot b^x$ with $b > 0$ , $b \ne 1$ .
Axis of symmetry
A line about which a graph is symmetric. For a parabola, it passes through the vertex.

Common Mistakes and Misconceptions — Non Linear Functions

The traps other students keep falling into on non linear functions questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Forgetting the minus sign in $x = -\dfrac{b}{2a}$ .
Why it happens
Slight formula error.
How to avoid it
Memorise WITH the minus. For $y = x^2 + 6x$ , $b = 6$ , so $x = -\tfrac{6}{2} = -3$ at the vertex.
✕Confusing $a^x$ growth (constant ratio) with $x^a$ power (constant difference of logs).
Why it happens
Both involve exponents.
How to avoid it
$2^x$ is EXPONENTIAL — base fixed, exponent is $x$ . $x^2$ is a POWER FUNCTION — exponent fixed, base is $x$ . These have very different shapes.
✕Saying the graph touches the asymptote at infinity.
Why it happens
Imprecise.
How to avoid it
Asymptote is APPROACHED but never reached. The graph gets arbitrarily close — but no actual touching point exists.

Practice questions

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

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Non Linear Functions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

AlgebraIB MYP Maths Extended Level

Non Linear Functions

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Short Study Notes — Non Linear Functions

Start with these resources to cover the key concepts, then work through the practice questions.

Page 1 / 0

Detailed Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Non Linear Functions, ready to print or save offline.

Non-Linear Functions — IB MYP Mathematics (Extended): quadratic, cubic, reciprocal, exponential

Beyond straight lines, functions come in characteristic shapes. This MYP Mathematics Extended note covers the main non-linear function families and their key features.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Identify each function family from its formula or graph.
MYP Mathematics A — Find intercepts and key points of non-linear graphs.
MYP Mathematics A — Sketch the four key non-linear functions.
MYP Mathematics D — Apply to real situations (projectiles, growth, decay).

Quadratic — the parabola

$y = ax^2 + bx + c$ — U or ∩.

A quadratic $y = ax^2 + bx + c$ graphs as a PARABOLA.

Key features:

Opening direction: $a > 0$ → U-shape (opens UP). $a < 0$ → ∩-shape (opens DOWN).
Vertex (turning point): at $x = -\dfrac{b}{2a}$ . Substitute back to find $y$ .
Y-intercept: at $(0, c)$ .
X-intercepts (roots): solve $ax^2 + bx + c = 0$ .
Axis of symmetry: vertical line through the vertex.

Worked example. Sketch $y = x^2 - 4x + 3$ .

Opens UP ( $a = 1 > 0$ ).
Vertex: $x = -\dfrac{-4}{2} = 2$ . $y = 4 - 8 + 3 = -1$ . Vertex at $(2, -1)$ .
Y-intercept: $(0, 3)$ .
X-intercepts: $x^2 - 4x + 3 = (x-1)(x-3) = 0$ , so $x = 1, 3$ .
Sketch: U-shape, min at $(2, -1)$ , crossing x-axis at $1$ and $3$ .

Four standard non-linear shapes. Recognising these instantly is a key skill.

$a > 0$ : U. $a < 0$ : ∩.
Vertex at $x = -\dfrac{b}{2a}$ .
Roots from factorisation or formula.

Cubic, reciprocal, exponential

Three more families to recognise.

Cubic. $y = ax^3 + bx^2 + cx + d$ . The standard cubic $y = x^3$ goes from bottom-left to top-right (S-curve through origin).

Key features:

One inflection point (where curvature changes).
May have 0, 1 or 2 turning points.
ALWAYS spans all real values (range $\mathbb{R}$ ) unless restricted.

Reciprocal. $y = \dfrac{k}{x}$ . Two-branch HYPERBOLA.

Key features:

Asymptotes at $x = 0$ (vertical) and $y = 0$ (horizontal).
For $k > 0$ : branches in Q1 and Q3. For $k < 0$ : Q2 and Q4.
Domain: $x \ne 0$ . Range: $y \ne 0$ .

