Parametric Models — AHL 2.10 Parametric models in context...

SECTION 1

Historical Context & Motivation

In everyday mathematics you are used to writing y as a function of x, such as y = 2x + 3. This works perfectly for simple lines and parabolas, but what happens when a curve loops back on itself, or when you need to know where an object is at every moment in time? A single equation y = f(x) struggles to capture these situations. Parametric equations were developed precisely to handle curves and motions that a simple y = f(x) relationship cannot describe. Instead of tying y directly to x, both coordinates are expressed through an independent third quantity — the parameter — which often represents time.

~300 BCE

Greek Conic Sections

Apollonius of Perga studied ellipses, parabolas, and hyperbolas by slicing cones — curves that cannot be written as a single y = f(x) over their full extent.

1637

Descartes & Coordinate Geometry

René Descartes introduced the Cartesian plane, linking algebra and geometry. This framework made it natural to express x and y separately in terms of a third variable.

1687

Newton's Principia

Isaac Newton described planetary orbits using time as a parameter, showing that x(t) and y(t) could model celestial motion with remarkable precision.

1900s

Engineering & Computer Graphics

Parametric curves became essential in CAD software, animation, and robotics, where smooth paths must be traced over time for design and control.

The central question this topic addresses is: How can we model paths, shapes, and real-world phenomena that depend on a changing quantity like time, angle, or distance? Parametric models give us a powerful and flexible answer.

SECTION 2

Core Principles & Definitions

A parametric model expresses the coordinates of a point as separate functions of a single independent variable called the parameter. Typically we write x = f(t) and y = g(t), where t is the parameter. As t changes, the point (x, y) traces out a curve in the Cartesian plane. Below are the foundational ideas you need to understand before working with these models.

The Parameter

An independent variable (often t for time or θ for angle) that drives both x and y. It does not appear on the final graph but controls the motion along the curve.

Parametric Equations

A pair of functions x = f(t) and y = g(t) that together describe every point on the curve. Each value of t produces one specific (x, y) point.

Domain of the Parameter

The set of allowable t-values determines which part of the curve is traced. Restricting or extending the domain changes the portion of the curve you see.

Eliminating the Parameter

Sometimes you can combine the two equations to remove t and obtain a single Cartesian equation y = f(x) or an implicit form. This is called elimination and reveals the overall shape.

Direction & Speed

Because the parameter often represents time, parametric models carry information about which direction the curve is traced and how fast the point moves — something a Cartesian equation alone cannot show.

✦ KEY TAKEAWAY

Think of a parametric model like a GPS tracker on a delivery drone. The GPS doesn't just record the drone's path on a map (that would be the Cartesian equation). It also records when the drone was at each location. The parameter t is like the timestamp — it tells you the drone's x-position and y-position at every instant, giving you the full story of the journey, not just the route.

SECTION 3

Visual Explanation — Tracing a Parametric Curve

The diagram below shows how the parametric equations x = 2cos(t) and y = sin(t) trace out an ellipse as the parameter t increases from 0 to 2π. Notice how the left panel shows the individual graphs of x(t) and y(t), while the right panel shows the resulting path in the xy-plane with arrows indicating direction.

Left: the individual component functions x(t) = 2cos(t) (solid purple) and y(t) = sin(t) (dashed pink) plotted against the parameter t. Right: the resulting ellipse in the xy-plane with key parameter values labelled. The arrow shows the anticlockwise direction of motion as t increases from 0 to 2π.

This visual makes a key point clear: the Cartesian curve (the ellipse) contains no information about timing or direction. But the parametric form tells you that the point starts at (2, 0) when t = 0, rises to (0, 1) at t = π/2, passes through (−2, 0) at t = π, dips to (0, −1) at t = 3π/2, and returns to the start at t = 2π. That extra layer of information — when and where — is what makes parametric models so useful in real-world contexts.

SECTION 4

Mathematical Framework

The mathematical heart of parametric models is straightforward. You define two functions of a single parameter and, when needed, eliminate the parameter to find the Cartesian equivalent. Below are the essential formulas and techniques.

PARAMETRIC FORM

x = f(t), y = g(t), t ∈ [a, b]

f and g are functions of the parameter t. The interval [a, b] specifies the domain of t that determines which portion of the curve is traced.

ELIMINATING THE PARAMETER (LINEAR)

x = a + bt, y = c + dt → solve for t from one equation, substitute into the other

For linear parametric equations, isolate t = (x − a)/b and substitute into y = c + d·[(x − a)/b], yielding a Cartesian line.

