1-4= Sketch the curve by using the parametric equations to...

$1-4=$ 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.
$$x=e^{-t}+t, \quad y=e^{t}-t, \quad-2 \leqslant t \leqslant 2$$

Key Concepts

Direction of Motion

The direction in which the curve is being traced can be determined by observing how the (x, y) coordinates change as the parameter increases. This is critical because it provides insight into the orientation of the curve and the progression of points along it. Indicating this progression with an arrow on the sketch is necessary to understand the dynamic behavior of the parameterized system.

Exponential Functions in Parametric Equations

Exponential functions often introduce rapid growth or decay into the behavior of curves described by parametric equations. Their properties, such as having distinctly different rates of increase or decrease, can significantly affect the shape of the curve. Recognizing and analyzing the influence of these exponential components is essential for accurately predicting the curvature, slope, and overall movement on the graph.

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of a third variable (typically called the parameter). This method allows for a more flexible description of curves that may not be functions in the traditional sense, meaning the relationship between x and y can loop or double back, and it is particularly useful in representing motion where the parameter can denote time.

Plotting Curves using Parametric Representations

Plotting a curve from parametric equations involves computing the coordinates for a range of parameter values and then graphing these (x, y) points. This process provides a visual representation of the curve and helps in understanding its overall shape, including any turning points or asymptotic behavior. It also involves connecting the calculated points in the order dictated by the parameter to accurately represent the curve.

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