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A2 Level · Pure Mathematics 4 WMA14

Parametric Equations

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Parametric Equations

Master curves that move: express x and y separately using a third variable (the parameter)

🎯 Quick Summary

The Big Idea

Instead of y = f(x), use a third variable t (the parameter): x = f(t) and y = g(t)

Why It Matters

Perfect for describing motion, curves that "loop back", and situations where x and y depend on time

Key Skill 1

Plot parametric curves by building a table of values as the parameter changes

Key Skill 2

Eliminate the parameter to find the standard Cartesian equation y = f(x)

Key Skill 3

Recognize circles and ellipses: use trig identities to spot them instantly

Key Skill 4

Sketch the curve by finding intercepts, asymptotes, and the shape of the path

1️⃣ The Basics: What Are Parametric Equations?

The Standard Form

A pair of equations where both x and y are written separately in terms of a third variable called the parameter:

Parametric Form

x = f(t)

y = g(t)

where t is the parameter (usually time, or an angle θ)

Why Use Them?

In real life, many curves are easiest to describe using a third variable:

Motion problems: A ball's horizontal position x and height y both depend on time t
Curves that loop: Some curves go up and down or left and right in a way that doesn't fit y = f(x) nicely
Circles and ellipses: Much simpler as x = r cos θ, y = r sin θ than as x² + y² = r²

Key Insight

A parametric curve is a path traced out over time. Each value of t gives you one point (x, y) on that path. As t increases, you move along the curve.

How to Plot a Parametric Curve

Step 1: Make a table. Choose values of t (usually evenly spaced, like t = −3, −2, −1, 0, 1, 2, 3).

Step 2: For each value of t, calculate the corresponding x and y.

Step 3: Plot the (x, y) points on a grid.

Step 4: Join them smoothly, respecting the direction of the path as t increases.

Worked Example: Plotting a Simple Parabola

Plot the curve given by:
x = 2t + 1 and y = t² − 4 for −3 ≤ t ≤ 3

✓ Solution:

Step 1: Build the table
When t = −3: x = 2(−3) + 1 = −5, y = (−3)² − 4 = 5 → point (−5, 5)
When t = −2: x = 2(−2) + 1 = −3, y = (−2)² − 4 = 0 → point (−3, 0)
When t = −1: x = 2(−1) + 1 = −1, y = (−1)² − 4 = −3 → point (−1, −3)
When t = 0: x = 2(0) + 1 = 1, y = 0² − 4 = −4 → point (1, −4)
When t = 1: x = 2(1) + 1 = 3, y = 1² − 4 = −3 → point (3, −3)
When t = 2: x = 2(2) + 1 = 5, y = 2² − 4 = 0 → point (5, 0)
When t = 3: x = 2(3) + 1 = 7, y = 3² − 4 = 5 → point (7, 5)

Step 2: Plot and join
Plot the seven points and join them with a smooth curve. The shape is a parabola opening upward (because y = t²). The lowest point is at (1, −4) when t = 0.

Parametric Equations of a Circle

the classic example

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Also in the full note

2️⃣ Eliminating the Parameter: From Parametric to Cartesian
3️⃣ Sketching Graphs of Parametric Curves
📝 What to Memorise
✅ Concepts Checklist
⚠️ Exam Tips & Common Mistakes
What Does "Eliminating the Parameter" Mean?
Method 1: Substitution (For Linear and Simple Equations)
Method 2: Using Trigonometric Identities

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Parametric Equations

📊 Parametric Equations

Parametric Equations

🎯 Quick Summary

The Big Idea

Why It Matters

Key Skill 1

Key Skill 2

Key Skill 3

Key Skill 4

1️⃣ The Basics: What Are Parametric Equations?

The Standard Form

Parametric Form

Why Use Them?

Key Insight

How to Plot a Parametric Curve

Worked Example: Plotting a Simple Parabola

Parametric Equations of a Circle

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