🎯⭐ INTERACTIVE LESSON

Parametric Equations

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Parametric Equations - Complete Interactive Lesson

Part 1: Parametric Basics

📈 Introduction to Parametric Equations

Part 1 of 7

What Are Parametric Equations?

Instead of $y = f(x)$ , we describe curves using a parameter $t$ :

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

As $t$ varies, the point $(x, y)$ traces a curve in the plane.

Why Use Parameters?

Direction & timing: parametric curves have a built-in direction (as $t$ increases)
Multiple $y$ values: can describe curves that fail the vertical line test (circles, loops)
Physical meaning: $t$ often represents time — the curve shows an object's path

Example: A Circle

$x = \cos t, \quad y = \sin t, \quad 0 \leq t \leq 2\pi$

Traces the unit circle counterclockwise starting at $(1, 0)$ .

At $t = 0$ : $(1, 0)$ . At $t = \frac{\pi}{2}$ : . At : .

📝 Common Parametric Representations

Lines

$x = x_0 + at, \quad y = y_0 + bt$ Direction: . Passes through at .

🔄 Orientation and Domain

Direction Matters

The same curve can have different orientations:

$x = t, \; y = t^2$ for $t \in [0, 2]$ → traces left to right

Parametric Basics 🎯

Evaluate & Eliminate 🧮

1) $x = 2t + 1, \; y = 3t - 2$ . At $t = 3$ : the -coordinate is?

Identify the Curve 🔽

Exit Quiz ✅

Part 2: Graphing Parametric Curves

📐 Slopes & Tangent Lines for Parametric Curves

Part 2 of 7

The Parametric Derivative

For $x = f(t), \; y = g(t)$ , the slope of the tangent line is:

$\frac{d y}{d x} = \frac{d y / d t}{d x / d t} = \frac{}{}$

Part 3: Eliminating the Parameter

📏 Arc Length of Parametric Curves

Part 3 of 7

The Arc Length Formula

For a smooth curve $x = f(t), \; y = g(t)$ from $t = a$ to :

Part 4: Parametric Motion

🎯 Projectile Motion & Applications

Part 4 of 7

Projectile Motion Equations

An object launched at angle $\alpha$ with initial speed $v_0$ from height $h_0$ :

Part 5: Applications

🔄 Parametric Curves & Eliminating the Parameter

Part 5 of 7

Techniques for Eliminating the Parameter

Parametric Form	Strategy	Rectangular Result
$x = at + b, \; y = ct + d$

Part 6: Problem-Solving Workshop

🌀 Special Parametric Curves

Part 6 of 7

Famous Curves with Parametric Equations

Cycloid — Point on rim of rolling circle (radius $a$ ): $x = a(t - \sin t), \quad y = a(1 - \cos t)$ Properties: arches from to , max height .

Part 7: Review & Applications

🧩 Parametric Equations — Full Synthesis

Part 7 of 7

Complete Skill Set

Topic	Key Idea
Parametrization	$x = f(t), y = g(t)$ ; direction from increasing $t$

$x = t, \quad y = t^2$ This is just $y = x^2$ parametrized with $t = x$ .

$x = a\cos t, \quad y = b\sin t, \quad 0 \leq t \leq 2\pi$ Semi-axes $a$ and $b$ . Since $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2 t + \sin^2 t = 1$ .

Eliminating the Parameter

To find the rectangular (Cartesian) equation:

Solve one equation for $t$
Substitute into the other

Example: $x = 2t + 1, \; y = t - 3$ . From $x$ : $t = \frac{x-1}{2}$ . Then $y = \frac{x-1}{2} - 3 = \frac{x-7}{2}$ → a line.

x = 2-t, \; y = (2-t)^2

for

t \in [0, 2]

→ traces right to left

Same parabolic arc, opposite direction!

Restricting the Parameter

The domain of $t$ controls which portion of the curve is drawn:

$x = \cos t, \; y = \sin t$ for $0 \leq t \leq \pi$ → upper semicircle only
$x = \cos t, \; y = \sin t$ for $0 \leq t \leq 4\pi$ → circle traced twice

💡 Key Insight: Different parametrizations can produce the same geometric curve but with different starting points, speeds, and directions.

