Parametric Equations

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

📝 Common Parametric Representations

Lines

$x = x_0 + at, \quad y = y_0 + bt$ Direction: $(a, b)$. Passes through $(x_0, y_0)$ at $t = 0$.

Parabolas

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

Ellipses

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

1. Solve one equation for $t$
2. 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.

🔄 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
• $x = 2-t, \; y = (2-t)^2$ 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.

Parametric Basics 🎯

Evaluate & Eliminate 🧮

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

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")

Identify the Curve 🔽

Exit Quiz ✅

📐 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:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)} \quad \text{(provided } f'(t) \neq 0\text{)}$

Key Cases

📝 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 $t = \frac{\pi}{2}$

$\frac{dy}{dt} = \sin t = 1$ 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:

1. Find $\frac{dy}{dx} = \frac{y'(t)}{x'(t)}$
2. Differentiate this with respect to $t$: $\frac{d}{dt}\left(\frac{dy}{dx}\right)$
3. Divide by $\frac{dx}{dt}$

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 $t = 0$ = ?

3) $x = t + \sin t, \; y = 1 - \cos t$. $\frac{dy}{dx}$ at $t = \frac{\pi}{2}$ = ? (Enter as a fraction)

Tangent Properties 🔽

Exit Quiz ✅

📏 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 $t = b$:

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

Intuition

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 $t = 0$ to $t = 3$. The integrand $\sqrt{1+t}$ gives $L = \frac{2}{3}[u^{3/2}]_1^4$. Evaluate: $L$ = ? (Enter as a fraction like "14/3")

Arc Length Concepts 🔽

Exit Quiz ✅

🎯 Projectile Motion & Applications

Part 4 of 7

Projectile Motion Equations

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

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

where $g \approx 9.8$ $m/s^{2}$ or $32$ $ft/s^{2}$.

Key Quantities

📝 Example: Baseball Problem

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

$x = 128\cos 30°\,t = 64\sqrt{3}\,t$ $y = 3 + 128\sin 30°\,t - 16t^2 = 3 + 64t - 16t^2$

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)

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

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
• $a = 2, b = 1$: figure-eight shapes

Applications Quiz 🎯

Projectile Calculations 🧮

A ball is launched at $v_0 = 80$ ft/s at $\alpha = 45°$ from ground level ($h_0 = 0$, $g = 32$ $ft/s^{2}$).

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 🔽

Exit Quiz ✅

🔄 Parametric Curves & Eliminating the Parameter

Part 5 of 7

Techniques for Eliminating the Parameter

📝 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 $e^t > 0 \implies y > 1$)

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 $-1 \leq y \leq 1$ and $0 \leq x \leq 1$

⚠️ 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: $x = 5\cos t, \; y = -5\sin t$
• Starting at top: $x = 5\sin t, \; y = 5\cos t$

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

Direction: $(3, 4)$. So $x = 2 + 3t, \; y = 3 + 4t$ with $t \in [0, 1]$ 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 $y$: $x^2 = y^n$. What is $n$? (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 $n$?

Matching Curves 🔽

Exit Quiz ✅

🌀 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 $(0,0)$ to $(2\pi a, 0)$, max height $2a$.

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:

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 🔽

Exit Quiz ✅

🧩 Parametric Equations — Full Synthesis

Part 7 of 7

Complete Skill Set

🎓 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!)
• Not checking $dx/dt \neq 0$ for horizontal tangents

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 ✅