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)

Verified step by step guidance

1. Identify the given curve equation: \(x = y^2\). Since we want the upper half of the parabola, this corresponds to the part where \(y \geq 0\).
2. 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.
3. Express \(x\) in terms of \(t\): since \(x = y^2\), substituting \(y = t\) gives \(x = t^2\).
4. Write the parametric equations as \(x = t^2\) and \(y = t\), where \(t\) represents the parameter.
5. 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 \(t \in [0, \infty)\).

Key Concepts

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.