37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The upper half of the parabola x=y ², originating at (0, 0)

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Identify the given curve equation: \(x = y^2\). Since we want the upper half of the parabola, this corresponds to the part where \(y \geq 0\).

Choose a parameter to express both \(x\) and \(y\) in terms of it. A natural choice is to let \(t = y\), since \(y\) is already isolated in the equation.

Express \(x\) in terms of \(t\): since \(x = y^2\), substituting \(y = t\) gives \(x = t^2\).

Write the parametric equations as \(x = t^2\) and \(y = t\), where \(t\) represents the parameter.

Determine the interval for \(t\): because we want the upper half of the parabola starting at \((0,0)\) and \(y \geq 0\), the parameter \(t\) should satisfy \(t \geq 0\). So, the interval is \(t \in [0, \infty)\).

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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, usually denoted t. Instead of y as a function of x, both x and y are defined in terms of t, allowing more flexible representations of curves, including those that are not functions in the traditional sense.

Parabola and Its Equation

A parabola is a conic section defined by a quadratic relationship between x and y. In this case, the parabola is given by x = y², which opens along the x-axis. Understanding this relationship helps in choosing a parameter that can generate points on the curve.

Parameter Interval and Curve Orientation

The parameter interval defines the range of values for the parameter t that traces the desired portion of the curve. For the upper half of the parabola x = y² starting at (0,0), the parameter must be chosen so that y ≥ 0, ensuring only the upper half is represented.

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.The upper half of the parabola x=y ², originating at (0, 0)