Sketch the curve by using the parametric equations to plot...

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases.

$ x = e^{-a} + t $, $ \quad y = e^a - t $, $ \quad -2 \leqslant t \leqslant 2 $

Key Concepts

Parametric Equations

Parametric equations define a curve by expressing the coordinates of its points as functions of one or more independent parameters. In this context, the equations x = constant + t and y = constant - t describe a curve by letting the parameter t vary over a specified interval, which allows one to calculate corresponding x and y values for plotting the curve.

Elimination of the Parameter

Eliminating the parameter involves rewriting the parametric equations to obtain a direct relationship between x and y. This process can yield a familiar Cartesian equation that represents the curve, such as a straight line or a conic section, and helps reveal the intrinsic nature of the curve without the intermediary parameter.

Graphical Representation and Plotting Points

Plotting points is a common method for sketching curves defined by parametric equations. By selecting various values of the parameter within the given range and calculating the corresponding (x, y) coordinates, one can obtain discrete points that, when connected, approximate the shape of the curve. This technique also facilitates understanding the curve's behavior and identifying key features.

Curve Orientation and Direction of Increase

The orientation or direction of a curve is determined by the way the coordinates change as the parameter increases. An arrow on the graph indicates the direction in which the curve is traced as the parameter grows, which is crucial when analyzing the movement along the curve and understanding how the curve is generated from the parametric representation.

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