Problem 38 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=t+1\\\ &y=\sqrt{2-t} \end{aligned}$$

Step-by-Step Solution

Answer

The curve starts at the point (0, \(\sqrt{2}\)) when t is at its minimum, moves down and to the right, reaches its lowest point at t=2 (3,0), then continues upwards as t increases. This is a graph of a downward-opening parabola.

Step 1: Understand the Parametric Equations

Identify the given parametric equations. In this case they are \(x = t + 1\) and \(y = \sqrt{2 - t}\).

Step 2: Pick Values for t

Choose a range of values for t, usually starting from a negative number and going up to a positive one. Discrete numbers can be selected in that range to plot individual points.

Step 3: Calculate Corresponding x and y values

Plugging the values of t into the parametric equations, calculate the corresponding x and y coordinates. For example, if t=0, then x=0+1=1 and y=\(\sqrt{2 - 0}\)=\(\sqrt{2}\).

Step 4: Plot Points on a Graph

Plot each pair of x and y as an individual point on a graph. If using a graphing utility, input the parametric equations and the range of t values directly into the tool.

Step 5: Connect the Points

Arrange the points in the order of the t values, and then connect them with a smooth curve to represent the path traveled as t increases.

Key Concepts

Graphing Utility

When faced with the challenge of graphing parametric equations, a graphing utility becomes an indispensable tool for students. Such utilities simplify the process by allowing the parameters to be entered into the system, which then automatically generates the corresponding graph. For the exercise at hand, where the parametric equations are given by
\(x=t+1\)
and
\(y=\sqrt{2-t}\),
a graphing utility efficiently handles the undertaking of plotting this curve.

The advantages of using a graphing utility include time-saving, providing visual understanding, and allowing for experimentation. Students can vary the range of parameter \(t\) and observe the resulting changes in the graph, gaining a deeper insight into the behavior of the parametric equations. Moreover, these utilities often come with features for zooming, panning, and adjusting the graph's visual representation, which is particularly useful when precise details or a broad overview is required.

Plotting Parametric Coordinates

Plotting parametric coordinates is a method of drawing a curve by marking out points that result from substituting a range of numbers into parametric equations. Unlike regular Cartesian coordinates, which provide direct correlations between \(x\) and \(y\) values, parametric equations introduce an independent parameter, typically denoted as \(t\), which defines a set of conditions for both \(x\) and \(y\). In our case, we see that \(x\) and \(y\) are both linked to \(t\) by their respective formulas.

By plugging in various values of \(t\), we generate pairs of \(x\) and \(y\) that we then plot on the coordinate plane. It's crucial to maintain the sequence of \(t\) values to properly represent the direction of the curve. This process is made easier with a graphing utility, but it can also be achieved manually. Especially for intricate or unusual curves, plotting points aids in visualizing the complex relationship between \(x\), \(y\), and \(t\) and can illuminate characteristics of the curve that might not be immediately obvious.

Parametric Equations

Parametric equations represent a set of equations where the graph's points are expressed as functions of a third variable, usually denoted by \(t\). This variable \(t\) is known as the parameter, which can represent anything from time to angle to any other independent variable of interest. Unlike traditional Cartesian equations that express \(y\) as a function of \(x\), parametric equations define \(x\) and \(y\) separately, each in terms of \(t\).

In our exercise, the parametric equations are \(x = t + 1\) and \(y = \sqrt{2 - t}\). As \(t\) varies, each value of \(t\) corresponds to a specific point \((x, y)\) that can be plotted on a graph. This allows us to trace the path of a moving point and is particularly useful in physics and engineering applications where the trajectory of an object needs to be visualized. Understanding parametric equations is fundamental for students, as it opens up a broader perspective on how different mathematical concepts can intersect and influence one another.