Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x=√(t+1), y=√t, t ≥ 0

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Start with the given parametric equations: \(x = \sqrt{t+1}\) and \(y = \sqrt{t}\), where \(t \geq 0\).

Express both \(x\) and \(y\) in terms of \(t\) squared to eliminate the square roots: \(x^2 = t + 1\) and \(y^2 = t\).

Use the expression for \(y^2\) to substitute for \(t\) in the equation for \(x^2\): \(x^2 = y^2 + 1\).

Rewrite the equation to isolate terms and get the Cartesian form: \(x^2 - y^2 = 1\).

Analyze the parameter interval \(t \geq 0\) to determine the portion of the graph traced by the particle and the direction of motion by considering how \(x\) and \(y\) change as \(t\) increases.

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주요 개념

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex paths and motions in the plane.

Eliminating the Parameter to Find Cartesian Equations

To find a Cartesian equation from parametric equations, solve one parametric equation for the parameter and substitute into the other. This process removes the parameter, yielding a direct relationship between x and y that describes the particle's path.

Graphing and Interpreting the Particle's Path and Direction

After finding the Cartesian equation, graph it to visualize the path. Use the parameter interval to determine which portion of the curve is traced and analyze how the parameter changes to indicate the direction of motion along the path.