91Ó°ÊÓ

When working with parametric equations, each variable is expressed as a function of one or more independent parameters. In our given exercise, the equations are:

• \( x = \sqrt{t} \)
• \( y = t^2 - 1 \)

Here, both \( x \) and \( y \) are functions of the parameter \( t \). Often in problems like this, the goal is to remove the parameter to find a direct relationship between \( x \) and \( y \), known as a rectangular equation.
This involves expressing \( y \) directly as a function of \( x \) by eliminating the parameter \( t \), leading to a more straightforward equation that uses just the Cartesian coordinates \( x \) and \( y \).
This conversion provides a clearer view of the path or curve described by the equations without explicitly involving the parameter.

Interval notation is a crucial part of understanding the range of values a variable can take. In our exercise, the parameter \( t \) is defined in the interval \([0, \infty)\), meaning \( t \) can be any non-negative value but not infinity itself.
This translates into understanding the permissible values of \( x \) once \( t \) is expressed in terms of \( x \).
After substituting \( t \) with \( x^2 \) to form the rectangular equation, we determine that \( x \) also follows the interval \([0, \infty)\), as the square root operation requires non-negative inputs.

• The notation \([0, \infty)\) represents all real numbers starting from 0 up to, but not including, infinity.
• This helps define the domain of functions when solving mathematical expressions.

Understanding such notations is essential for accurately describing the range over which a function or equation is valid.

A function of a variable expresses one quantity directly in terms of another. In the context of this exercise, this means rewriting equations to eliminate the parameter \( t \) and showing \( y \) as a function of \( x \).
Initially, we had:

• \( x = \sqrt{t} \)
• \( y = t^2 - 1 \)

By identifying \( t = x^2 \), we substitutely formed the rectangular equation:
\( y = x^4 - 1 \).
This expression shows \( y \) directly as a function of \( x \), offering a simpler representation of the relationship between \( x \) and \( y \) without the intermediate parameter.

• Such expressions help to analyze how output variables depend on input variables.
• This helps us understand the behavior of equations in a more intuitive manner.

Recognizing functions in variables is key to solving real-world problems where relationships between variables are involved.