Logarithmic, sinusoidal, logistic and piecewise models (HL)

Mathematics - Applications and Interpretation · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Advanced Modeling

In your mathematical journey so far, you have likely used linear and simple exponential functions to describe the world. However, real life is rarely that simple! Sometimes things grow and then level off, like the height of a sunflower. Other things repeat in cycles, like the tides of the ocean. In this chapter, we explore Logarithmic, Sinusoidal, Logistic, and Piecewise models. These are the specialized tools you need to model complex, real-world phenomena accurately.

Note: While we focus on the shapes and formulas of these models here, you can find information on how to transform these graphs or linearize data in the surrounding chapters of Topic 2: Functions.

1. Exponential Half-Life Models

We know that exponential growth happens when something increases quickly. Half-life is the "flip side"—it describes how quickly something decays or disappears. It is specifically used to measure the time it takes for a quantity to reduce to exactly half of its initial value.

The Concept

If you start with $100$ grams of a radioactive substance and its half-life is $10$ years, after $10$ years you will have $50$ grams. After another $10$ years, you will have $25$ grams. It never actually reaches zero; it just keeps getting smaller!

The Math

The model usually looks like this:
$N(t) = N_0(0.5)^{\frac{t}{h}}$
Where:
$N_0$ is the initial amount.
$t$ is the time passed.
$h$ is the half-life (the time constant).
Alternatively, it can be written using the natural base $e$: $N(t) = N_0 e^{-kt}$.

Quick Tip: If a question asks for the "rate of decay," they are usually looking for the value of $k$. If they ask "how long until half is gone," they want the value of $t$ that makes the exponent result in a factor of $0.5$.

2. Natural Logarithmic Models

Logarithmic models are the "slow growers." They increase quickly at first but then slow down significantly as $x$ increases. These are perfect for modeling things like sound intensity (decibels) or the Richter scale for earthquakes.

The Formula

$f(x) = a + b \ln x$

Key things to remember about this shape:
1. The domain is $x > 0$. You cannot take the logarithm of a negative number or zero!
2. There is a vertical asymptote at $x = 0$.
3. If $b > 0$, the graph rises. If $b < 0$, the graph falls.

Real-World Example: The time it takes to learn a new skill often follows a logarithmic curve. You make huge progress in the first few days ($x$ is small), but then it takes much longer to make smaller gains as you become an expert ($x$ is large).

3. Sinusoidal Models (HL)

If something repeats in a regular cycle (periodic motion), we use a sinusoidal model. This could be anything from a Ferris wheel's height to the temperature throughout a year.

The Formula

$f(x) = a \sin(b(x - c)) + d$

Each letter (parameter) changes the graph in a specific way:
$a$ (Amplitude): This is the distance from the middle of the graph to the top (or bottom). It tells you the "intensity" of the wave.
$b$ (Frequency factor): This relates to the period (the time for one full cycle). The period is calculated as $\frac{2\pi}{b}$.
$c$ (Phase shift): This moves the graph left or right. It tells you where the cycle starts.
$d$ (Vertical shift): This is the mean value or the "midline" of the graph. It is the average value of the function.

Common Mistake: Students often confuse the range with the amplitude. If a wave goes from a height of $2$ to $10$, the midline ($d$) is $6$ and the amplitude ($a$) is $4$, not $8$!

4. Logistic Models

In many real scenarios, exponential growth cannot last forever because resources run out. This is where the Logistic Model comes in. It looks like an "S-curve." It starts with exponential growth, hits an inflection point, and then levels off at a maximum value.

The Formula

$f(x) = \frac{L}{1 + C e^{-kx}}$

Where:
$L$ is the carrying capacity (the upper limit or horizontal asymptote).
$k$ is the growth rate ($k > 0$).
$C$ is a constant determined by the initial conditions.

Did you know? This model is used by biologists to predict how many fish can live in a pond. The pond only has so much food and space ($L$), so the population can't grow forever!

Key Takeaway: As $x$ becomes very large, the term $C e^{-kx}$ goes to zero, leaving you with $f(x) = \frac{L}{1} = L$. This is why the graph levels off at $L$.

5. Piecewise Models

Sometimes, a single formula isn't enough to describe a situation because the "rules" change. A piecewise model uses different functions for different parts of the domain.

Example Structure

$f(x) = \begin{cases} 2x + 1, & x < 0 \\ x^2, & x \ge 0 \end{cases}$

This means: "If $x$ is less than $0$, use the line $2x + 1$. If $x$ is $0$ or more, use the parabola $x^2$."

Real-World Application: Income Tax

Governments often use piecewise functions for tax. You might pay $10\%$ on your first $\$20,000$, and $15\%$ on everything over that. The "rule" changes at the $\$20,000$ mark.

Step-by-Step for Piecewise Graphs:
1. Identify the "boundary" values where the function changes.
2. Draw each section only within its allowed $x$-range.
3. Check the endpoints. A filled-in circle means the point is included ($\le$ or $\ge$), and an open circle means it is not included ($<$ or $>$).

Chapter Summary

Half-life: Used for decay; how long to lose half the amount.
Logarithmic: Fast start, slow finish; vertical asymptote at $x=0$.
Sinusoidal: Cycles and waves; defined by amplitude, period, and shifts.
Logistic: S-shaped; growth that reaches a limit (carrying capacity).
Piecewise: Different rules for different intervals; perfect for taxes or tiered pricing.

Don't worry if these formulas look intimidating! On the IB exam, you will often use your Graphic Display Calculator (GDC) to find these parameters using "Regression" (fitting a curve to data points). Focus on understanding what each model represents in the real world.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of half-life in the context of exponential decay models?

Half-life is the time it takes for a quantity to reduce to exactly half of its initial value.

What is the standard formula for an exponential half-life model?

$N(t) = N_0(0.5)^{\frac{t}{h}}$, where $N_0$ is the initial amount and $h$ is the half-life.

What are the key characteristics of a natural logarithmic model's graph?

The graph has a vertical asymptote at $x = 0$, a domain of $x > 0$, and increases quickly at first before slowing down.

In the sinusoidal model $f(x) = a \sin(b(x - c)) + d$, what does the parameter $a$ represent?

The parameter $a$ is the amplitude, which is the distance from the midline to the maximum (or minimum) value.

How do you calculate the period of a sinusoidal function from the frequency factor $b$?

The period is calculated using the formula $P = \frac{2\pi}{b}$.

What does the parameter $d$ represent in a sinusoidal model?

$d$ represents the vertical shift, also known as the midline or the mean value of the function.

What is the "carrying capacity" in a logistic model?

The carrying capacity, denoted as $L$, is the maximum value or horizontal asymptote that the function approaches as $x$ becomes very large.

Why does the logistic function level off at $L$ as $x$ increases?

As $x \to \infty$, the term $C e^{-kx}$ approaches zero, causing the denominator to approach 1 and the function to approach $L$.

What defines a piecewise model?

A piecewise model uses different functions (rules) for different specific intervals of the domain.

How are inclusive and non-inclusive boundaries represented on a piecewise graph?

A filled-in circle represents an included point ($\le$ or $\ge$), while an open circle represents a non-included point ($ < $ or $ > $).

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