91Ó°ÊÓ

Eliminating the parameter in parametric equations means converting them into a single equation without the parameter. Here, the parameter is \( t \). Start by finding \( t \) from one of the given parametric equations. In this case, from the equation \( x = \ln(2t) \), solve for \( t \) to get \( t = \frac{e^x}{2} \).
Next, substitute this expression for \( t \) into the second parametric equation \( y = t^2 \). This gives us \( y = \left(\frac{e^x}{2}\right)^2 \), which simplifies to \( y = \frac{e^{2x}}{4} \).
This equation, \( y = \frac{e^{2x}}{4} \), is referred to as the Cartesian equation, derived by eliminating the parameter \( t \). No longer do we need \( t \) to understand the relationship between \( x \) and \( y \).

A Cartesian equation describes a curve using two coordinates, typically \( x \) and \( y \), without an external parameter. By eliminating the parameter \( t \) from a parametric set of equations, you obtain such an equation.
In our example, once we solved for \( t \) and substituted into the equation for \( y \), the resulting Cartesian equation was: \( y = \frac{e^{2x}}{4} \).
This equation depicts a continuous relationship where the output \( y \) is dependent on \( x \). In essence, Cartesian equations simplify the complexity of parametric representations, providing a clearer picture of how \( y \) depends on \( x \) directly.

Asymptotes are lines that a graph approaches but never touches or crosses. They help understand the behavior of a function as it moves towards infinity in either direction.
To find the asymptotes, examine the function's behavior at the extremes of its domain. In the equation \( y = \frac{e^{2x}}{4} \), as \( x \) approaches negative infinity, the value of \( y \) approaches zero.
This behavior indicates the presence of a horizontal asymptote at \( y = 0 \). This means the graph gets closer and closer to this line but doesn't actually touch it as \( x \) decreases indefinitely.
There are no vertical asymptotes in this particular equation, as there are no values of \( x \) that lead to undefined \( y \)-values.

Graph sketching is the process of visually representing a function based on its equation. Having the Cartesian equation, \( y = \frac{e^{2x}}{4} \), serves as a guide to this process.
Begin by identifying key features such as intercepts, symmetry, and asymptotes, which were discussed earlier. This equation yields a graph that is symmetric about the y-axis because \( e^{2x} \) remains positive for all values of \( x \).
When sketching, note that all \( y \)-values are positive, as \( \frac{e^{2x}}{4} \) is always greater than zero. The closest the graph gets to the x-axis (\( y = 0 \)) is defined by its horizontal asymptote.

• The graph starts near the horizontal asymptote and curves upwards as \( x \) increases.
• No parts of the curve dip below the x-axis.

With these points clear, you can confidently sketch the curve to visualize its growth and limitations.