Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Graph each circle by hand if possible. Give the domain and range. $$x^{2}+y^{2}=0$$

Step-by-Step Solution

Answer

The graph is a point at (0, 0) with domain and range both [0, 0].

Step 1: Analyze the Circle Equation

The given equation is \(x^2 + y^2 = 0\). This is the equation of a circle centered at the origin \((0, 0)\) with radius \( r \). To find the radius, we compare this to the standard form of a circle's equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \( r \) is the radius. Here, we have \(h = 0\), \(k = 0\), and \(r^2 = 0\), which means \(r = 0\). Thus, it's a circle with radius zero, essentially a point at the origin.

Step 2: Determine the Domain

The domain of a function or graph refers to all possible \(x\)-values. For the given equation, since the circle is actually just a single point \((0, 0)\), the only \(x\)-value present is \(x = 0\). Therefore, the domain is \([0, 0]\).

Step 3: Determine the Range

The range of a function or graph refers to all possible \(y\)-values. Since the circle is reduced to the point \((0, 0)\), the only \(y\)-value present is \(y = 0\). Hence, the range is \([0, 0]\).

Step 4: Graph the Circle

Since the equation represents a degenerate circle (a point), the graph consists of just a single point at the origin "(0, 0)". There are no other points to plot in this case.

Key Concepts

Domain and Range of a Circle

When dealing with circles, understanding the domain and range is crucial for defining the set of all possible inputs and outputs.
The **domain** in mathematics refers to all possible values the variable can take. For a typical circle equation, this would encompass every possible x-coordinate where the circle exists on the graph. With the special equation provided, the circle simplifies to a single point at the origin.
Thus, the domain is specifically the set that contains only this x-coordinate. For our given equation, this domain is \[ [0, 0] \] as there are no other x-values that satisfy the equation.
Similarly, the **range** defines all potential y-values. In most circle graphs, this includes every y-coordinate within the circle's vertical span. In our scenario of the degenerate circle, which is a point, the range mirrors the domain. It is also \[ [0, 0] \]. Understanding these aspects is a stepping stone when graphing more complex circle equations.

Graphing Circles

Graphing a standard circle usually involves plotting points around a center at a particular radius.
However, with our unique scenario, we effectively deal with a degenerate circle, which simplifies graphing significantly.
For the given equation \[ x^2 + y^2 = 0 \], the solution represents a circle of radius zero centered at the origin. Simply put, the graph is just the point at \( (0, 0) \).
While a typical circle would include multiple points, this degenerate form only requires marking a single dot. - Accomplish this by aligning your graph grid and marking the origin.- Verify the coordinates are at x = 0 and y = 0.This process highlights the uniqueness of our circle: it shows the fundamental principles of graphing circles, although very simplified in this case.

Standard Form of a Circle

Understanding the standard form of a circle's equation is fundamental to grasp how variations affect circle properties. The standard form is: \[ (x - h)^2 + (y - k)^2 = r^2 \],where:

• \( (h, k) \)

  represents the circle's center.

• \( r \)

  is the radius.

In our instance of \[ x^2 + y^2 = 0 \], we see that \( (h, k) \) naturally becomes \( (0, 0) \), indicating an origin-centered circle. Interesting here is that \( r^2 = 0 \), thus \( r = 0 \). This equation's significance lies in helping us comprehend its breakdown into simpler parts. In learning to manipulate these elements, we gain control over how circles are framed, graphed, and interpreted.