91Ó°ÊÓ

In mathematics, transforming parametric equations into a Cartesian equation is a useful technique. This process helps us express the relationship between variables without introducing a third, intermediary parameter.

To convert a parametric equation to a Cartesian equation, we follow a few steps:

• Determine a way to express the parameter in terms of either variable.
• Substitute that expression into the other equation.
• Simplify the resulting expression to reveal a relationship solely between the two main variables.

In our exercise example, we began with the parametric equations:
\( x = 1 + t \) and \( y = t^2 - 1 \).
First, we solved for \( t \) in the \( x \) equation, giving us \( t = x - 1 \). Substituting \( t = x - 1 \) into the \( y \) equation yields the Cartesian form:
\[ y = (x-1)^2 - 1 \].
After simplifying, we arrive at the Cartesian equation \( y = x^2 - 2x \). This expresses the curve completely in terms of \( x \) and \( y \) and illustrates the foundational principle of removing parameters.

A parabola is a symmetrical curve on a plane, which can open upwards, downwards, leftwards, or rightwards. In a Cartesian coordinate system, a basic parabola can be represented by the equation:
\( y = ax^2 + bx + c \).
This equation indicates how the parabola is oriented and where it is located. The values of \( a \), \( b \), and \( c \) determine the direction and shape of the parabola.

• If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• The vertex of the parabola, where it turns direction, plays a crucial role in its shape. This can be found using \( x = -\frac{b}{2a} \).
• The y-intercept occurs where the parabola crosses the y-axis, at point \( (0, c) \).

In our specific example, the Cartesian equation \( y = x^2 - 2x \) demonstrates a parabola opening upwards because the coefficient of \( x^2 \) is positive (1). By analyzing the terms, we infer the vertex formula reveals the vertex is at \( x = 1 \). This helps in sketching an accurate representation.

Eliminating the parameter from parametric equations is a standard method to simplify and clarify relationships between variables. This approach focuses on removing the parameter, often denoted as \( t \), that we use to define a curve. By doing so, we derive what is known as a Cartesian or implicit equation.

The general steps include:

• Isolating the parameter in one of the parametric equations.
• Substituting this expression into the second equation.
• Simplifying the resulting equation to connect the primary variables.

For the exercise provided, we had \( x = 1+t \) and \( y = t^2 - 1 \). Upon solving for \( t \) in the first equation, \( t = x - 1 \) was substituted into the second. This led us to simplify to find the Cartesian equation \( y = x^2 - 2x \). By following this process, the parameter \( t \) is effectively eliminated, leaving a direct connection between \( x \) and \( y \). This makes it easier to visualize and analyze the curve's properties.