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Graph sketching is a critical step in visualizing mathematical equations. By graphing equations, we can better understand their properties and behaviors. In this exercise, we consider the equation derived from parametric forms: \( y = \frac{1}{x} \). This equation represents a rectangular hyperbola. To sketch, identify key characteristics:

• The hyperbola has two main branches located symmetrically about both axes.
• The x-axis and y-axis act as asymptotes, which means the branches approach them but never actually touch or cross.

Begin by plotting a few points for clarity. For instance, when \( x = 1 \), \( y = 1 \) resulting in the point \( (1, 1) \). Conversely, when \( x = -1 \), \( y = -1 \) resulting in the point \( (-1, -1) \). Notice the curve's symmetry in the first and third quadrants. This visual representation allows one to interpret the relationship between \( x \) and \( y \) more intuitively.

Equation derivation involves eliminating parameters to form a direct relationship between variables. In this situation, starting with the parametric equations \( x = e^t \) and \( y = e^{-t} \), our objective was to express \( y \) directly in terms of \( x \). First, recognize that expressing \( t \) as a function of \( x \) helps eliminate the parameter. Since \( x = e^t \), rewrite it using logarithms to isolate \( t \): \( t = \ln x \). Substituting back into the second equation \( y = e^{-t} \), we replace \( t \) with \( \ln x \), simplifying it to \( y = e^{-\ln x} = \frac{1}{e^{\ln x}} = \frac{1}{x} \).Thus, the parameter-free equation is \( y = \frac{1}{x} \). This equation signifies a hyperbola and simplifies the relationship between \( x \) and \( y \) without the intermediary variable \( t \), making analysis easier and more direct.

A hyperbola is a type of curve on a plane, defined by its distinct geometric properties. In this context, the equation \( y = \frac{1}{x} \) describes a rectangular hyperbola. A hyperbola consists of two disconnected curves, or branches, which reflect symmetrically relative to the origin and their respective asymptotes.Key features of the hyperbola:

• It has asymptotes, lines that the branches of the hyperbola approach but do not intersect.
• The curve is symmetric about the lines \( y = x \) and \( y = -x \), especially visible in rectangular hyperbolas.
• As \( x \) moves towards zero, \( y \) tends towards infinity.
• Similarly, as \( x \) becomes very large or very small, \( y \) tends towards zero.

Recognizing these elements supports deeper comprehension of the functional relationship and the behavior of equations associated with hyperbolas.

The orientation of curves describes the direction in which a curve is traced as the parameter increases. For the parametric equations \( x = e^t \) and \( y = e^{-t} \), understanding orientation helps to predict and comprehend movement along the curve. When plotting the hyperbola \( y = \frac{1}{x} \), relate

• The orientation is from (1, 1) moving towards the fourth quadrant while \( x \) increases in the first quadrant.
• Likewise, for negative values, it begins at (-1, -1) leading to the second quadrant in the third quadrant.

Understanding this tracing pathway is crucial for visualizing how the curve develops over its domain. These directions emerge naturally by following the path as the parameter \( t \) changes, offering a complete picture of the curve's behavior and structure.