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Parametric Equations

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Parametric Equations - Complete Interactive Lesson

Part 1: What Is a Parameter?

🧭 Parametric Equations

Part 1 of 7 — What Is a Parameter?

Topics in This Part

Section
Two Equations, One Curve
Building a Table of Values
Reading a Point Off the Curve

🔑 Key Concept: A parametric curve describes $x$ and $y$ separately, each as a function of a third variable called the parameter (usually $t$ ). Instead of $y = f(x)$ , you get $x = f(t)$ and $y = g(t)$ .

Two Equations, One Curve

A parametric equation gives the coordinates of a point as functions of a parameter $t$ :

$x = f(t), \qquad y = g(t)$

As $t$ runs through its values, the point traces out a curve.

Plug In the Parameter 🧮

For the curve $x = 2t + 1,\quad y = t^2 - 3$ , find the coordinates at each $t$ .

Concept Check 🎯

Match the Point 🔽

For $x = t^2,\quad y = 2t$ , choose the point produced by each $t$ -value.

Why Parametrize?

A plain equation $y = f(x)$ can't do everything:

It can't describe a curve where one $x$ has two $y$ -values (like a full circle) — that fails the vertical line test.
It throws away time information — you can't tell how fast or in what direction a point moves.

Parametric equations fix both problems. They let a curve loop, cross itself, or backtrack, and they record exactly when the point reaches each spot.

💡 In Part 2 we'll plot a parametric curve and track the direction of motion with arrows.

Part 2: Plotting & Direction of Motion

🧭 Parametric Equations

Part 2 of 7 — Plotting & Direction of Motion

🔑 The Idea: To graph a parametric curve, build a table of $t$ -values, plot the points in order, and draw arrows showing the direction the point travels as $t$ increases. That direction is called the orientation.

Plotting Step by Step

Graph $x = t + 1,\quad y = 2t$ for .

Part 3: Eliminating the Parameter

🧭 Parametric Equations

Part 3 of 7 — Eliminating the Parameter

🔑 The Goal: Convert $x = f(t),\; y = g(t)$ into a single relationship between $x$ and by . This reveals the of the curve (line, parabola, circle, …) in familiar Cartesian form.

Part 4: Modeling Motion

🧭 Parametric Equations

Part 4 of 7 — Modeling Motion

🔑 Big Payoff: Parametric equations are how we describe objects moving in a plane. With $t$ as time, $x(t)$ gives horizontal position and $y(t)$ gives vertical position — together they capture the full path and the timing.

Position as a Function of Time

A particle moves so that at time $t$ (seconds) its position is

Part 5: Parametrizing Lines & Circles

🧭 Parametric Equations

Part 5 of 7 — Parametrizing Lines & Circles

🔑 The Reverse Skill: Instead of being handed parametric equations, you'll now build them — writing parametric equations for a line through two points, or for a circle of a given radius.

Parametrizing a Line

To parametrize the line starting at $(x_0, y_0)$ and moving in a fixed direction:

Part 6: Ellipses, Speed & Transformations

🧭 Parametric Equations

Part 6 of 7 — Ellipses, Speed & Transformations

🔑 Beyond Circles: Changing the coefficients on $\cos t$ and $\sin t$ stretches a circle into an ellipse, and changing the coefficient inside (like $\cos 2t$ ) changes how fast the curve is traced.

From Circle to Ellipse

Use different radii in the $x$ - and -directions:

Part 7: Mixed Practice & Exit Quiz

🧭 Parametric Equations

Part 7 of 7 — Mixed Practice & Exit Quiz

You can now (1) evaluate parametric equations, (2) plot with orientation, (3) eliminate the parameter, (4) model motion in time, (5) parametrize lines and circles, and (6) handle ellipses and tracing speed. Let's put it all together.

Quick Reference

Goal	Key move
Find a point	Plug $t$ into both $x(t)$ and $y(t)$

(x, y) = \big(f(t),\, g(t)\big)

Think of $t$ as time. At each instant $t$ , the two formulas tell you where the moving point is.

