Problem 47 - College Algebra Essentials

Question

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+2)^{2}+(y+2)^{2}=4 $$

Step-by-Step Solution

Answer

The circle described by the equation \( (x+2)^{2}+(y+2)^{2}=4 \) has a center at (-2, -2) and a radius of 2. The domain and range of this circle are both \(-4 <= x,y <= 0\).

Step 1: Identify the Center and Radius

In the equation \( (x+2)^{2}+(y+2)^{2}=4 \), it can be seen that this represents a circle with its center at (-2, -2) and a radius of 2 (given by the root of 4). This matches the form \( (x - h)^2 + (y - k)^2 = r^2 \) where h = -2, k = -2 and r = 2.

Step 2: Graphing the Equation

Plot the center at point (-2, -2) and draw a circle with a radius of 2 units. Make sure to label the center and radius on the graph.

Step 3: Determining the Domain and Range

The domain of this circle is the set of all x-values that the circle covers. Since the center of the circle is at x = -2 and the radius is 2, the circle reaches out 2 units in either direction along x-axis. Therefore, the domain is \(-2 - 2 <= x <= -2 + 2\) or \(-4 <= x <= 0\).\n\nSimilarly, the range covers all y-values that the circle ranges over. The center of the circle is at y = -2 and reaches out 2 radius units in either direction along the y-axis. So the range is \(-2 - 2 <= y <= -2 + 2\) or \(-4 <= y <= 0\).

Key Concepts

Center and Radius of a Circle

When we talk about the geometry of circles, two of the most essential attributes we need to identify are the center and the radius. The center of a circle is the single point from which all points on the circle are equidistant, essentially the 'middle' of the circle. The radius is the distance from the center of the circle to any point on the circumference of the circle.

In the given equation, we see it in the form of \((x+2)^{2}+(y+2)^{2}=4\). By comparing this with the standard form of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\), we can easily identify the center at \((-2, -2)\) along with the radius of 2 units (since the square root of 4 is 2). It's crucial for students to be comfortable with translating between this algebraic expression and the visual representation of a circle's properties.

Domain and Range of a Relation

The concepts of domain and range are fundamental in understanding how relations are graphically represented. The domain of a relation consists of all the possible x-values, or input values, while the range represents all possible y-values, or output values. When graphing the domain and range, we essentially look for the extents to which the graph spreads horizontally and vertically, respectively.

Referring to our circle's equation, the domain is determined by subtracting and adding the radius to the x-coordinate of the center. This gives us the interval \([-4, 0]\), meaning that the circle covers all x-values within that range. Similarly, the range is found by applying the same logic to the y-coordinate of the center, resulting in \([-4, 0]\). Understanding these concepts is vital for students to grasp the breadth of any equation's representation on a graph.

Plotting Equations on a Graph

Plotting an equation on a graph visually conveys the relation defined by the equation. This graph can show us not just the shape but also the location and size of geometric figures like circles. With our equation \((x+2)^{2}+(y+2)^{2}=4\), plotting the circle involves drawing a point at the center \((-2, -2)\) and then using a compass or a ruler to draw a circle with a radius that extends 2 units from this center in all directions.

To provide additional clarity while plotting, it can be helpful to also mark the domain and range along the axes. This visual aid supports the student's understanding of how the algebraic expression translates into a graphical one. Remember, accuracy in plotting ensures that the subsequent analysis of domain and range is correct.

Standard Form of a Circle's Equation

A circle's equation in standard form is a concise and specific way to convey all of the necessary information about a circle's graph. It is usually expressed as\((x - h)^2 + (y - k)^2 = r^2\). Here, \(h\) and \(k\) represent the x and y coordinates of the center of the circle, respectively, and \(r\) stands for the radius.

Understanding this form is crucial because it allows us to find the center and radius quickly, which are instrumental in graphing the circle and determining its domain and range. The equation provided in the exercise is a variation of this standard form, adjusted for the circle's particular center and radius, and shows the utility of having a standard form to simplify the graphing and analysis of circles.