Parametric equations of curves

Welcome to Parametric Equations!

In your math journey so far, you’ve mostly seen curves described by a single equation like \(y = x^2 + 2\). This is called a Cartesian equation. But sometimes, it’s much easier to describe a journey by looking at the \(x\) and \(y\) coordinates separately over time.

In this chapter, we will learn how to use a third variable—a "middleman"—to link \(x\) and \(y\) together. Don't worry if this seems a bit strange at first! By the end of these notes, you'll see that parametric equations are just a different way of telling the same story.

1. What are Parametric Equations?

Instead of one equation connecting \(x\) and \(y\) directly, we use two equations. Both \(x\) and \(y\) are defined in terms of a third variable called the parameter.

The parameter is usually represented by the letter \(t\) (often standing for time) or \(\theta\) (often representing an angle).

An Everyday Analogy

Imagine you are tracking a drone flying through the air.
- The \(x\)-equation tells you how far east the drone is at any time \(t\).
- The \(y\)-equation tells you how high up the drone is at that same time \(t\).
Together, these two separate pieces of information tell you exactly where the drone is located.

Key Terms to Remember

1. Parametric Equations: A set of equations that express a group of quantities as functions of one or more independent variables called parameters.
2. Parameter: The independent variable (like \(t\)) that both \(x\) and \(y\) depend on.
3. Cartesian Equation: The "standard" equation involving only \(x\) and \(y\) (and constants).

Quick Review Box:
- Parametric: \(x = f(t)\) and \(y = g(t)\)
- Cartesian: \(y = f(x)\)

2. Converting Parametric to Cartesian

One of the most common tasks you will face is "eliminating the parameter." This means turning your two parametric equations back into one single \(x\) and \(y\) equation. There are two main ways to do this:

Method A: Substitution (The Algebraic Way)

This works best when the equations involve simple algebra.

Step 1: Rearrange the simplest equation (usually the \(x\) one) to make \(t\) the subject.
Step 2: Substitute this expression for \(t\) into the other equation.
Step 3: Simplify the result.

Example:
If \(x = t + 3\) and \(y = t^2\)
From the first equation: \(t = x - 3\)
Substitute into the second: \(y = (x - 3)^2\)
Now you have a Cartesian equation!

Method B: Using Trig Identities

When you see \(\sin\) or \(\cos\) in your parametric equations, you usually need your trigonometric identities. The most famous one is:
\(\sin^2\theta + \cos^2\theta = 1\)

Step 1: Rearrange the equations to isolate \(\sin\theta\) and \(\cos\theta\).
Step 2: Square both expressions.
Step 3: Add them together and set them equal to 1.

Memory Aid: Think of "S.O.S." — Segregate the trig, Operate by squaring, Sum them to 1.

Key Takeaway: Converting is just about "getting rid of the middleman" (\(t\) or \(\theta\)) to see the direct relationship between \(x\) and \(y\).

3. Sketching Parametric Curves

Sketching a parametric curve is a lot like plotting coordinates in middle school, just with one extra step.

Step-by-Step Sketching

1. Create a Table: Make three columns: \(t\), \(x\), and \(y\).
2. Choose Values for \(t\): If the question gives a range (like \(0 \le t \le 5\)), use those.
3. Calculate Coordinates: Plug each \(t\) value into your \(x\) and \(y\) equations to get pairs of \((x, y)\).
4. Plot and Join: Plot the \((x, y)\) points on a standard grid and join them with a smooth curve.
5. Show Direction: Draw small arrows on the curve to show the "direction of increasing \(t\)." This shows the path an object would take as time goes forward.

Common Mistake to Avoid: Don't plot \(t\) on your axes! Your graph axes should always be \(x\) and \(y\). \(t\) is the "ghost" variable that helps you find the points but doesn't appear on the final grid.

4. Parametric Equations in Context (Modelling)

Why do we bother with this? Because the real world moves in paths! Parametric equations are perfect for modelling.

Real-World Examples

- Projectiles: When you kick a football, its horizontal distance (\(x\)) and vertical height (\(y\)) change independently as time (\(t\)) passes.
- Robotics: A robotic arm moving to a specific coordinate might have its motor angles controlled by a parameter.
- Planetary Orbits: The position of a planet around a sun is often easier to describe using an angle \(\theta\) as the parameter.

Did you know?
Computer animators use parametric curves (often called Bezier curves) to create smooth shapes for characters in movies and video games!

5. Prerequisite Concepts Check

If you are struggling with this chapter, make sure you are comfortable with these "building block" skills:
- Substitution: Plugging one equation into another.
- Trigonometry: Knowing \(\sin\), \(\cos\), and the identity \(\sin^2\theta + \cos^2\theta = 1\).
- Rearranging Formulae: Making a specific letter the "subject" of an equation.

Chapter Summary & Key Takeaways

- Parametric equations use a parameter (\(t\) or \(\theta\)) to link \(x\) and \(y\).
- To turn them into a Cartesian equation, eliminate the parameter using substitution or trig identities.
- To sketch the curve, create a table of values for \(t\), find the \((x, y)\) points, and plot them.
- Always include arrows on your sketch to show the direction of movement as \(t\) increases.
- These equations are vital for modelling real-life movement like projectiles and orbits.

Keep practicing! Parametric equations are like learning a new language—at first, it's confusing, but soon you'll be "speaking" in curves and parameters fluently!