Modelling with standard functions

Q: When is a linear model used in real-world scenarios?

A linear model, $f(x) = mx + c$, is used when a variable changes by a constant amount for every unit of change in the independent variable.

Q: What are the key features of a quadratic model graph?

The key features include the vertex (maximum or minimum point), the axis of symmetry ($x = -\frac{b}{2a}$), the zeros/roots (x-intercepts), and the y-intercept ($c$).

Q: What defines an exponential model compared to a linear model?

Exponential models are used when a quantity changes by a percentage or ratio rather than a flat amount, resulting in rapid growth or decay.

Q: What is a horizontal asymptote in an exponential model?

A horizontal asymptote is a horizontal line $y = c$ that the graph approaches as $x$ becomes very large (or very small) but never actually reaches.

Q: What is the difference between direct and inverse variation in power models?

In direct variation ($n > 0$), $y$ increases as $x$ increases; in inverse variation ($n < 0$), $y$ decreases as $x$ increases.

Q: How many "turns" can a cubic model have?

A cubic model ($y = ax^3 + bx^2 + cx + d$) can have up to two "turns," consisting of one local maximum and one local minimum.

Q: What do the parameters $a$ and $d$ represent in a sinusoidal model?

$a$ represents the amplitude (distance from the middle to the peak), and $d$ represents the midline or principal axis (the average value).

Q: What is a common calculator setting mistake when working with sinusoidal models?

Students often forget to check if their calculator is in Radians or Degrees ; IB modelling questions usually require Radians .

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

Introduction to Modelling with Standard Functions

In the real world, things rarely happen in a perfectly straight line. To understand the world around us—from how a virus spreads to how a ball flies through the air—we use mathematical models. A model is simply using a function to represent a real-life situation. In this chapter, we will explore the "menu" of standard functions available to you in the IB Applications and Interpretation course. Think of these as different tools in a toolkit; your job is to learn which tool fits which job!

Note: For help with drawing these on your calculator or finding where they cross, check out the chapters on "Graphs, key features and intersections" and "Modelling skills".

1. Linear Models: $y = mx + c$

The simplest model is the linear model. We use this when something changes by a constant amount for every unit of time or distance.

The Equation: $f(x) = mx + c$

$m$ (The Gradient): The rate of change. For every 1 unit $x$ increases, $y$ changes by $m$.
$c$ (The y-intercept): The starting value when $x = 0$.

Real-world example: A taxi ride that costs a fixed booking fee of \$5 plus \$2 per kilometer. The model would be $C(x) = 2x + 5$.

2. Quadratic Models: $y = ax^2 + bx + c$

Quadratic models create a "U" shape (a parabola). They are perfect for situations that have a maximum or a minimum point.

The Equation: $f(x) = ax^2 + bx + c$

Key Features to Know:

The Vertex: The highest or lowest point of the curve.
Axis of Symmetry: The vertical line that cuts the parabola in half. It is always at $x = -\frac{b}{2a}$.
Zeros/Roots: Where the graph hits the x-axis (where $y = 0$).
y-intercept: The value of $c$.

Real-world example: Projectile motion. If you kick a football, it goes up, reaches a peak (the vertex), and comes back down. Its height over time is a quadratic model.

3. Exponential Models: Growth and Decay

Exponential functions are used when something changes by a percentage or a ratio rather than a flat amount. These functions grow or shrink very quickly!

The Equations:
1. $f(x) = k a^x + c$
2. $f(x) = k a^{-x} + c$
3. $f(x) = k e^{rx} + c$

Key Features:

Horizontal Asymptote: This is the line $y = c$. The graph will get closer and closer to this value but will never actually touch it. It represents a "floor" or "ceiling" for the model.
Growth vs. Decay: If the base is greater than 1 (or the exponent is positive), it's growth. If the base is between 0 and 1 (or the exponent is negative), it's decay.

Real-world example: Compound interest in a bank account (growth) or the cooling of a cup of coffee toward room temperature (decay). The room temperature would be the horizontal asymptote $c$.

4. Power Models: Direct and Inverse Variation

These models describe how one variable changes in proportion to another raised to a specific power.

