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Questions and Answers
What is the Cartesian equation for the parametric equations x = sin(πt/2), y = t, where 0 ≤ t ≤ 6?
x = sin(πy/2)
What is the Cartesian equation for the parametric equations x = t², y = t + 1?
x = (y - 1)²
What is the Cartesian equation for the parametric equations x = cos(t), y = sin(t)?
x² + y² = 1
Identify the Cartesian equation for the parametric equations x = 3t, y = 9t.
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Which parametric equations represent a circular path?
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Study Notes
Parametric Equations
- A parametric equation is an equation that uses a parameter (usually represented by 't') to define the x and y coordinates of a curve.
- Parametric equations describe the motion of a particle in the x-y plane.
- The value of the parameter 't' corresponds to a specific point on the curve.
Cartesian Equation
- A Cartesian equation is a conventional equation that defines a curve in terms of x and y only.
- To obtain the Cartesian equation from a parametric equation, eliminate the parameter 't'.
Examples
- Each example shows a set of parametric equations and their corresponding Cartesian equation.
- By eliminating 't' from the parametric equations, a Cartesian equation that describes the same curve is obtained.
- Each example explores different types of curves, such as parabolas, circles, and other shapes.
Exercises
- The exercises provide parametric equations and parameter intervals describing the motion of a particle in the x-y plane.
- Students are required to:
- Identify the particle’s path by finding the Cartesian equation.
- Graph the obtained Cartesian equation.
- Identify the portion of the graph traced by the particle and the direction of motion.
Key Points
- Understanding parametric equations is essential for describing motion and defining curves in the x-y plane.
- The Cartesian equation provides a standard representation for curves in terms of x and y but lacks information about the particle's movement.
- The process of eliminating the parameter 't' is crucial for converting parametric equations into Cartesian equations and vice versa.
- The exercises provide hands-on practice with applying parametric equations to real-world scenarios and emphasize the connection between parametric and Cartesian representations.
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Description
Test your understanding of parametric and Cartesian equations with this quiz. Explore how to convert parametric equations into Cartesian form and examine various types of curves. Solve exercises that involve the motion of a particle in the x-y plane.