ELIMINATING THE PARAMETER (TRIGONOMETRIC)

x = A cos(t), y = B sin(t) → (x/A)² + (y/B)² = 1

Use the identity cos²(t) + sin²(t) = 1. Replace cos(t) = x/A and sin(t) = y/B, then square and add. This reveals the Cartesian equation of an ellipse with semi-axes A and B.

GRADIENT OF A PARAMETRIC CURVE

dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0

To find the slope at any point on the parametric curve, differentiate y and x separately with respect to t and then divide. This is derived from the chain rule and is extremely useful for finding tangent lines.

💡 IB Exam Tip

In IB Applications and Interpretation HL, you are expected to use your GDC to graph parametric equations. Make sure you know how to switch your calculator to parametric mode and set the t-range and t-step appropriately. A t-step that is too large will produce a jagged curve.

SECTION 5

Real-World Parametric Models

Parametric equations are not just a mathematical curiosity — they model real phenomena that you encounter in science, engineering, and everyday life. The table below summarises the most common parametric models you will see in IB Maths AI HL and the contexts where they arise.

Common parametric models encountered in IB Maths AI HL

Model Type	Parametric Equations	Real-World Context
Straight-line motion	x = x₀ + v_xt, y = y₀ + v_yt	An aircraft flying at constant velocity; a ship crossing a strait
Circular motion	x = r cos(t), y = r sin(t)	A satellite orbiting Earth; a Ferris wheel rider's position over time
Elliptical path	x = a cos(t), y = b sin(t)	Planetary orbits; an oval running track
Projectile motion	x = v₀ cos(α)·t, y = v₀ sin(α)·t − ½gt²	A basketball shot; a water fountain jet; a firework trajectory
Lissajous figures	x = A sin(at + δ), y = B sin(bt)	Oscilloscope patterns; vibration analysis in engineering

A projectile launched at angle α with initial speed v₀. The parabolic trajectory is traced as t increases. Each labelled point shows the position at a specific time. Notice how the horizontal spacing is uniform (constant horizontal velocity) while the vertical motion shows the effect of gravity.

The projectile motion model is one of the most important parametric models you will encounter. The horizontal equation is linear in t because no force acts horizontally (ignoring air resistance), while the vertical equation is quadratic because gravity creates constant downward acceleration. The beauty of the parametric approach is that you can determine the position at any moment — information that the Cartesian parabola alone cannot provide.

SECTION 6

Worked Example — Ferris Wheel

A Ferris wheel has a radius of 20 m and its centre is 22 m above the ground. It completes one full revolution every 60 seconds, rotating anticlockwise. A rider boards at the lowest point. Write parametric equations for the rider's position and find their height after 10 seconds.

Ferris Wheel Parametric Model

Step 1 — Set up the coordinate systemPlace the origin at the centre of the wheel. The centre is 22 m above the ground, and the radius is 20 m, so the lowest point is at 22 − 20 = 2 m above the ground. In our coordinate system, the rider starts at the point (0, −20) relative to the centre.

Step 2 — Determine the angular speedOne full revolution is 2π radians in 60 seconds, so the angular speed is ω = 2π/60 = π/30 radians per second.

ω = π/30 rad/s

Step 3 — Write the parametric equations relative to the centreFor anticlockwise rotation starting at the bottom, the rider's position relative to the centre is x(t) = 20 sin(πt/30) and y(t) = −20 cos(πt/30). The sine and cosine are chosen so that at t = 0, x = 0 and y = −20 (the bottom).

x(t) = 20 sin(πt/30), y(t) = −20 cos(πt/30)

Step 4 — Adjust for the height above groundSince the centre is 22 m above the ground, the actual height h(t) above the ground is h(t) = 22 − 20 cos(πt/30). The x-equation stays the same if we keep the horizontal origin at the centre of the wheel.

h(t) = 22 − 20 cos(πt/30)

Step 5 — Find the height at t = 10 sSubstitute t = 10: h(10) = 22 − 20 cos(10π/30) = 22 − 20 cos(π/3) = 22 − 20 × 0.5 = 22 − 10 = 12 m.

h(10) = 12 m above the ground

🔑 Why start with −cos?

We use −cos(t) for the vertical component because the rider starts at the bottom of the wheel. The cosine function starts at its maximum value (1), so placing a negative sign shifts the starting point to the minimum (−1 × 20 = −20), which corresponds to the lowest point of the circle.

