91Ó°ÊÓ

Rectangular equations are mathematical expressions that describe a relationship between two variables, typically x and y, without involving any other variable. They are called 'rectangular' because they can be plotted directly on Cartesian coordinates, forming shapes on a two-dimensional plane. In this exercise, the given rectangular equation is:

y = x^2 + 1

To break it down, y is expressed as a function of x. This is a simple quadratic equation that represents a parabola opening upwards with its vertex at (0,1). Learning to convert rectangular equations into parametric equations allows us to describe pathways and motion in more flexible ways.

Choosing a parameter is essential to convert a rectangular equation into parametric equations. A parameter, commonly denoted as t, represents a third variable that both x and y depend on. This transforms our equations into a set of functions based on t.

For the first pair, if we choose x equal to t (i.e., x = t), we substitute this back into the original equation to get y:
y = t^2 + 1

Hence, the first set of parametric equations is:

• x = t
• y = t^2 + 1

For the second pair, we select a different relationship for x with respect to t to ensure a meaningful parametric transformation. Let’s choose x equal to t - 2:
x = t - 2

Plugging this into the rectangular equation, we get:
y = (t - 2)^2 + 1
Simplifying, this becomes:
y = t^2 - 4t + 5

Therefore, our second pair of parametric equations is:

• x = t - 2
• y = t^2 - 4t + 5

Choosing different parameters gives us varied representations of the same curve, useful in many applied mathematics problems.

Equation transformation is the process of altering the structure of an equation to make it more suitable for specific applications while preserving the nature of the original equation. When we transform rectangular equations into parametric form, we effectively introduce a new variable, simplifying the complexity involved in graphing and solving.

To transform the given rectangular equation y = x^2 + 1 into parametric equations, we:

• First, select an appropriate parameter t.
• Next, define x and y in terms of this parameter.

The transformation process enables describing the same curve in different ways, as seen with the two chosen sets of parametric equations:

• For x = t:
  
• For x = t - 2:
  x = t - 2, y = t^2 - 4t + 5

This reformulation can be very useful in physics and engineering, especially when dealing with motion and paths where time or another external variable is involved.