91Ó°ÊÓ

Parametric equations allow us to describe curves using a third variable, usually denoted as \( t \). In the context of parabolas, parametric equations are often in the form \( x = at + b \) and \( y = ct^2 + dt + e \). Here, the parameter \( t \) acts as a ‘time’ variable that helps trace out the path of the curve. For parabolas, this setup often involves a quadratic expression in the equation for \( y \).

In the given problem, the parametric equations are \( x = 2t + 1 \) and \( y = t^2 - 3 \). Observing the equation for \( y \), we see a quadratic term \( t^2 \). This quadratic nature immediately suggests that the curve traced is a parabola. The linear equation for \( x \) simply adjusts the horizontal positioning but doesn't affect the parabolic nature. The beauty of parametric equations is that they provide a dynamic way to represent these classic shapes.

An essential part of working with parametric equations is transforming them to a single relation between \( x \) and \( y \), often called the algebraic form. To identify the curve precisely and simplify understanding, we eliminate the parameter \( t \).

From the problem, we start with \( x = 2t + 1 \). Solving for \( t \) gives us \( t = \frac{x - 1}{2} \). Substituting this expression into \( y = t^2 - 3 \) results in \( y = \left(\frac{x - 1}{2}\right)^2 - 3 \). This step is crucial as it converts parametric form into a standard algebraic form \( y = a(x-h)^2 + k \), typical for conic sections. Such transformation helps predicate the curve's shape and orientation from known algebraic patterns.

Once the parametric equations are transformed into a single equation in \( x \) and \( y \), the task becomes identifying the curve type based on its algebraic structure.

The derived equation \( y = \frac{(x - 1)^2}{4} - 3 \) resembles the standard parabola form \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola and \( a \) dictates its width and direction. Since the expression \( (x - 1)^2 \) is squared and the equation is linear in \( y \), we can confirm it's a vertical parabola. Being able to identify these features is key in understanding and categorizing the curve in question as a parabola, given its orientation and position.