Problem 14 - Algebra and Trigonometry

Question

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \( t.\) \(x=\sqrt{t}, y=t-1 ; t \geq 0\)

Step-by-Step Solution

Answer

The plane curve described by the given parametric equations is a curve starting at the point (0,-1) and moving upwards in an elliptical shape with increasing values of \(t\).

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = \sqrt{t}\) and \(y = t - 1\) for \(t \geq 0\). This means the value of 'x' will be the square root of 't', and 'y' will be 't' subtracted by 1. You can generate pairs of (x, y) values by choosing different values for 't'.

Step 2: Generating Pairs of (x,y) values

Begin with simple values for \(t\), like 0, 1, 2, and 3. Substituting these into the parametric equations, we get the pairs (0,-1), (1,0), (1.41, 1) and (1.73, 2) respectively. These pairs represent points on the plane curve described by the equations.

Step 3: Plotting the graph

Plot the points (0,-1), (1,0), (1.41, 1) and (1.73, 2) on a graph, then join them using arrows to show the orientation of the curve corresponding to increasing values of \(t\). The arrows should go from the point with the lowest 't' value to the point with the highest 't' value.

Step 4: Visualizing the Graph

By drawing a smooth curve through these points and extending the arrows, we get the complete graph of the parametric equations. As \(t\) increases, the curve moves upwards, following the direction of the arrows.

Key Concepts

Understanding Parametric Equations in Algebra

Parametric equations offer a versatile way of representing curves in algebra. Unlike traditional Cartesian equations that express y as a function of x, parametric equations define both x and y in terms of a third variable, typically called a parameter, which is often denoted as t. This method allows for a more flexible representation of curves, such as circles and ellipses, which are difficult to represent with a single function in Cartesian form.

In the context of our exercise, the parameter t determines the position on the curve through the equations \(x = \sqrt{t}\) and \(y = t - 1\) for \(t \geq 0\). As t varies, x and y trace out the path of a curve on a graph. Understanding the relationship between t and the x and y coordinates is essential to graphing parametric equations accurately.

Plotting Points on a Graph

Plotting points is a fundamental skill in graphing. When dealing with parametric equations, plotting points involves calculating the x and y coordinates separately for different values of the parameter t.

To do this effectively, choose a range of t values, especially those that are easy to compute, like integers or simple fractions. After calculating the x and y values, you can plot these points on a Cartesian plane. For instance, when t = 0, the point will be (0, -1), and as t increases, succeeding points are found. This step-by-step approach ensures that each coordinate pair accurately represents a point on the curve defined by the parametric equations.

Orientation of a Curve

The orientation of a curve indicates the direction in which the curve proceeds as the parameter increases. In graphing parametric equations, orientation is shown using arrows drawn on the curve in the direction of increasing t values.

It's crucial to visualize this orientation as it demonstrates how the curve is traced over time. In our case, as t increases, the curve moves upwards, and each plotted point corresponds to a specific moment in time. By following the journey of these points on the graph, one can gain insight into the dynamic nature of the parametrically defined curve. Orientation plays a significant role in understanding the behavior of moving objects and the flow of a parametrically described system.

Square Root Function

The square root function is an essential concept in both algebra and geometry. It is represented as \( f(x) = \sqrt{x} \), where each non-negative input x produces a non-negative output, the square root of x. This function is crucial in our parametric equations where x is defined as \(x = \sqrt{t}\).

Graphically, the square root function forms a curve that starts at the origin (0,0) and increases slowly at first, then more rapidly as x increases. This behavior is reflected in the shape and behavior of the parametric curve we are exploring. As t increases and hence the x-value through the square root function, the curve moves outward in a particular fashion that characterizes the square root's progressive increase.