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 left half of the parabola y=x ² +1, originating at (0, 1)

Verified step by step guidance

1. Identify the given curve: the parabola is defined by the equation \(y = x^2 + 1\).
2. Since we want parametric equations, choose a parameter, typically \(t\), to represent \(x\). Let \(x = t\).
3. Express \(y\) in terms of \(t\) using the original equation: \(y = t^2 + 1\).
4. Determine the interval for \(t\) to represent the left half of the parabola. The left half corresponds to \(x \leq 0\), so \(t \leq 0\).
5. Write the parametric equations and specify the interval: \(x = t\), \(y = t^2 + 1\), with \(t \in (-\infty, 0]\).

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 flexibility in describing curves, especially those that are not functions in the traditional sense.

Parabola and Its Properties

A parabola is a curve defined by a quadratic equation, such as y = x² + 1. Understanding its shape and symmetry is essential; here, the parabola opens upward with vertex at (0,1). The 'left half' refers to the portion where x ≤ 0, which guides the choice of parameter intervals.

Parameter Interval and Curve Orientation

Choosing an appropriate interval for the parameter ensures the parametric equations trace the desired portion of the curve. For the left half of the parabola, the parameter should cover values corresponding to x ≤ 0, and the interval defines the start and end points, such as originating at (0,1).