2) $x = \cos t, \; y = \sin t$ . At $t = \frac{\pi}{3}$ : $x$ = ? (Enter as a fraction like "1/2")

3) Eliminate $t$ from $x = t + 1, \; y = t^2 + 2t$ . Express $y$ in terms of $x$ : $y = x^2 +$ ? $x +$ ? (Enter the constant term, like "-1")

\frac{d y}{d x} = \frac{d y / d t}{d x / d t} = \frac{g ^{'} ( t )}{f ^{'} ( t )} (provided f^{'} (t) 0)

Condition	Geometric Meaning
$\frac{dy}{dt} = 0, \; \frac{dx}{dt} \neq 0$	Horizontal tangent
$\frac{dx}{dt} = 0, \; \frac{dy}{dt} \neq 0$
Both zero	Need further analysis (possible cusp)

📝 Example: Tangent Line to a Cycloid

A cycloid is traced by a point on a rolling circle:

$x = t - \sin t, \quad y = 1 - \cos t$

Find the slope at $t = \frac{\pi}{2}$ .

$\frac{dx}{dt} = 1 - \cos t = 1 - 0 = 1$ at

$\frac{dy}{dt} = \sin t = 1$ at $t = \frac{\pi}{2}$

$\frac{dy}{dx} = \frac{1}{1} = 1$

Point: $x = \frac{\pi}{2} - 1, \; y = 1$ .

Tangent line: $y - 1 = 1\left(x - (\frac{\pi}{2}-1)\right)$

Where are horizontal tangents? $\sin t = 0 \implies t = n\pi$ ( $n$ integer).

At $t = \pi$ : point is $(\pi, 2)$ — top of the arch, slope = 0. ✓

📊 Second Derivative & Concavity

The second derivative for parametric curves:

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$

Steps:

Find $\frac{dy}{dx} = \frac{y'(t)}{x'(t)}$

Example: $x = t^2, \; y = t^3$

$\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2}$

$\frac{d}{dt}\left(\frac{3t}{2}\right) = \frac{3}{2}$

$\frac{d^2y}{dx^2} = \frac{3/2}{2t} = \frac{3}{4t}$

Concave up when $t > 0$ , concave down when $t < 0$ .

Derivative Quiz 🎯

Compute Slopes 🧮

1) $x = 3t, \; y = t^2 - 1$ . What is $\frac{dy}{dx}$ at $t = 3$ ? (Enter as a fraction or whole number)

2) $x = e^t, \; y = e^{-t}$ . $\frac{dy}{dx}$ at = ?

3) $x = t + \sin t, \; y = 1 - \cos t$ . $\frac{dy}{dx}$ at = ? (Enter as a fraction)

Tangent Properties 🔽

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

At each instant, the point moves by:

$\Delta x \approx \frac{dx}{dt}\Delta t$ (horizontal)
$\Delta y \approx \frac{dy}{dt}\Delta t$ (vertical)

By the Pythagorean theorem: $\Delta s \approx \sqrt{(\Delta x)^2 + (\Delta y)^2}$

Summing up → the integral.

📝 Example 1: Circle Circumference

$x = R\cos t, \; y = R\sin t, \; 0 \leq t \leq 2\pi$

$\frac{dx}{dt} = -R\sin t, \quad \frac{dy}{dt} = R\cos t$

$L = \int_0^{2\pi}\sqrt{R^2\sin^2 t + R^2\cos^2 t}\,dt = \int_0^{2\pi}R\,dt = 2\pi R \; ✓$

Example 2: Line Segment

$x = 1 + 3t, \; y = 2 + 4t, \; 0 \leq t \leq 1$

$\frac{dx}{dt} = 3, \quad \frac{dy}{dt} = 4$

$L = \int_0^1 \sqrt{9 + 16}\,dt = \int_0^1 5\,dt = 5$

This matches the distance formula: $\sqrt{3^2 + 4^2} = 5$ . ✓

Example 3: $x = t^2, \; y = t^3, \; 0 \leq t \leq 1$

$L = \int_0^1\sqrt{4t^2 + 9t^4}\,dt = \int_0^1 t\sqrt{4+9t^2}\,dt$

Let $u = 4+9t^2$ : $du = 18t\,dt$

$= \frac{1}{18}\int_4^{13}\sqrt{u}\,du = \frac{1}{27}\left[u^{3/2}\right]_4^{13} = \frac{13\sqrt{13}-8}{27}$

🚀 Speed Along a Parametric Curve

The speed at time $t$ is the rate of change of arc length:

$\text{speed} = \frac{ds}{dt} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$

Example: Projectile Motion

$x = v_0 t, \; y = h_0 + v_0 t - \frac{1}{2}gt^2$

Speed: $\sqrt{v_0^2 + (v_0 - gt)^2}$

At $t = 0$ : speed $= \sqrt{v_0^2 + v_0^2} = v_0\sqrt{2}$

💡 Key: Speed is ALWAYS non-negative. It equals the magnitude of the velocity vector $(\frac{dx}{dt}, \frac{dy}{dt})$ .