$x = t, \qquad y = t^2$

$t$	$x = t$	$y = t^2$	Point $(x, y)$
$-2$	$-2$	$4$	$(-2, 4)$
$-1$

The points fall on the familiar parabola — but now we also know the order they're visited and the speed of travel.

🔑 Key Idea: A single $t$ -value produces a single point. Plug $t$ into both equations to get the $x$ - and $y$ -coordinates.

1) At $t = 0$ : $x = \,?$ 2) At $t = 0$ : $y = \,?$ 3) At $t = 3$ : $x = \,?$

$t$	$x = t+1$	$y = 2t$	Point
$-2$	$-1$	$-4$	$(-1, -4)$
$-1$	$0$	$-2$	$(0, -2)$
$0$	$1$	$0$	$(1, 0)$
$1$	$2$	$2$	$(2, 2)$
$2$	$3$	$4$	$(3, 4)$

Plot these in order. The curve is a line, and as $t$ increases the point moves up and to the right — that's the orientation.

⚠️ Don't forget the arrows. Two different parametrizations can trace the same line but in opposite directions. The orientation is part of the answer.

Direction Check 🎯

Build the Table 🧮

For $x = t^2,\quad y = t + 1$ , complete the missing coordinates.

1) At $t = -2$ : $x = \,?$ 2) At $t = -2$ : $y = \,?$ 3) At $t = 2$ : $x = \,?$ 4) At $t = 2$ : $y = \,?$

Trace the Curve 🔽

Consider $x = t,\quad y = t^2$ for $-2 \le t \le 2$ . Fill in the trip.

Method 1: Solve and Substitute

When one equation is easy to solve for $t$ , do that, then substitute into the other.

Example: $x = t + 2,\quad y = t^2$

Solve the simpler equation for $t$ : from $x = t + 2$ we get $t = x - 2$ .
Substitute into the other: $y = (x - 2)^2$ .

So the curve is the parabola

$y = (x - 2)^2$

with vertex $(2, 0)$ .

✅ Check: at $t = 0$ , the parametric point is $(2, 0)$ ; plugging $x = 2$ into $y = (x-2)^2$ gives . ✓

Concept Check 🎯

Method 2: The Pythagorean Identity

When $x$ and $y$ involve $\cos t$ and $\sin t$ , use the identity

$\cos^2 t + \sin^2 t = 1$

Example: $x = \cos t,\quad y = \sin t$

Then $x^2 + y^2 = \cos^2 t + \sin^2 t = 1$ , which is the :

$x^2 + y^2 = 1$

Example: $x = 3\cos t,\quad y = 3\sin t$

Here $\cos t = \dfrac{x}{3}$ and $\sin t = \dfrac{y}{3}$ , so

$\left(\tfrac{x}{3}\right)^2 + \left(\tfrac{y}{3}\right)^2 = 1 \;\Longrightarrow\; x^2 + y^2 = 9$

a circle of radius $3$ .

💡 Pattern: $x = r\cos t,\; y = r\sin t$ always gives the circle $x^2 + y^2 = r^2$ .

Pick the Right Method 🔽

Choose the best first step for eliminating the parameter, then the resulting curve.

Eliminate the Parameter 🧮

1) $x = 5\cos t,\; y = 5\sin t$ gives $x^2 + y^2 = \,?$ 2) $x = t + 4,\; y = t$ . Eliminating $t$ gives $y = x - \,?$ (enter the number subtracted) 3) $x = 2t,\; y = t$ . Eliminating $t$ gives $y = \dfrac{x}{?}$ (enter the denominator)

$x(t) = 1 + 2t, \qquad y(t) = 5 - t$

To find where it is at a given time, plug in $t$ :

$t$ (s)	$x = 1+2t$	$y = 5-t$	Position
$0$	$1$	$5$	$(1, 5)$
$1$

Each second, $x$ increases by $2$ and $y$ decreases by $1$ — steady motion to the right and down.

🔑 Reading the model: the coefficient of $t$ in each equation is the constant rate of change of that coordinate. Here the horizontal rate is $+2$ and the vertical rate is $-1$ unit per second.

