Home / IB DP Maths 2026, 2027 & 2028 / Application and Interpretation HL / IB Mathematics AI SL Functions models MAI Study Notes

IB Mathematics AI SL Functions models MAI Study Notes - New Syllabus

IB Mathematics AI SL Functions models MAI Study Notes

LEARNING OBJECTIVE

  • Modelling with the Different kind of functions.

Key Concepts: 

  • Linear Models

  • Quadratic Models

  • Exponential Growth and Decay Models

  • Direct and Inverse Variation

  • Cubic Models

  • Sinusoidal Models

  • Choosing and Applying Models

 

MAI HL and SL Notes – All topics

LINEAR MODELS

A linear model represents a relationship between two variables where the rate of change is constant. The graph is a straight line, and the model is used when changes in one quantity result in proportional changes in another.

Form:

$y = mx + c$

$m$: Slope (rate of change)
$c$: y-intercept

Usage:

Constant rate of change.

Example:

A taxi charges \$3 base fare and \$2 per km.

$\text{Cost} = 2x + 3$

Example:

A taxi company charges a flat fee of \$3 plus \$2 per kilometer.

Find the cost of a 10 km ride.

▶️Answer/Explanation

Solution:

$ \text{Cost} = 2x + 3 $

$ x = 10 \Rightarrow \text{Cost} = 2(10) + 3 = 20 + 3 = \boxed{23} $

QUADRATIC MODELS

A quadratic model describes a relationship where the rate of change itself changes at a constant rate. The graph is a parabola that opens upwards or downwards. It’s often used to model situations involving acceleration or area.

Form:

$y = ax^2 + bx + c$

Parabola shape
Vertex: $x = -\frac{b}{2a}$

Usage:

Projectile motion, area problems, acceleration.

Example:

Height $h$ of a ball thrown upward:

$h(t) = -4.9t^2 + 20t + 1.5$

By GDC:

TI-nspire $\implies$ MENU $\implies$ 6: ANALYZE GRAPH
$\implies$ 1: ZERO $\implies$ Choose lower & upper bound
$\implies$ ENTER $\implies$ Root given

TI-84 $\implies$ 2nd CALC $\implies$ 2: ZERO
$\implies$ Choose left & right bound
$\implies$ ENTER ENTER

Example: 

Find y-intercept , Axis of symmetry ,Vertex , Roots (by factoring) of  $f(x) = x^2 – 6x + 8$:

Sketch the graph using GDC

▶️Answer/Explanation

Solution:

y-intercept:

$
f(0) = 0^2 – 6(0) + 8 = 8 \Rightarrow (0, 8)
$

Axis of symmetry:

$
x = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3
$

Vertex:

$
x = 3, \quad f(3) = 3^2 – 6(3) + 8 = 9 – 18 + 8 = -1
\Rightarrow \text{Vertex at } (3, -1)
$

Roots (by factoring):

$
x^2 – 6x + 8 = (x – 2)(x – 4)
\Rightarrow \text{Roots at } (2, 0) \text{ and } (4, 0)
$

EXPONENTIAL GROWTH AND DECAY MODELS

Exponential models show situations where quantities grow or decrease at rates proportional to their current value. These models are used for population growth, radioactive decay, interest, and more.

Form:

Growth: $y = a(1 + r)^t$
Decay: $y = a(1 – r)^t$
or
General: $y = ab^x$

Where:

$a$: initial amount
$r$: growth/decay rate
$t$: time
$b > 1$ for growth, $0 < b < 1$ for decay

Example (growth): Population doubles every 5 years:

$P(t) = 500 \cdot 2^{t/5}$

Example: 

A population of bacteria doubles every 3 hours.

Initial population: 500.

Model:

$ P(t) = 500 \cdot 2^{t/3} $

What is the population after 6 hours?

▶️Answer/Explanation

Solution:

$ P(6) = 500 \cdot 2^{6/3} = 500 \cdot 2^2 = 500 \cdot 4 = \boxed{2000} $

DIRECT AND INVERSE VARIATION

Direct Variation –

A direct variation describes a linear relationship where one variable is a constant multiple of another (i.e., they increase or decrease together at the same ratio).

