Problem 31 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

Graph is a point at (0,0); domain and range are \{0\}.

Step 1: Identify the Standard Form

The equation of a circle in standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. The given equation is \(x^2 + y^2 = 0\).

Step 2: Determine the Center and Radius

By comparing \(x^2 + y^2 = 0\) to the standard form, we identify that \(h = 0\), \(k = 0\), and \(r^2 = 0\). This gives a center at \((0,0)\) and a radius \(r = 0\).

Step 3: Graph the Circle

A circle with a radius of 0 is essentially a single point at its center, \((0,0)\). Thus, the graph consists of the point \((0, 0)\) only.

Step 4: Find the Domain

The domain of the circle is the set of all \(x\)-values that the circle covers. Since the circle is just the point \((0,0)\), the only \(x\)-value in the domain is \(x = 0\). Thus, the domain is \([0, 0]\), equivalently \{0\}.

Step 5: Find the Range

The range of the circle is the set of all \(y\)-values that the circle covers. Again, since the circle is just the point \((0,0)\), the only \(y\)-value in the range is \(y = 0\). Thus, the range is \([0, 0]\), equivalently \{0\}.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is a concise way to represent all the necessary elements of a circle. It is expressed as \[(x-h)^2 + (y-k)^2 = r^2\] where:

• \((h,k)\) represents the center of the circle.
• \(r\) is the radius of the circle which is always a non-negative value.

Given the equation \(x^2 + y^2 = 0\), comparing it to the standard form reveals that the center \((h, k)\) is at \((0, 0)\) and the radius squared is zero. This means the radius \(r\) is zero, indicating the circle is just a point. Understanding the arrangement in this form helps in easily identifying the dimensions and placement of the circle on the coordinate plane. This is especially useful for detecting any deviations or translations from the origin or default positions.

Graphing Circles

Graphing circles involves plotting them in a coordinate plane according to their equation in standard form. Normally, you would start by marking the center \((h, k)\) of the circle. Then, using the radius \(r\), determine the boundary of the circle by measuring this distance from the center in all directions. However, in the unique case of our equation \(x^2 + y^2 = 0\), where the radius \(r = 0\), the usual boundary doesn't exist.Since the radius is zero, the graph doesn't show an extended circle, but rather, just its center, the point \((0, 0)\). This is because a radius of zero means that there is no distance from the center to any outer circle, hence no extended circle surface or edge to plot. Thus, the graph of this equation is simply the origin point, giving a special instance of a "circle" being reduced to a solitary point.

Domain and Range

In mathematical terms, the domain and range help define the scope of the circle on the graph.Domain generally refers to all the possible \(x\)-values that can occur in your circle's equation. For typical circles, this is given by the horizontal stretch defined by the radius. But in the current context of the equation \(x^2 + y^2 = 0\), with the radius being zero, the domain is a single value, \([0,0]\) or simply \{0\}. This indicates that the only \(x\)-coordinate covered by this 'point-circle' is at the origin.Range, on the other hand, involves all the possible \(y\)-values involved. Again, under normal circle conditions, this would be dictated by the vertical stretch of the circle through the center. Here with our point at origin, the range is likewise \([0,0]\) or \{0\}. This reflects that there are no other vertical points covered besides the \(y = 0\). Understanding domain and range in this way shows exactly how the circle relates to the coordinate plane for any radius or center location.