Arc Length Quiz 🎯

Arc Length Calculations 🧮

1) $x = 5t, \; y = 12t$ from $t = 0$ to $t = 3$ . Arc length = ?

2) The speed of a particle with $x = 3\cos t, \; y = 3\sin t$ is constant at what value?

3) $x = t, \; y = \frac{2}{3}t^{3/2}$ from to . The integrand gives . Evaluate: = ? (Enter as a fraction like "14/3")

Arc Length Concepts 🔽

$x = (v_0\cos\alpha)\,t, \quad y = h_0 + (v_0\sin\alpha)\,t - \frac{1}{2}gt^2$

where $g \approx 9.8$ m/s² or $32$ ft/s².

Quantity	Formula
Time of flight	Solve $y = 0$ (quadratic in $t$ )
Maximum height	At $t = \frac{v_0\sin\alpha}{g}$ (when $\frac{dy}{dt} = 0$ )
Range	$x$ at landing time
Maximum range	At $\alpha = 45°$ (on level ground)

📝 Example: Baseball Problem

A ball is hit at $v_0 = 128$ ft/s at angle $\alpha = 30°$ from $h_0 = 3$ ft.

$x = 128\cos 30°\,t = 64\sqrt{3}\,t$

Maximum height: $\frac{dy}{dt} = 64 - 32t = 0 \implies t = 2$ s

$y_{\max} = 3 + 64(2) - 16(4) = 3 + 128 - 64 = 67$ ft

Time of flight: $3 + 64t - 16t^2 = 0 \implies 16t^2 - 64t - 3 = 0$

$t = \frac{64 + \sqrt{4096 + 192}}{32} = \frac{64 + \sqrt{4288}}{32} \approx 4.05$ s

Range: $x \approx 64\sqrt{3}(4.05) \approx 449$ ft

⚾ That is a home run in most ballparks!

🔧 Other Applications of Parametric Equations

Particle on a Ferris Wheel

Center at $(0, h)$ , radius $R$ , angular speed $\omega$ :

$x = R\cos(\omega t), \quad y = h + R\sin(\omega t)$

Spirograph (Epitrochoid)

$x = (R+r)\cos t - d\cos\left(\frac{R+r}{r}t\right)$

Lissajous Figures

$x = A\sin(at + \delta), \quad y = B\sin(bt)$

The shape depends on the frequency ratio $a:b$ and phase shift $\delta$ .

$a = b, \delta = 0$ : line
$a = b, \delta = \frac{\pi}{2}$ : ellipse

Applications Quiz 🎯

Projectile Calculations 🧮

A ball is launched at $v_0 = 80$ ft/s at $\alpha = 45°$ from ground level ( $h_0 = 0$ , $g = 32$ ft/s²).

1) The horizontal component of velocity $v_0\cos 45°$ = ? (Enter like "40sqrt2")

2) Time to reach max height = $\frac{v_0\sin\alpha}{g}$ = ? (Enter as a fraction like "5/2")

3) Maximum height = $\frac{(v_0\sin\alpha)^2}{2g}$ = ? (whole number in ft)

Projectile Properties 🔽

📝 Worked Examples

Example 1: Trig Elimination

$x = 3 + 2\cos t, \; y = -1 + 2\sin t$

$(x-3) = 2\cos t, \; (y+1) = 2\sin t$

$(x-3)^2 + (y+1)^2 = 4\cos^2 t + 4\sin^2 t = 4$

Circle centered at $(3, -1)$ , radius $2$ .

Example 2: Exponential

$x = e^{2t}, \; y = e^t + 1$

Since $x = e^{2t} = (e^t)^2$ and $e^{t} = y - 1$ :

$x = (y-1)^2$ , i.e., $y = 1 + \sqrt{x}$ (since )

Example 3: Watch the Domain!

$x = t^2, \; y = t$ → $x = y^2$ (full sideways parabola)

$x = \sin^2 t, \; y = \sin t$ → also $x = y^2$ , but only and

⚠️ The parametrization restricts which part of the Cartesian curve is actually traced!

✏️ Creating Parametric Equations

Given a Cartesian curve, find parametric equations. Multiple answers exist!

For $y = x^2 - 4x + 3$ :

Simple: $x = t, \; y = t^2 - 4t + 3$
Right to left: $x = -t, \; y = t^2 + 4t + 3$
Shifted: $x = t + 2, \; y = t^2 - 1$ (completing the square)

For a circle $x^2 + y^2 = 25$ :

Standard: $x = 5\cos t, \; y = 5\sin t$ (CCW from $(5,0)$ )
Clockwise: $x = 5 \cos t$

For a line through $(2, 3)$ and $(5, 7)$ :

Direction: $(3, 4)$ . So $x = 2 + 3t, \; y = 3 + 4t$ with for the segment.

