Problem 45 - 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. $$ (x+3)^{2}+(y-2)^{2}=4 $$

Step-by-Step Solution

Answer

The center of this circle is (-3,2) and the radius is 2. The domain of this function is [-5,-1] and the range is [0,4].

Step 1: Identify Center and Radius

The given equation is \((x+3)^2 + (y-2)^2 = 4\). By comparing it to the general form, you can see that the center (h,k) of the circle is (-3,2) and the radius r is the square root of 4, which is 2.

Step 2: Plot The Graph

Since the center and radius have been identified, the circle can be plotted on a graph. Position the center point at (-3,2) and draw a circle with a radius of 2 around this point. Make sure your circle passes through the points (-3+2, 2), (-3-2, 2), (-3, 2+2), and (-3, 2-2) because these points are directly 2 units to the right, left, up and down from the center. You should end up with a circle that is symmetrical about the point (-3,2).

Step 3: Determine Domain and Range

Looking at the graph, the circle extends 2 units to left and right of -3 on the x-axis, and 2 units down and up from 2 on the y-axis. Therefore, the domain (values of x) is from -5 to -1 while the range (values of y) is from 0 to 4.

Key Concepts

Center of a Circle

In the world of geometry, the center of a circle is a fundamental concept. It's the fixed point from which every point on the circle is equidistant. When we look at the equation of a circle in its standard form \[(x-h)^2 + (y-k)^2 = r^2\]we find that \(h\) and \(k\) are the coordinates of the center. In our given equation, \((x+3)^2 + (y-2)^2 = 4\), we equate \((h,k)\) to \((-3,2)\). This tells us precisely where the center of our circle is located on the Cartesian plane.
Remember, identifying the center is key to graphing the circle and analyzing its properties. Once you know the center, graphing becomes about plotting around this fixed location.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. In our circle's equation \((x+3)^2 + (y-2)^2 = 4\), the radius can be identified as the square root of the constant on the right-hand side of the equation. This is because it represents \(r^2\). So, taking the square root of \(4\) gives us a radius \(r = 2\).
With this radius, we know the circle extends 2 units in every direction from the center. This distance is significant when plotting because it defines how large the circle is. Being sure of the radius helps in drawing a precise and accurate circle.

Graphing Circles

Graphing circles involves plotting them accurately on a coordinate plane using the center and radius. Begin by marking the center \((-3, 2)\) on your plane. From this point, you will measure outwards in all directions using the radius. For this specific circle, with a radius of 2, you'll mark points 2 units away: right at \((-1, 2)\), left at \((-5, 2)\), up at \((-3, 4)\), and down at \((-3, 0)\). These points are crucial because they allow the circle to be symmetrical around the center.
The circle should smoothly pass through these points in a round fashion. Remember, the circle should look even and smooth. Use these techniques to ensure your representation is geometrically sound.

Domain and Range

In mathematics, domain and range offer insights into the limits of a function or relation, in this case, a circle. The domain refers to all possible \(x\)-values a circle can have, while the range is the set of possible \(y\)-values. For the circle in our equation, the center is at \((-3, 2)\) with a radius of 2, so it stretches 2 units left to right, and 2 units up and down.
Thus, the domain of this circle stretches from \(x = -5\) to \(x = -1\). Similarly, it encompasses a range from \(y = 0\) to \(y = 4\). This is derived through the fact the circle covers 2 units in these directions. Understanding domain and range helps in situating the circle within the broader graph context and predicting its interaction with axes and other geometric figures.