Asymptote = a line the graph approaches but NEVER touches.

Exponential. $y = a \cdot b^x$ where $b > 0, b \ne 1$ .

If $b > 1$ : GROWTH. Curve rises rapidly. Real-life: compound interest, bacterial growth.
If $0 < b < 1$ : DECAY. Curve falls rapidly. Real-life: radioactive decay, cooling.
Y-intercept: $(0, a)$ (since $b^0 = 1$ ).
Horizontal asymptote at $y = 0$ .

Worked examples:

$y = 2^x$ : grows from $(0, 1)$ .
$y = (1/2)^x$ : decays from $(0, 1)$ .
$y = 3 \cdot 4^x$ : starts at $(0, 3)$ , grows fast.

Cubic: S-shape; range $\mathbb{R}$ .
Reciprocal: hyperbola with asymptotes at $x = 0$ and $y = 0$ .
Exponential: growth ( $b > 1$ ) or decay ( $0 < b < 1$ ).
Exponential y-intercept at $(0, a)$ .

Real-life applications

Each family models particular real-world phenomena.

Quadratic models:

Projectile motion (height as function of time): $h = -5t^2 + ut + h_0$ .
Optimisation: max area for a given perimeter, max profit for a given price.

Cubic models:

Volume of a cube as function of side: $V = s^3$ .
Some growth processes with thresholds.

Reciprocal models (inverse proportion):

Speed vs time at fixed distance: $s = \dfrac{d}{t}$ .
Pressure vs volume in gases (Boyle's law).
Brightness of a light source vs distance (more precisely $\dfrac{1}{d^2}$ ).

Exponential models:

Compound interest: $A = P (1 + r)^t$ .
Population growth (when unconstrained).
Radioactive decay.
Cooling of a hot object (Newton's law of cooling).

Projectile → quadratic.
Compound interest → exponential.
Inverse proportion → reciprocal.
Pick the model whose SHAPE matches your data.

How it’s examined

Criterion A: identify and sketch each family. Criterion C: clear sketch with key points labelled. Criterion D: choose appropriate model for real data.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

Master this Topic

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

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Non Linear Functions, ready to print or save as PDF.

Step-by-step worked examples — Non Linear Functions

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

1Find the vertex of a quadratic
Getting started• vertex
Question
Find the vertex of $y = x^2 - 6x + 5$ .
Step-by-step solution
1. Step 1
  Vertex at $x = -\dfrac{b}{2a} = -\dfrac{-6}{2} = 3$ .
2. Step 2
  $y$ -value: $y = 9 - 18 + 5 = -4$ .
3. Step 3
  Vertex at $(3, -4)$ .
Answer
Vertex $(3, -4)$ .
2Asymptotes of a reciprocal
Building confidence• asymptotes
Question
State the asymptotes of $y = \dfrac{3}{x - 2} + 1$ .
Step-by-step solution
1. Step 1
  Vertical asymptote: where denominator is zero. $x - 2 = 0 \Rightarrow x = 2$ .
2. Step 2
  Horizontal asymptote: as $x \to \pm\infty$ , $\dfrac{3}{x-2} \to 0$ , so $y \to 1$ .
3. Step 3
  Asymptotes: $x = 2$ (vertical) and $y = 1$ (horizontal).
Answer
$x = 2$ and $y = 1$ .
3Compound interest (exponential)
Building confidence• exponential
Question
$2000 invested at 4% per year. Find the amount after 10 years.
Step-by-step solution
1. Step 1
  Exponential model: $A = P (1 + r)^t = 2000 \times 1.04^{10}$ .
2. Step 2
  $1.04^{10} \approx 1.4802$ .
3. Step 3
  $A \approx 2000 \times 1.4802 = 2960.49$ .
Answer
$\approx \$ 2960.49$
4Identify the function family
Stretch• classification
Question
The graph passes through $(0, 1)$ , $(1, 3)$ , $(2, 9)$ , $(3, 27)$ . Which family does it belong to and what's the equation?
Step-by-step solution
1. Step 1
  Check ratio: $3/1 = 3$ , $9/3 = 3$ , $27/9 = 3$ . Constant ratio → EXPONENTIAL.
2. Step 2
  $y$ -intercept is 1 at $x = 0$ , so $a = 1$ and $b = 3$ .
3. Step 3
  Equation: $y = 3^x$ .
Answer
Exponential: $y = 3^x$ .

