Problem 34 - 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 \(x.\) $$\begin{aligned} &x=10-0.01 e^{t}\\\ &y=0.4 t^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The answer will be determined by the result of the Vertical Line Test. If for any vertical line, it intersects the graph at more than one point, \[y\] is not a function of \[x\]. However, the specific answer is dependent on the visual result of the graphing process and vertical line test.

Step 1: Graph the Parametric Equations

Utilize the capabilities of a graphing utility to sketch the curve based on the parametric equations. Input each respective function into the graphing utility, setting t as the parameter. Adjust the range of 't' if desired to get a more detailed graph. Once the equations are placed, generate the graph.

Step 2: Apply the Vertical Line Test

Once the graph is plotted, apply the Vertical Line Test to check if \(y\) is a function of \(x\). This is done by imagining or drawing vertical lines along the graph. If any vertical line intersects the graph at more than one point, then \(y\) is not a function of \(x\).

Step 3: Analyze and Conclude

Based on the graph and result of the Vertical Line Test, conclude whether \(y\) is a function of \(x\) or not. This should be clear from the vertical line test.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visually represent mathematical functions and equations. Here, it helps us graph the parametric equations like the ones given in the original exercise. These often come in the form of calculators or software, which allow you to input equations with a parameter (in this case, 't') and produce a graph accordingly.
To use a graphing utility effectively, you need to:

• Input the parametric equations as given, specifying the parameters correctly.
• Set a suitable range for the parameter 't' to ensure a complete and accurate graph.
• Explore different ranges for 't' if the graph does not appear as expected.

In the exercise, when you graph the functions \(x = 10 - 0.01 e^{t}\) and \(y = 0.4 t^{2}\), it's crucial to observe the behavior and shape of the curve, as it guides you in applying further tests, like the Vertical Line Test.

Vertical Line Test

The Vertical Line Test is a simple yet effective way to determine if a curve in a plane is a representation of a function of \(x\). In order to understand how the Vertical Line Test works, imagine drawing vertical lines across your graph.
This is the process:

• Identify arbitrary vertical lines along the \(x\) axis on your graph.
• Check whether any of these lines intersect the graph at more than one point.

If a vertical line intersects the graph at more than one point, the graph does not represent \(y\) as a function of \(x\). This is because, in a function, each \(x\) must correspond to exactly one \(y\). In this exercise, after graphing \(x = 10 - 0.01 e^{t}\) and \(y = 0.4 t^{2}\), you apply this test to see if a vertical line will cross the curve in multiple places.

Function of x

A function of \(x\) is one where each \(x\)-value has only one corresponding \(y\)-value. This is fundamental in mathematics for ensuring clear and predictable relationships.
When dealing with parametric equations, determining if \(y\) is a function of \(x\) involves both graphical and analytical methods. Graphically, we've used the Vertical Line Test, and now:

• Consider the behavior described by the parametric equations: \(x = 10 - 0.01 e^{t}\) and \(y = 0.4 t^{2}\).
• Notice how changes in the parameter 't' impact both \(x\) and \(y\).

By carefully examining the plotted points and their configuration on the graph, you can conclude about the function's properties. In essence, if no vertical line crosses the curve more than once, \(y\) is indeed a function of \(x\). This exercise teaches how crucial it is to visualize and test parametric equations in verifying functional relationships.