Problem 29 - 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 \(x.\) $$\begin{array}{l} x=4+3 \cos \theta \\ y=-2+\sin \theta \end{array}$$

Step-by-Step Solution

Answer

'y' is not a function of 'x' as per the Vertical Line Test.

Step 1: Graphing the parametric equations

First, plot the curve represented by the given parametric equations \(x=4+3\cos\theta\) and \(y=-2+\sin\theta\). To do this, use a graphing utility to generate multiple pairs of (x, y) values for a range of \(\theta\) values. Typically, \(\theta\) is varied from 0 to \(2\pi\) and the corresponding values of x and y are computed using the given expressions. The resultant pairs of x, y values will provide the points on the graph.

Step 2: Vertical Line Test

After plotting the points and drawing the curve, apply the Vertical Line Test. This test is used to determine if a curve represents a function. If any vertical line intersects the curve at more than one point, then 'y' is not a function of 'x'. If every vertical line intersects the curve at most one point, then 'y' is a function of 'x'.

Step 3: Conclude

Analyze the graph. You should see that the curve appears to trace out an ellipse, which means vertical lines will intersect the curve at two points. Thus 'y' is not a function of 'x' according to vertical line test.

Key Concepts

Graphing Utility

A graphing utility is a digital tool that simplifies the plotting of equations. It can be a calculator or a computer software program. These tools help visualize mathematical concepts by generating graphs quickly and accurately. When working with parametric equations like \(x=4+3\cos \theta \) and \( y=-2+\sin \theta \), a graphing utility allows us to compute the (x, y) coordinates for various \( \theta \) values.

• Enter the parametric equations into the utility.
• Set the range for \( \theta \), typically from 0 to \(2\pi \).
• Generate points for each value of \( \theta \).

Graphing utilities offer various plotting options like adjusting the viewing window, focusing on specific graph sections, and changing axes. This can be extremely useful when examining details or behavior of the graph you might miss by plotting manually.

Vertical Line Test

The Vertical Line Test is a simple method for determining whether a curve on a graph represents a function of \( x \). The basic rule is:

• If a vertical line intersects the graph at more than one point at any location, then \( y \) is not a function of \( x \).
• If each vertical line intersects the graph at most one point, then \( y \) is a function of \( x \).

This concept is critical when analyzing graphs of parametric equations or plotting any graph to determine if it meets the definition of a function. For the given problem where the graph traces an ellipse, any vertical line that cuts through the edges of the ellipse will hit two points on the graph, demonstrating that the graph does not pass the Vertical Line Test. Therefore, \( y \) is not a function of \( x \) for this graph.

Function

A function is a special relationship in mathematics where each input has a single output. In terms of graphs, this means that for each \( x \)-value, there is only one corresponding \( y \)-value. This uniqueness is what the Vertical Line Test checks.
When considering parametric equations, the concept of a function becomes a bit complex. With traditional \( y = f(x) \) relations, it’s clear how functions operate. However, parametric equations introduce another variable \( \theta \), creating a set of points that might not maintain the unique \( y \) for each \( x \). This is why understanding the nature of functions is essential when dealing with parametric plots.

• Functions must pass the Vertical Line Test.
• Each \( x \) is associated with one and only one \( y \).

In the problems and exercises concerning graphs, recognizing when a graph represents a function helps clarify how we can manipulate and use the graphed data.