Form:

$y = kx$

$k$: constant of proportionality

Example:

If $y$ varies directly with $x$ and $y = 10$ when $x = 2$, then:

$k = \frac{y}{x} = 5 \Rightarrow y = 5x$

Inverse Variation –

An inverse variation describes a relationship where the product of two variables is constant. As one increases, the other decreases proportionally.

Form:

$y = \frac{k}{x}$

Example:

If $y$ varies inversely with $x$ and $y = 4$ when $x = 3$:

$k = yx = 12 \Rightarrow y = \frac{12}{x}$

(a) Direct Variation)

Example:

$y \propto x$, and $y = 12$ when $x = 4$.

Find: $y$ when $x = 6$

▶️Answer/Explanation

Solution:

$ y = kx \Rightarrow 12 = k(4) \Rightarrow k = 3 $

$ y = 3x \Rightarrow y = 3(6) = \boxed{18} $

(b) Inverse Variation)

Example:

$y \propto \frac{1}{x}$, and $y = 5$ when $x = 2$.

Find: $y$ when $x = 10$

▶️Answer/Explanation

Solution:

$ y = \frac{k}{x} \Rightarrow 5 = \frac{k}{2} \Rightarrow k = 10 $

$ y = \frac{10}{10} = \boxed{1} $

CUBIC MODELS

A cubic model represents relationships involving three-degree polynomials. The graph can have one or two turning points and is used to model more complex growth patterns or changes in direction.

Form:

$y = ax^3 + bx^2 + cx + d$

Can have up to 3 real roots and 2 turning points

Usage:

Non-linear motion, volume growth, economics.

Example:

$y = x^3 – 6x^2 + 11x – 6$

Example:

A function is defined by $ f(x) = x^3 – 3x^2 + 2x $

Find $f(2)$

▶️Answer/Explanation

Solution:

$ f(2) = (2)^3 – 3(2)^2 + 2(2) = 8 – 12 + 4 = \boxed{0} $

SINUSOIDAL MODELS

Sinusoidal models use sine or cosine functions to represent periodic phenomena that repeat at regular intervals, such as seasonal temperatures, sound waves, or tides.

Form:

$y = a \sin(bx + c) + d \quad \text{or} \quad y = a \cos(bx + c) + d$

Where:

$a$: amplitude
$b$: affects period $= \frac{2\pi}{|b|}$
$c$: phase shift
$d$: vertical shift

Usage:

Seasonal data, waves, pendulum motion.

Example: Temperature variation during a day

$T(t) = 10 \cos\left(\frac{\pi}{12}t – \frac{\pi}{2}\right) + 20$

Example:

Describe the key characteristics and transformations of the graph of the function $y = 2\sin(x – \frac{\pi}{4}) + 3$.

▶️Answer/Explanation

Solution:

Graph $y = 2\sin(x – \frac{\pi}{4}) + 3$.

Period: $\frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi$
Amplitude: $|A| = |2| = 2$
$C = \frac{\pi}{4}$, so the graph shifts right $\frac{\pi}{4}$
$D = 3$, so the graph shifts up 3

CHOOSING AND APPLYING MODELS

This refers to the process of selecting the most appropriate mathematical model to represent a set of data or real-world situation based on its behavior and characteristics.

Steps:

  • Understand the context: Is the relationship linear, curved, periodic?
  • Plot the data: Use scatter plots to visually identify the pattern.
  • Choose a model: Match the pattern (e.g., parabolic = quadratic, periodic = sinusoidal).
  • Find parameters: Use regression or algebraic manipulation.
  • Test the model: Compare predicted vs actual values.
  • Interpret results: Ensure they make sense in context.

Using Technology: This involves the use of tools like graphing calculators, software, or online platforms to visualize, analyze, and fit mathematical models to data accurately.

  • Plot data points
  • Use regression tools (LinReg, QuadReg, ExpReg, etc.)
  • Analyze residuals to assess model fit
  • Desmos or GeoGebra:
  • Visualize function behavior dynamically

  • Fit curves to data interactively

Example:

A population grows according to the model

$ P(t) = 1200 \cdot 1.03^t $

Estimate the population after 5 years.

▶️Answer/Explanation

Solution:

$ P(5) = 1200 \cdot 1.03^5 \approx 1200 \cdot 1.159 = \boxed{1390.8} $

Scroll to Top