Key Definitions and Keywords — Non Linear Functions

Definitions to memorise and the exact keywords mark schemes credit for non linear functions answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Vertex (of a parabola)
Examiner keyword
The turning point of a parabola — minimum if opens up, maximum if opens down.
Asymptote
Examiner keyword
A line that a graph approaches but never touches.
Exponential function
Examiner keyword
A function of the form $f(x) = a \cdot b^x$ with $b > 0$ , $b \ne 1$ .
Axis of symmetry
A line about which a graph is symmetric. For a parabola, it passes through the vertex.

Common Mistakes and Misconceptions — Non Linear Functions

The traps other students keep falling into on non linear functions questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Forgetting the minus sign in $x = -\dfrac{b}{2a}$ .
Why it happens
Slight formula error.
How to avoid it
Memorise WITH the minus. For $y = x^2 + 6x$ , $b = 6$ , so $x = -\tfrac{6}{2} = -3$ at the vertex.
✕Confusing $a^x$ growth (constant ratio) with $x^a$ power (constant difference of logs).
Why it happens
Both involve exponents.
How to avoid it
$2^x$ is EXPONENTIAL — base fixed, exponent is $x$ . $x^2$ is a POWER FUNCTION — exponent fixed, base is $x$ . These have very different shapes.
✕Saying the graph touches the asymptote at infinity.
Why it happens
Imprecise.
How to avoid it
Asymptote is APPROACHED but never reached. The graph gets arbitrarily close — but no actual touching point exists.

Practice questions

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

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Non Linear Functions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Non Linear Functions

Short Study Notes in the form of Slides

Short Study Notes — Non Linear Functions

Detailed Notes

What you’ll learn

How it’s examined

1Find the vertex of a quadratic

2Asymptotes of a reciprocal

3Compound interest (exponential)

4Identify the function family

Vertex (of a parabola)

Asymptote

Exponential function

Axis of symmetry

✕Forgetting the minus sign in x=−b2ax = -\dfrac{b}{2a}x=−2ab​.

✕Confusing axa^xax growth (constant ratio) with xax^axa power (constant difference of logs).

✕Saying the graph touches the asymptote at infinity.

Practice questions

Past paper style quiz

Assess your understanding

Non Linear Functions — frequently asked questions

Non Linear Functions

Short Study Notes in the form of Slides

Short Study Notes — Non Linear Functions

Detailed Notes

What you’ll learn

How it’s examined

1Find the vertex of a quadratic

2Asymptotes of a reciprocal

3Compound interest (exponential)

4Identify the function family

Vertex (of a parabola)

Asymptote

Exponential function

Axis of symmetry

✕Forgetting the minus sign in x=−b2ax = -\dfrac{b}{2a}x=−2ab​.

✕Confusing axa^xax growth (constant ratio) with xax^axa power (constant difference of logs).

✕Saying the graph touches the asymptote at infinity.

Practice questions

Past paper style quiz

Assess your understanding

Non Linear Functions — frequently asked questions

✕Forgetting the minus sign in $x = -\dfrac{b}{2a}$ .

✕Confusing $a^x$ growth (constant ratio) with $x^a$ power (constant difference of logs).

✕Forgetting the minus sign in $x = -\dfrac{b}{2a}$ .

✕Confusing $a^x$ growth (constant ratio) with $x^a$ power (constant difference of logs).