📈 Parametric Equations

Part 1 of 7 — Introduction to Parametric Equations

1. Parametric equations

x = f(t) and y = g(t) define a curve

2. Parameter t often represents time

Parameter t often represents time

3. Each value of t gives a point (x, y) on the curve

Each value of t gives a point (x, y) on the curve

4. Direction of motion is determined by increasing t

Direction of motion is determined by increasing t

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Key Concepts Summary

• Parametric equations: x = f(t) and y = g(t) define a curve
• Parameter t often represents time
• Each value of t gives a point (x, y) on the curve
• Direction of motion is determined by increasing t

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Graphing Parametric Curves

Part 2 of 7 — Graphing Parametric Curves

1. Make a table of t, x, y values and plot points

Make a table of t, x, y values and plot points

2. Indicate direction with arrows

Indicate direction with arrows

3. The same curve can have different parametric representations

The same curve can have different parametric representations

4. Restrict t-domain to show only part of the curve

Restrict t-domain to show only part of the curve

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Key Concepts Summary

• Make a table of t, x, y values and plot points
• Indicate direction with arrows
• The same curve can have different parametric representations
• Restrict t-domain to show only part of the curve

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Eliminating the Parameter

Part 3 of 7 — Eliminating the Parameter

1. Solve one equation for t, substitute into the other

Solve one equation for t, substitute into the other

2. For x = cos t, y = sin t

use cos²t + sin²t = 1 → x² + y² = 1

3. Eliminating the parameter gives the rectangular equation

Eliminating the parameter gives the rectangular equation

4. May need to restrict domain/range after elimination

May need to restrict domain/range after elimination

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Key Concepts Summary

• Solve one equation for t, substitute into the other
• For x = cos t, y = sin t: use cos²t + sin²t = 1 → x² + y² = 1
• Eliminating the parameter gives the rectangular equation
• May need to restrict domain/range after elimination

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Parametric Equations for Conics

Part 4 of 7 — Parametric Equations for Conics

1. Circle

x = h + r cos t, y = k + r sin t

2. Ellipse

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

3. Line

x = x₁ + at, y = y₁ + bt

4. Parabola

x = t, y = at² + bt + c (or other parameterizations)

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Key Concepts Summary

• Circle: x = h + r cos t, y = k + r sin t
• Ellipse: x = h + a cos t, y = k + b sin t
• Line: x = x₁ + at, y = y₁ + bt
• Parabola: x = t, y = at² + bt + c (or other parameterizations)

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Projectile Motion

Part 5 of 7 — Projectile Motion

1. Horizontal

x = v₀ cos(θ) · t

2. Vertical

y = v₀ sin(θ) · t - ½gt² + h₀

3. Maximum height at t = v₀ sin(θ)/g

Maximum height at t = v₀ sin(θ)/g

4. Range (horizontal distance) = v₀² sin(2θ)/g

Range (horizontal distance) = v₀² sin(2θ)/g

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Key Concepts Summary

• Horizontal: x = v₀ cos(θ) · t
• Vertical: y = v₀ sin(θ) · t - ½gt² + h₀
• Maximum height at t = v₀ sin(θ)/g
• Range (horizontal distance) = v₀² sin(2θ)/g

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

1. Horizontal

x = v₀ cos(θ) · t

2. Vertical

y = v₀ sin(θ) · t - ½gt² + h₀

3. Maximum height at t = v₀ sin(θ)/g

Maximum height at t = v₀ sin(θ)/g

4. Range (horizontal distance) = v₀² sin(2θ)/g

Range (horizontal distance) = v₀² sin(2θ)/g

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Key Concepts Summary

• Horizontal: x = v₀ cos(θ) · t
• Vertical: y = v₀ sin(θ) · t - ½gt² + h₀
• Maximum height at t = v₀ sin(θ)/g
• Range (horizontal distance) = v₀² sin(2θ)/g

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Review & Applications

Part 7 of 7 — Review & Applications

1. Horizontal

x = v₀ cos(θ) · t

2. Vertical

y = v₀ sin(θ) · t - ½gt² + h₀

3. Maximum height at t = v₀ sin(θ)/g

Maximum height at t = v₀ sin(θ)/g

4. Range (horizontal distance) = v₀² sin(2θ)/g

Range (horizontal distance) = v₀² sin(2θ)/g

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Key Concepts Summary

• Horizontal: x = v₀ cos(θ) · t
• Vertical: y = v₀ sin(θ) · t - ½gt² + h₀
• Maximum height at t = v₀ sin(θ)/g
• Range (horizontal distance) = v₀² sin(2θ)/g

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