The Equation: $f(x) = a x^n$

Direct Variation ($n > 0$): As $x$ goes up, $y$ goes up. For example, the area of a square $A = s^2$.
Inverse Variation ($n < 0$): As $x$ goes up, $y$ goes down. A common form is $y = \frac{a}{x}$ (where $n = -1$).
Vertical Asymptote: When $n$ is negative (inverse variation), the y-axis ($x = 0$) becomes a vertical asymptote because you cannot divide by zero!

Quick Tip: If you see a graph that looks like it's "hugging" the y-axis and the x-axis, it's likely an inverse variation model.

5. Cubic Models: $y = ax^3 + bx^2 + cx + d$

Cubic models can have two "turns" (a local maximum and a local minimum). They are useful for more complex data that changes direction twice.

Real-world example: The volume of a shape where all dimensions are changing, or certain complex cost functions in economics.

6. Sinusoidal Models: Periodic Waves

Sinusoidal models are used for periodic behavior—things that repeat in a regular cycle over time.

The Equations:
$f(x) = a \sin(bx) + d$
$f(x) = a \cos(bx) + d$

Key Features:

$a$ (Amplitude): The height from the middle of the wave to the top. It shows how far the values "stretch" from the center.
$d$ (The Principal Axis / Midline): The average value. The wave oscillates above and below this line $y = d$.
$b$: This affects the period (how long it takes for one full cycle to complete).

Real-world example: The rising and falling of tides, the height of a person on a Ferris wheel, or seasonal temperature changes.

Did you know? Your heart rate on an EKG monitor is a real-life version of a periodic (sinusoidal-style) function!

Summary Table for Quick Review

Linear: Constant rate of change. Constant slope.
Quadratic: Has a "turning point" (Vertex). Symmetric.
Exponential: Rapid change. Has a horizontal asymptote.
Inverse: One value increases as the other decreases. Hugs the axes.
Sinusoidal: Repetitive, wave-like motion.

Common Mistakes to Avoid

Confusing $x^2$ and $2^x$: Remember, if the $x$ is the "base" (bottom), it's Quadratic. If the $x$ is the "exponent" (top), it's Exponential.
Forgetting the Asymptote: When sketching exponential or inverse functions, always draw the dotted line for the asymptote and label it.
Calculator Radians: For Sinusoidal models, always check if your GDC (calculator) is in Radians or Degrees. Usually, IB modelling questions use Radians unless degrees are specified!

Key Takeaway: Choosing a model is about looking at the shape of the data. Does it go up forever (Linear/Exponential)? Does it turn back (Quadratic)? Does it repeat (Sinusoidal)? Once you identify the shape, you've found your model!

Quick check

Can you answer these now?

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

When is a linear model used in real-world scenarios?

A linear model, $f(x) = mx + c$, is used when a variable changes by a constant amount for every unit of change in the independent variable.

What are the key features of a quadratic model graph?

The key features include the vertex (maximum or minimum point), the axis of symmetry ($x = -\frac{b}{2a}$), the zeros/roots (x-intercepts), and the y-intercept ($c$).

What defines an exponential model compared to a linear model?

Exponential models are used when a quantity changes by a percentage or ratio rather than a flat amount, resulting in rapid growth or decay.

What is a horizontal asymptote in an exponential model?

A horizontal asymptote is a horizontal line $y = c$ that the graph approaches as $x$ becomes very large (or very small) but never actually reaches.

What is the difference between direct and inverse variation in power models?

In direct variation ($n > 0$), $y$ increases as $x$ increases; in inverse variation ($n < 0$), $y$ decreases as $x$ increases.

How many "turns" can a cubic model have?

A cubic model ($y = ax^3 + bx^2 + cx + d$) can have up to two "turns," consisting of one local maximum and one local minimum.

What do the parameters $a$ and $d$ represent in a sinusoidal model?

$a$ represents the amplitude (distance from the middle to the peak), and $d$ represents the midline or principal axis (the average value).

What is a common calculator setting mistake when working with sinusoidal models?

Students often forget to check if their calculator is in Radians or Degrees; IB modelling questions usually require Radians.

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