Projectile Motion

A classic application: a ball launched horizontally. With horizontal speed $v$ and gravity, a common model is

$x(t) = v\,t, \qquad y(t) = h - 16t^2$

(using feet, with initial height $h$ ). The $x$ -equation is linear (constant horizontal speed); the $y$ -equation is quadratic (gravity pulls it down faster and faster).

Example: $x = 20t,\quad y = 100 - 16t^2$

Starting height: at $t = 0$ , $y = 100$ ft.
When does it hit the ground? Set $y = 0$ : $100 - 16 t^{2} = 0 \Rightarrow t^{2} = \frac{100}{}$ s.

💡 The curve traced is a parabola, but the parametric form tells us the time of every event — something $y = f(x)$ alone can't.

Modeling Check 🎯

Read the Rates 🔽

A particle moves with $x(t) = 4 - 3t,\quad y(t) = 2 + 5t$ . Interpret the motion.

Analyze the Motion 🧮

A ball is launched: $x(t) = 30t,\quad y(t) = 144 - 16t^2$ (feet, seconds).

1) Starting height (at $t = 0$ ): $y = \,?$ ft 2) Time to hit the ground ( $y = 0$ ): $t = \,?$ s 3) Horizontal distance when it lands: $x = \,?$ ft

x = x_0 + at, \qquad y = y_0 + bt

Here $(a, b)$ is the direction: each unit of $t$ adds $a$ to $x$ and $b$ to $y$ . At $t = 0$ you're at the start point.

Example: line through $(2, 1)$ and $(5, 7)$

Start at $(2, 1)$ : so $x_0 = 2,\; y_0 = 1$ .
Direction = (change in $x$ , change in $y$ ) = $(5 - 2,\; 7 - 1) = (3, 6)$ .

$x = 2 + 3t, \qquad y = 1 + 6t$

✅ Check: $t = 0$ gives $(2, 1)$ ✓ and $t = 1$ gives $(2+3,\, 1+6) = (5, 7)$ ✓.

Parametrize the Line 🧮

Write parametric equations for the line through $(1, 4)$ and $(7, 8)$ , starting at $(1, 4)$ with $t = 0$ there and $t = 1$ at $(7, 8)$ .

Form: $x = 1 + a\,t,\quad y = 4 + b\,t$ .

1) $a = \,?$ (the $x$ -direction) 2) $b = \,?$ (the $y$ -direction)

Parametrizing a Circle

A circle of radius $r$ centered at the origin is parametrized by

$x = r\cos t, \qquad y = r\sin t, \qquad 0 \le t \le 2\pi$

As $t$ goes from $0$ to $2\pi$ , the point sweeps counterclockwise once around, starting at $(r, 0)$ .

To shift the center to $(h, k)$ , just add the center coordinates:

$x = h + r\cos t, \qquad y = k + r\sin t$

Example: circle of radius $4$ centered at $(2, -3)$

$x = 2 + 4\cos t, \qquad y = -3 + 4\sin t$

💡 At $t = 0$ : $x = 2 + 4 = 6$ , $y = -3 + 0 = -3$ , so it starts at — the rightmost point of the circle.

Concept Check 🎯

Build the Parametrization 🔽

Match each curve to its parametric equations.

$x = a\cos t, \qquad y = b\sin t$

Eliminate the parameter with the Pythagorean identity. Since $\cos t = \dfrac{x}{a}$ and $\sin t = \dfrac{y}{b}$ :

$\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1$

That's the standard ellipse: it stretches $a$ units horizontally and $b$ units vertically.

Example: $x = 4\cos t,\quad y = 2\sin t$

$\frac{x^2}{16} + \frac{y^2}{4} = 1$

an ellipse $4$ wide (semi-axis) horizontally and $2$ tall vertically.

🔑 When $a = b$ the ellipse is just a circle of radius $a$ — the circle is the special "equal-axes" ellipse.

