Problem 18 - 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 \( t.\) \(x=t^{2}+1, y=t^{3}-1 ;-\infty< t<\infty\)

Step-by-Step Solution

Answer

The plane curve described by the parametric equations \(x=t^{2}+1\) and \(y=t^{3}-1\) over the interval from negative infinity to positive infinity will appear to be a curve, with an orientation indicated by arrows that move in the direction of the curve from smaller to larger t values.

Step 1: Understanding Parametric Equations

Begin by understanding that parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as 'parameters.' For example, the equations \(x = t^{2}+1\) and \(y = t^{3}-1\) are parametric equations. They describe a curve on a plane. Where \(x\) and \(y\) are dependent on the parameter \(t\).

Step 2: Identify a Set of Values for t

Given that \( t \) can be any real number from negative infinity to positive infinity, choose a reasonable range of values to plot. For example, you could choose values of \( t \) from -2 to 2.

Step 3: Substitute Values for t into Parametric Equations and Solve

Substitute the chosen values for \( t \) into the parametric equations. For example, if you choose \( t = -2 \), substitute \( -2 \) into both equations to get \( x = (-2)^{2}+1 = 5 \) and \( y = (-2)^{3}-1 = -9 \). Repeat this step for all selected values of \( t \).

Step 4: Plot the Points

Plot the points obtained from the previous step on a graphing paper or using a graphing tool. Ensure to label each point with their corresponding \( t \) value.

Step 5: Draw the Curve and Indicate Direction

Join the plotted points to form a curve. Next, use arrows to show the direction of the curve based on increasing \( t \) values. This indicates the orientation of the curve.

Key Concepts

Plane Curve

A plane curve refers to a curve that lies on a flat surface, known as a plane. In mathematics, plane curves are often described using parametric equations, which allow us to represent the curve by two separate equations that define the coordinates on the plane. These equations give the x and y coordinates as functions of a third variable, typically a parameter like t. For the given problem, the parametric equations are:

• \( x = t^2 + 1 \)
• \( y = t^3 - 1 \)

These functions, with the parameter \( t \), describe a path traced on the plane as t varies. Since both equations are continuous and differentiable over all real numbers, they form a smooth plane curve without any interruptions or breaks.

Graphing

Graphing is the process of visually representing mathematical equations or functions on a coordinate plane. When dealing with parametric equations, it often involves drawing the path traced by the equations over a range of parameter values. For our problem, graphing involves several steps:

• Choosing a set of \( t \) values, such as from \(-2\) to \(2\), to plug into the equations.
• Calculating the corresponding \( x \) and \( y \) values for each selected \( t \).
• Plotting these (x, y) points on the graph.

Once plotted, the collected points show the physical path of the plane curve. As you plot each point, you can start to see the overall shape and direction in which the graph is developing.

Orientation

Orientation in parametric curves refers to the direction in which the curve is traced as the parameter increases. In the exercise, the orientation is given by arrows along the curve. These arrows show that as \( t \) increases, the curve moves in a particular direction.Understanding orientation helps in comprehending how the curve evolves with changing values of \( t \). It's especially useful when mapping or comparing how various segments of the curve relate to one another. To illustrate orientation while graphing, simply draw small arrows along the curve at various points, indicating the progression of \( t \) values.

Point Plotting

Point plotting is critical for understanding the shape and position of a parametric curve on the plane. It involves manually computing and marking key points determined by the parametric equations. For illustration:

• Select a range of \( t \) values, e.g., \(-2, -1, 0, 1, 2\).
• Substitute each \( t \) value into the equations \( x = t^2 + 1 \) and \( y = t^3 - 1 \) to calculate the corresponding \( (x, y) \) points.
• Plot these points on a coordinate plane.

By following these steps, each point represents a specific position on the plane that corresponds to a specific \( t \) value. Once all points are plotted, connecting them reveals the progress and shape of the curve. This technique provides a tangible way to visualize and better understand the behavior of the plane curve.