SECTION 7

Parametric vs. Cartesian — Strengths & Limitations

Both parametric and Cartesian forms have their place. The table below helps you understand when each approach is most appropriate and where each form falls short.

Comparing parametric and Cartesian representations

Feature	Parametric Form	Cartesian Form
Direction of travel	✓ Naturally encodes direction as t increases	✗ No directional information
Timing / speed	✓ t often represents time, so speed is built in	✗ Does not capture when a point is reached
Curves that loop	✓ Easily represents loops, spirals, and self-intersecting curves	✗ Fails the vertical line test; must split into pieces
Simplicity	✗ Requires two equations instead of one	✓ Single equation is often simpler to manipulate
Graphing by hand	✗ Need a table of t-values; slower to sketch	✓ Standard techniques (intercepts, symmetry) work directly
Technology / GDC	✓ Most calculators have a parametric mode	✓ Standard function mode is well-supported

✦ KEY TAKEAWAY

Choosing between parametric and Cartesian is like choosing between a video and a photograph. A Cartesian equation gives you the final picture — the shape of the curve. A parametric model gives you the full video — you see how the curve is traced out over time, frame by frame. When the motion matters, go parametric.

SECTION 8

Connection to Further Topics

The parametric models introduced here lay the groundwork for several advanced topics you will encounter later in IB HL and beyond. Understanding how x and y depend on a third variable prepares you for work in calculus, vectors, and applied modelling.

How parametric models connect to future learning

This Lesson (AHL 2.10 Intro)	Where It Leads
Writing parametric equations from context	AHL 2.10 continued: fitting parametric models to data using technology
Eliminating the parameter	Connecting parametric curves to known Cartesian forms (conics, polynomials)
Gradient dy/dx from parametric form	AHL 5.13: Differentiation of parametric functions and related rates
Position as (x(t), y(t))	AHL 3.12–3.13: Vector functions of time, velocity and acceleration vectors
Modelling circular / elliptical motion	Physics HL: uniform circular motion, Kepler's laws of planetary orbits

In particular, the move from parametric curves to vector-valued functions is quite natural: instead of writing x = f(t) and y = g(t) separately, you can package them into a single vector r(t) = (f(t), g(t)). Differentiating this vector with respect to t gives the velocity vector, and differentiating again gives the acceleration vector. This is the language of motion in physics and engineering, and it all starts with the parametric thinking you are developing right now.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain, in your own words, why a parametric model provides more information about a curve than a Cartesian equation. Give one example where this extra information is important.

PROBLEM 2 — BASIC CALCULATION

A particle moves along a path defined by x = 3t + 1 and y = 2t − 4. Find the Cartesian equation of the path by eliminating the parameter t.

PROBLEM 3 — INTERMEDIATE

A curve is defined by x = 4 cos(t) and y = 3 sin(t) for 0 ≤ t ≤ 2π. (a) Eliminate the parameter to find the Cartesian equation. (b) State the lengths of the semi-major and semi-minor axes. (c) Find the coordinates of the point when t = π/4.

PROBLEM 4 — APPLIED

A football is kicked from ground level at an initial speed of 25 m/s at an angle of 40° to the horizontal. Using g = 9.8 m/s², write parametric equations for the ball's position and find: (a) the time when the ball reaches its maximum height, (b) the maximum height, and (c) the horizontal distance (range) when it returns to the ground.

PROBLEM 5 — CRITICAL THINKING

Two boats leave different points at the same time. Boat A starts at (0, 10) with x = 5t and y = 10 − 2t. Boat B starts at (50, 0) with x = 50 − 3t and y = 4t. (a) Do the paths of the two boats intersect? (b) If so, do the boats actually collide — meaning they are at the same point at the same time? Justify your answer.

SUMMARY

Lesson Summary

A parametric model expresses the coordinates of a curve as separate functions of a parameter — typically time — using the form x = f(t), y = g(t). This approach captures not only the shape of the curve but also the direction and timing of motion along it. You can eliminate the parameter to convert back to a Cartesian equation, revealing the overall geometric shape — for instance, using the Pythagorean identity to turn x = a cos(t), y = b sin(t) into the equation of an ellipse.

Common real-world applications include projectile motion (where horizontal position is linear in t and vertical position is quadratic), circular and elliptical motion (modelled with trigonometric functions), and straight-line paths (modelled with linear functions). On the IB exam, remember to use your GDC in parametric mode and always define the domain of t to match the context of the problem.