Question

Find a set of parametric equations for the parabola \(y=x^{2}\)

Short answer

Question: Write the set of parametric equations for the parabola y = x^2. Answer: The set of parametric equations for the parabola y = x^2 are x(t) = t and y(t) = t^2.

Step-by-step solution

Decide on a parameter for x

Let's assume x = t, where t can be any real number.

Determine the equation for y in terms of t

Since we already have the equation y = x^2, we can substitute x = t to determine y in terms of t: y = t^2

Write the parametric equations for the parabola

The parametric equations for the parabola y = x^2 are given by: x(t) = t y(t) = t^2 These parametric equations represent the parabola for any value of the parameter t.

Key concepts

Parabola Insight: The U-Shaped Curve

A parabola is a unique, U-shaped curve that represents the graph of a quadratic function. This shape is not only found in algebra but also appears in physics when describing phenomena like projectile motion.
A parabola is defined mathematically by a quadratic equation, commonly expressed in the form:\[ y = ax^2 + bx + c \]In this equation, if \(a\) is greater than zero, the parabola opens upwards, resembling a smile. Conversely, when \(a\) is less than zero, it opens downwards, akin to a frown. A specific and simple example is the equation \(y = x^2\), a standard parabola that opens upwards with its vertex at the origin (0,0).
This curve is characterized by its symmetry with respect to its vertical axis, known as the axis of symmetry. Understanding the properties of a parabola, such as its vertex, focus, and directrix, can be essential in coordinate geometry problems. Recognizing a parabola's graph helps in visualizing equations and understanding the underlying mathematical relationships.

Understanding Parametrization in Algebra

Parametrization is a method used to describe a curve using parameters. Instead of expressing all variables directly, one variable is redefined using a parameter which can simplify the equations.
For the case of the parabola \(y = x^2\), parametrization can be achieved by introducing a parameter, typically denoted as \(t\). When we set \(x = t\), it allows us to redefine \(y\) based on \(t\), which results in:

• \(x(t) = t\)
• \(y(t) = t^2\)

By showing both \(x\) and \(y\) in terms of \(t\), we construct a set of parametric equations. These equations represent the entire curve for any value of \(t\), providing a more flexible way to analyze and generate points on the parabola.
Using parameters is advantageous in complex geometry problems and is especially useful in physics to describe curves with respect to time or another variable.

Coordinate Geometry: Linking Algebra and Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe curves and shapes in a coordinate plane. This branch of mathematics enables the conversion of geometric problems into algebraic equations, making complex problems more manageable.
By assigning coordinates to geometric locations, problems become easier to solve analytically. In the context of a parabola, coordinate geometry allows us to use algebraic techniques to find important properties like intercepts, focus, and latus rectum.
Analyzing the relationship between the variables \(x\) and \(y\), as seen in parametric equations, assists in translating these properties into visual representations. It simplifies the understanding of the path described by a parabolic function in a two-dimensional space.

• It provides tools to calculate distances and angles.
• It helps in determining the nature of curves and intersections.
• It serves as a bridge between numerical computations and graphical understanding.

Coordinate geometry remains a fundamental aspect of modern mathematics education and is indispensable for those exploring calculus and higher-level mathematics.