Question

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{aligned} x &=t \\ y &=2 t+1 \end{aligned} $$ $$ \text { (b) } \begin{aligned} x &=\cos \theta \\ y &=2 \cos \theta+1 \end{aligned} $$ $$ \text { (c) } \begin{aligned} x &=e^{-t} \\ y &=2 e^{-t}+1 \end{aligned} $$ (d) \(x=e^{t}\) $$ y=2 e^{t}+1 $$

Short answer

The curves of parametric equations (a) and (d) are the same, their orientations are also same and these curves are smooth. The curve of parametric equations (b) is the same line as in the other cases but with a different orientation and smoothness. The curve of parametric equations (c) remains similar to previous, but with varying orientation and same smoothness.

Step-by-step solution

Plot the graph of the parametric equations (a)

The graph plotted for this set of equations represents a straight line. For different values of \( t \), \( x = t \) and \( y = 2t + 1 \). A line is formed starting from the point (0,1) and increasing in the direction of the arrows.

Plot the graph of the parametric equations (b)

The graph plotted for this set of equations represents a straight line with the same y-intercept as in case (a), but lying along the x-axis as the range of the cosine function is between -1 and 1. The curve is smooth but the orientation is different as it traverses back and forth between -1 and 1 in x direction.

Plot the graph of the parametric equations (c)

For these equations, as \( t \) increases, \( x \) and \( y \) become smaller approaching the values (0,1). As \( t \) decreases, \( x \) and \( y \) increase with \( y \) always being 1 unit above \( x \). This curve is smooth and has a different orientation to (a) and (b). The curve decreases as \( t \) increases.

Plot the graph of the parametric equations (d)

The plotted graph for this set of equations represents an increasing line starting from point (1,3). As \( t \) increases, both \( x \) and \( y \) increase. This curve is smooth and has the same orientation as (a).

Key concepts

Graphing Parametric Equations

Understanding how to graph parametric equations provides insight into a variety of mathematical concepts and offers a powerful tool for visualizing the relationship between two variables. Generally, parametric equations involve two equations, one for each coordinate, typically denoted as x(t) and y(t), where t is the parameter. To graph these equations, one plots the coordinates (x(t), y(t)) for various values of t.

In the example provided, solutions (a) and (d) are linear functions, whereas solutions (b) and (c) are non-linear due to the trigonometric and exponential functions involved. The straight-line graph in solution (a) indicates a direct proportionality between x and y, while solutions (b) involve periodicity due to the cosine function. Conversely, the exponential functions in (c) and (d) suggest growth or decay in the value of the variables as the parameter t changes. To plot these graphs correctly, one needs to calculate several pairs of (x, y) values by substituting values of t and then connect these points smoothly to reveal the curve. The graph's shape can divulge much about the system it represents, such as the direction of motion if the parameter t represents time.

Orientation of Parametric Curves

When analyzing the orientation of parametric curves, we are referring to the direction in which the curve progresses as the parameter increases. Usually, this is indicated with arrows on the graph. The orientation can reveal real-world application patterns, such as the direction of an object's movement in physics.

In the given step-by-step solution, the orientation of curve (a) is straightforward as it moves to the right as t increases. On the other hand, curve (b) oscillates back and forth, which is depicted by the change in direction as cosine values transition from positive to negative and vice versa. This behavior demonstrates a key difference in orientation compared to the other linear curves. For curves like (c), where the values decrease exponentially as t increases, the orientation is towards the point (0,1), representing a decay, which is precisely opposite of the orientation in solution (d) where the values rise exponentially. Identifying the orientation of curves helps in understanding the dynamics of the system being studied, such as whether a process is reversible and how it evolves over time.

Properties of Parametric Curves

The properties of parametric curves encompass their smoothness, concavity, and points of intersection or inflection. A curve is smooth if it has a continuous tangent line at every point, and these properties can profoundly affect the interpretation of a curve's shape and behavior.

In the original solutions, notice that the curves derived from equations (a) and (d) are linear and hence smooth without any breaks or sharp turns. Curves (b) and (c) are also smooth; however, they demonstrate different characteristics, such as periodicity in (b) and asymptotic behavior in (c), as they never cross certain lines (x or y-axis). These properties help in understanding not just the static graph but also how it behaves as the parameters change. For instance, a smooth curve indicates that the system may proceed without any abrupt changes, which could be important in engineering applications where sudden shifts could cause stress or failure.