Problem 35 - Precalculus

Question

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

Step-by-Step Solution

Answer

The graph of this set of parametric equations comes out to be a parabola, opening upwards and symmetric about the y-axis. The graph reflects that as \(t\) moves from negative infinity to positive infinity, the corresponding \(x\) value moves along with \(t\) while \(y\) increases steadily, because \(y = t^{2}\) produces \(y\) as a square of \(t\), hence always positive.

Step 1: Table of Values

First, make a table of values. Choose a range of values for \(t\) which will subsequently be used to calculate their corresponding \(x\) and \(y\) values. It's pragmatic to choose values of \(t\) equally distributed about the origin.

Step 2: Compute Coordinates

Once a table of values for \(t\) is made, compute corresponding \(x\) and \(y\) values. For \(x\), it's a direct assignment of values (since \(x = t\)). For \(y\), square each value of \(t\) (since \(y = t^{2}\)). This will yield a set of coordinate (\(x,y\)) pairs.

Step 3: Plot the Graph

Having obtained a set of (\(x,y\)) pairs, plot these points on the graph, marking each point distinctly. Interpolate these points with a smooth curve to build the graph of the parametric equations.

Key Concepts

Graphing Utility

Graphing utilities are digital tools that assist in plotting graphs of equations or inequalities accurately and efficiently. These tools can range from graphing calculators to sophisticated software available on computers and smartphones. Utilizing a graphing utility can save time and help visualize complex mathematical concepts, especially those involving multiple variables or parametric equations.

When working with parametric equations like \(x = t\) and \(y = t^2\), a graphing utility helps you visualize how the values of \(t\) in the equations create a path or curve in the coordinate plane. It plots each coordinate pair, generated by the equations based on values of \(t\), and connects them to represent the overall shape of the graph.

• Input the parametric equations directly into the graphing utility.
• Specify the range for \(t\), ensuring it covers enough values to accurately represent the curve.
• View the plotted curve, and use utilities' features like zooming and connecting points for better clarity.

By using a graphing utility, students can better understand the behavior of parametric equations and practice curve sketching dynamically.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics provides a connection between algebraic equations and geometric curves. Parametric equations, like \(x = t\) and \(y = t^2\), exemplify this connection by representing a geometric curve using a set of equations that define the coordinates of points on the curve.

In coordinate geometry, you can:

• Understand how algebraic equations map to geometric shapes.
• Manipulate equations to find properties like slope, distance, and areas.
• Utilize the coordinate system to bridge different mathematical concepts.

For parametric equations, coordinate geometry allows us to understand how changing the parameter \(t\) affects the position of points on a curve. Coordinates \((x, y)\) are calculated for different values of \(t\), constructing the shape on a coordinate plane, which is crucial for interpreting physical real-world trajectories.

Plotting Curves

Plotting curves from parametric equations is an essential skill, as it helps to visualize and interpret the relationship between mathematical functions and their graphical representations. The curve described by the parametric equations \(x = t\) and \(y = t^2\) is a parabola that opens upward, clearly illustrating the role each component of the parametric equations plays in shaping the graph.

The steps to plot curves using parametric equations include:

• Choose a suitable range for the parameter \(t\), such as negative to positive values covering interesting portions of the curve.
• Calculate a set of coordinate \((x, y)\) pairs from the equations.
• Plot these \((x, y)\) pairs on the coordinate plane.
• Connect the dots with a smooth curve to fully reveal the shape.

Plotting curves not only involves calculating and plotting points but also understanding the behavior of the function conveyed by the curve. This skill strengthens the comprehension of continuous changes in variables and their real-world implications.