Elimination Quiz 🎯

Eliminate Parameters 🧮

1) $x = 2t - 1, \; y = 4t + 3$ . Express $y$ in terms of $x$ : $y = 2x +$ ? (Enter the constant)

2) $x = t^3, \; y = t^2$ . Express $x$ in terms of $y$ : . What is ? (whole number)

3) $x = 2\cosh t, \; y = 3\sinh t$ . The curve is $\frac{x^2}{4} - \frac{y^2}{n} = 1$ . What is ?

Matching Curves 🔽

Astroid — Point inside rolling circle: $x = a\cos^3 t, \quad y = a\sin^3 t$ Rectangular: $x^{2/3} + y^{2/3} = a^{2/3}$

Involute of a Circle — Unwinding string from circle: $x = a(\cos t + t\sin t), \quad y = a(\sin t - t\cos t)$

📝 The Cycloid in Depth

The cycloid $x = a(t - \sin t), \; y = a(1 - \cos t)$ has remarkable properties.

Slope

$\frac{dy}{dx} = \frac{a\sin t}{a(1-\cos t)} = \frac{\sin t}{1-\cos t} = \cot\frac{t}{2}$

At $t = \pi$ (top of arch): slope = $\cot\frac{\pi}{2} = 0$ → horizontal tangent ✓

At $t \to 0^+$ : slope $\to \infty$ → vertical tangent (cusp) ✓

Arc Length of One Arch

$L = \int_0^{2\pi}\sqrt{a^2(1-\cos t)^2 + a^2\sin^2 t}\,dt$

Simplifies using $1-2\cos t + \cos^2 t + \sin^2 t = 2(1-\cos t) = 4\sin^2\frac{t}{2}$ :

$L = \int_0^{2\pi}2a\sin\frac{t}{2}\,dt = \left[-4a\cos\frac{t}{2}\right]_0^{2\pi} = -4a(-1-1) = 8a$

🎯 One arch of the cycloid has length exactly $8a$ — eight times the radius!

🎵 Lissajous Figures

$x = A\sin(at + \delta), \quad y = B\sin(bt)$

The frequency ratio $a:b$ determines the shape:

Ratio $a:b$	Shape
$1:1, \delta = 0$	Line segment (diagonal)
$1:1, \delta = \frac{\pi}{2}$

The number of lobes: up to $a$ lobes horizontally and $b$ lobes vertically.

Visualization Tip

Set $A = B = 1, \delta = \frac{\pi}{2}$ , and increment the ratio $a:b$ . The complexity increases — these patterns appear in oscilloscope traces and physics demonstrations.

Special Curves Quiz 🎯

Special Curve Calculations 🧮

1) Cycloid with $a = 3$ : arc length of one arch = $8a$ = ?

2) Cycloid with $a = 5$ : maximum height above baseline = $2a$ = ?

3) Astroid with $a = 8$ : at $t = 0$ , the point is at $(x, y) = (?, 0)$ . What is $x$ ?

Curve Identification 🔽

🎓 Problem-Solving Flowchart

Given parametric equations, find...

Cartesian equation? → Eliminate $t$ (algebraic or trig identity)

Don't forget domain restrictions!

Slope at a point? → $\frac{dy/dt}{dx/dt}$ at that $t$ value

Horizontal tangent? → $dy/dt = 0$ (and $dx/dt \neq 0$ )

Vertical tangent? → $dx/dt = 0$ (and $dy/dt \neq 0$ )

Arc length? → $\int\sqrt{(dx/dt)^2 + (dy/dt)^2}\,dt$

Concavity? → Compute $\frac{d^2y}{dx^2}$ using the parametric formula

Common Mistakes

Forgetting that eliminating $t$ may lose domain information
Using $\frac{d^2y}{dx^2} = \frac{d^2y/dt^2}{d^2x/dt^2}$ (WRONG!)

Comprehensive Quiz 🎯

Mixed Calculations 🧮

1) $x = t^2, y = t^3$ . Find $\frac{dy}{dx}$ at $t = 2$ . (whole number)

2) $x = 5t, y = 12t$ from $t = 0$ to $t = 4$ . Arc length = ?

3) Cycloid $x = 4(t-\sin t), y = 4(1-\cos t)$ . Arc length of one arch ( $8a$ ) = ?

