Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the curve represented by the parametric equations. Use the graph and the Vertical Line Test to determine whether \(y\) is a function of \(x.\) $$\begin{aligned} &x=t / 2\\\ &y=\ln \left(t^{2}+1\right) \end{aligned}$$

Step-by-Step Solution

Answer

After graphing the curves and applying the Vertical Line Test, conclude whether or not \(y\) is a function of \(x\).

Step 1: Graphing the parametric equations

Use any graphing utility (like Desmos, GeoGebra, or a graphing calculator) to plot the parametric equations. In the settings, set the \(t\) to vary within a reasonable range (like -10 to 10) for proper viewing. The horizontal values (\(x\)) are given by \(t/2\) and the vertical values (\(y\)) are given by \(\ln(t^{2}+1)\). Plot these points to create the graph.

Step 2: Observing the generated graph

Observe the graph generated by the graphing tool used in Step 1. It's important to note if every x-value corresponds to exactly one y-value.

Step 3: Applying the Vertical Line Test

Apply the Vertical Line Test to your graph, which involves imagining or drawing vertical lines across the graph. If any line intersects the graph in more than one place, then \(y\) is not a function of \(x\). If no vertical line can intersect the graph more than once, then \(y\) is a function of \(x\).

Key Concepts

Vertical Line Test

When dealing with graphs, especially those from parametric equations, it's crucial to determine if the resulting equation is a function. The Vertical Line Test is a simple way to do this. It's like drawing imaginary vertical lines from top to bottom on your graph.
A graph represents a function if no vertical line can cut it at more than one point.

• If a vertical line intersects the graph only once, it's a function of \(x\).
• If a vertical line crosses the graph in more than one spot, it's not a function of \(x\).

This test is a visual tool that helps quickly identify if each \(x\)-value corresponds to only one \(y\)-value. By applying this test to the graph of any parametric equation, you can easily discern if there is an overlap or if it satisfies the function condition.

graphing utilities

Graphing utilities are powerful tools for visualizing mathematical equations, especially for parametric equations. Examples include Desmos, GeoGebra, and traditional graphing calculators. These tools allow you to plot points derived from equations efficiently.
When using a graphing utility with parametric equations:

• Set a suitable range for the parameter, \(t\), e.g., from -10 to 10, to capture a comprehensive view of the graph.
• Enter the formulas for \(x\) and \(y\) as functions of \(t\).
• The utility will then generate a graph, showcasing how \(x\) and \(y\) change as \(t\) varies.

These utilities not only display the mathematical relationships but also provide an interactive way to experiment with and understand complex equations. They are indispensable for students learning about parametric equations.

function of x

A function of \(x\) is a specific type of relationship where each input \(x\) has exactly one output \(y\). For parametric equations, determining if \(y\) is a function of \(x\) involves analyzing if each \(x\) value maps to a unique \(y\) value.

• This is crucial in graphs where both \(x\) and \(y\) depend on a third variable, \(t\).
• A graph will represent a function of \(x\) if after eliminating \(t\), each \(x\) results in one unique \(y\).

The Vertical Line Test serves as a quick method of checking this criterion. When dealing with parametric forms, verifying that the plotted graph does not contradict the conditions of function mapping ensures that mathematical principles are maintained. Understanding this concept helps in distinguishing between functions and more complex relations.