Problem 30 - 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 \(x.\) $$\begin{aligned} &x=4+3 \cos \theta\\\ &y=-2+2 \sin \theta \end{aligned}$$

Step-by-Step Solution

Answer

The graph is a circle centered at `(4,-2)` with radius 2. Since vertical lines will intersect the graph more than once, \(y\) is not a function of \(x\).

Step 1: Graphing the parametric equations

Enter the parametric equations into the graphing utility. These are \(x = 4 + 3\cos(\theta)\) and \(y = -2 + 2\sin(\theta)\). Obtain a graph representing the curve defined by these equations.

Step 2: Perform the Vertical Line Test

Draw various vertical lines through the graph. If any of these lines intersect the graph at more than one point, then \(y\) is not a function of \(x\). Conversely, if none of the vertical lines intersect the graph more than once, \(y\) is a function of \(x\).

Step 3: Observations and Conclusions

Observe the interactions between the vertical lines and the graph. Derive a conclusion based on these interactions regarding whether or not \(y\) is a function of \(x\).

Key Concepts

Graphing Utility

To visualize parametric equations like \(x = 4 + 3\cos(\theta)\) and \(y = -2 + 2\sin(\theta)\), a graphing utility is extremely helpful. This tool allows you to enter the equations and generate the graph directly on a digital interface.
Here’s why using a graphing utility can be advantageous:

• **Accurate Plotting**: It automatically calculates the coordinates defined by the parametric equations, ensuring precision.
• **Visualization**: You can easily visualize complex curves and paths which might be tricky to draw by hand.
• **Interactivity**: Graphing utilities often allow you to manipulate the plot to zoom and pan for a better view of specific sections.
• **Time Efficiency**: Saves time compared to manually calculating and plotting each point.

These features are particularly useful when dealing with parametric equations that describe curves which do not resemble traditional graphs.

Vertical Line Test

The Vertical Line Test is a simple way to determine if a curve on a graph represents a function. This is applicable to typical \(y = f(x)\) graphs, where every input \(x\) has exactly one output \(y\).
How it works:

• **Draw Vertical Lines**: Imagine drawing vertical lines (parallel to the \(y\)-axis) across your graph.
• **Single Intersection**: If each line intersects the curve at most once, then \(y\) passes the Vertical Line Test and is a function of \(x\).
• **Multiple Intersections**: If any vertical line intersects the graph more than once, \(y\) is not a function of \(x\).

The test is applicable even when dealing with parametric equations. Once you have the graph, simply apply the Vertical Line Test to determine the function status.

Function of x

In calculus and algebra, a **function** essentially maps every element from one set (the domain) to a single element in another set (the codomain). When we say \(y\) is a function of \(x\), it implies that each \(x\) value has one corresponding \(y\) value.
For parametric equations:

• **Define the relationship**: Parametric equations express both \(x\) and \(y\) in terms of another variable, often \(\theta\) or \(t\), creating a path or a curve.
• **Direct Mapping**: To check function conditions, see if you can convert the parametric equations into an explicit \(y = f(x)\) form holding one \(y\) for each \(x\).
• **Visual and algebraic check**: Besides the graphical Vertical Line Test, functions can also be verified through algebraic manipulation to confirm the one-to-one correspondence.

Understanding whether \(y\) is a function of \(x\) helps in analyzing the behavior and characteristics of the curve formed by the parametric equations.