Final Concepts 🔽

Exit Quiz — Final ✅

Differentiate this with respect to

t

\frac{d}{dt}\left(\frac{dy}{dx}\right)

Divide by

\frac{dx}{dt}

L = \frac{2}{3}[u^{3/2}]_1^4

y = 3 + 128\sin 30°\,t - 16t^2 = 3 + 64t - 16t^2

y = 3 + 128 sin 30° t - 16 t^{2} = 3 + 64 t - 16 t^{2}

y = (R+r)\sin t - d\sin\left(\frac{R+r}{r}t\right)

a = 2, b = 1

: figure-eight shapes

e^t > 0 \implies y > 1

x = 5 cos t, y = - 5 sin t

Starting at top:

x = 5\sin t, \; y = 5\cos t

Not checking

dx/dt \neq 0

for horizontal tangents

Parametric Equations

Parabolas

Ellipses

Eliminating the Parameter

Restricting the Parameter

Key Cases

📝 Example: Tangent Line to a Cycloid

📊 Second Derivative & Concavity

Steps:

Example: $x = t^2, \; y = t^3$

Intuition

📝 Example 1: Circle Circumference

Example 2: Line Segment

Example 3: $x = t^2, \; y = t^3, \; 0 \leq t \leq 1$

🚀 Speed Along a Parametric Curve

Example: Projectile Motion

Key Quantities

📝 Example: Baseball Problem

🔧 Other Applications of Parametric Equations

Particle on a Ferris Wheel

Spirograph (Epitrochoid)

Lissajous Figures

📝 Worked Examples

Example 1: Trig Elimination

Example 2: Exponential

Example 3: Watch the Domain!

✏️ Creating Parametric Equations

For $y = x^2 - 4x + 3$ :

For a circle $x^2 + y^2 = 25$ :

For a line through $(2, 3)$ and $(5, 7)$ :

📝 The Cycloid in Depth

Slope

Arc Length of One Arch

🎵 Lissajous Figures

Visualization Tip

🎓 Problem-Solving Flowchart

Given parametric equations, find...

Common Mistakes

Parametric Equations

Parametric Equations - Complete Interactive Lesson

Part 1: Parametric Basics

📈 Introduction to Parametric Equations

What Are Parametric Equations?

Why Use Parameters?

Example: A Circle

📝 Common Parametric Representations

Lines

🔄 Orientation and Domain

Direction Matters

Part 2: Graphing Parametric Curves

📐 Slopes & Tangent Lines for Parametric Curves

The Parametric Derivative

Part 3: Eliminating the Parameter

📏 Arc Length of Parametric Curves

The Arc Length Formula

Part 4: Parametric Motion

🎯 Projectile Motion & Applications

Projectile Motion Equations

Part 5: Applications

🔄 Parametric Curves & Eliminating the Parameter

Techniques for Eliminating the Parameter

Part 6: Problem-Solving Workshop

🌀 Special Parametric Curves

Famous Curves with Parametric Equations

Part 7: Review & Applications

🧩 Parametric Equations — Full Synthesis

Complete Skill Set

Parabolas

Ellipses

Eliminating the Parameter

Restricting the Parameter

Key Cases

📝 Example: Tangent Line to a Cycloid

📊 Second Derivative & Concavity

Steps:

Example: x=t2, y=t3x = t^2, \; y = t^3x=t2,y=t3

Intuition

📝 Example 1: Circle Circumference

Example 2: Line Segment

Example 3: x=t2, y=t3, 0≤t≤1x = t^2, \; y = t^3, \; 0 \leq t \leq 1x=t2,y=t3,0≤t≤

🚀 Speed Along a Parametric Curve

Example: Projectile Motion

Key Quantities

📝 Example: Baseball Problem

🔧 Other Applications of Parametric Equations

Particle on a Ferris Wheel

Spirograph (Epitrochoid)

Lissajous Figures

📝 Worked Examples

Example 1: Trig Elimination

Example 2: Exponential

Example 3: Watch the Domain!

✏️ Creating Parametric Equations

For y=x2−4x+3y = x^2 - 4x + 3y=x2−4x+3:

For a circle x2+y2=25x^2 + y^2 = 25x2+y2=25:

For a line through (2,3)(2, 3)(2,3) and (5,7)(5, 7)(5,7):

📝 The Cycloid in Depth

Slope

Arc Length of One Arch

🎵 Lissajous Figures

Visualization Tip

🎓 Problem-Solving Flowchart

Given parametric equations, find...

Common Mistakes

Example: $x = t^2, \; y = t^3$

Example 3: $x = t^2, \; y = t^3, \; 0 \leq t \leq 1$

For $y = x^2 - 4x + 3$ :

For a circle $x^2 + y^2 = 25$ :

For a line through $(2, 3)$ and $(5, 7)$ :