Concept Check 🎯

Shifting and Speeding

Translation. Adding constants moves the whole curve. The ellipse centered at $(h, k)$ is

$x = h + a\cos t, \qquad y = k + b\sin t$

Speed of tracing. Replacing $t$ with $2t$ inside makes the point go around twice as $t$ runs over the same interval. Compare over $0 \le t \le 2\pi$ :

Equations	Times around the circle
$x = \cos t,\; y = \sin t$	once
$x = \cos 2t,\; y = \sin 2t$

💡 The shape is identical (the unit circle in all three), but the timing/speed of the trace differs — another thing only parametric form can express.

Shape vs. Speed 🔽

All three trace the unit circle. Over $0 \le t \le 2\pi$ , identify how each behaves.

Read the Ellipse 🧮

For $x = 6\cos t,\quad y = 4\sin t$ :

1) The eliminated equation is $\dfrac{x^2}{?} + \dfrac{y^2}{16} = 1$ . Find the missing denominator. 2) Horizontal semi-axis length $= \,?$ 3) Vertical semi-axis length $= \,?$

⚠️ Remember: parametric form keeps orientation and timing information that $y = f(x)$ throws away.

Mixed Practice 🧮

1) For $x = 3t,\; y = t - 2$ , the position at $t = 4$ is $(\,?,\, 2)$ . Find the $x$ -coordinate. 2) Parametrize the circle of radius $7$ centered at the origin as $x = 7\cos t,\; y = ?\sin t$ . Enter the missing coefficient. 3) For the ellipse $x = 5\cos t,\; y = 3\sin t$ , the eliminated form is $\dfrac{x^2}{25} + \dfrac{y^2}{?} = 1$ . Find the denominator.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.

100 - 16 t^{2} = 0 \Rightarrow t^{2} = \frac{100}{16} = 6.25 \Rightarrow t = 2.5

How far horizontally? at

t = 2.5

x = 20(2.5) = 50

ft.

Parametric Equations

Parametric Equations - Complete Interactive Lesson

Part 1: What Is a Parameter?

🧭 Parametric Equations

Topics in This Part

Two Equations, One Curve

Why Parametrize?

Part 2: Plotting & Direction of Motion

🧭 Parametric Equations

Plotting Step by Step

Part 3: Eliminating the Parameter

🧭 Parametric Equations

Part 4: Modeling Motion

🧭 Parametric Equations

Position as a Function of Time

Part 5: Parametrizing Lines & Circles

🧭 Parametric Equations

Parametrizing a Line

Part 6: Ellipses, Speed & Transformations

🧭 Parametric Equations

From Circle to Ellipse

Part 7: Mixed Practice & Exit Quiz

🧭 Parametric Equations

Quick Reference

A First Example

Method 1: Solve and Substitute

Example: x=t+2,y=t2x = t + 2,\quad y = t^2x=t+2,y=t2

Method 2: The Pythagorean Identity

Example: x=cos⁡t,y=sin⁡tx = \cos t,\quad y = \sin tx=cost,y=sint

Example: x=3cos⁡t,y=3sin⁡tx = 3\cos t,\quad y = 3\sin tx=3cost,y=3sint

Projectile Motion

Example: x=20t,y=100−16t2x = 20t,\quad y = 100 - 16t^2x=20t,y=100−16t2

Example: line through (2,1)(2, 1)(2,1) and (5,7)(5, 7)(5,7)

Parametrizing a Circle

Example: circle of radius 444 centered at (2,−3)(2, -3)(2,−3)

Example: x=4cos⁡t,y=2sin⁡tx = 4\cos t,\quad y = 2\sin tx=4cost,y=2sint

Shifting and Speeding

Example: $x = t + 2,\quad y = t^2$

Example: $x = \cos t,\quad y = \sin t$

Example: $x = 3\cos t,\quad y = 3\sin t$

Example: $x = 20t,\quad y = 100 - 16t^2$

Example: line through $(2, 1)$ and $(5, 7)$

Example: circle of radius $4$ centered at $(2, -3)$

Example: $x = 4\cos t,\quad y = 2\sin t$