Problem 52 - College Algebra

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. $$(x+2)^{2}+y^{2}=16$$

Step-by-Step Solution

Answer

The circle's center is (-2,0) and its radius is 4. The domain is [-6, 2] and the range is [-4, 4].

Step 1: Identify the Center and Radius

Examine the equation \((x+2)^{2}+y^{2}=16\). In this case, h corresponds to -2 (because of the required format \(x-h\)), and k is 0 (because there's no y - k in the equation). The radius can be given by \( \sqrt{16} \), which results in 4.

Step 2: Create the Graph

Sketch a circle on the graph with center at (-2,0) and a radius of 4.

Step 3: Identifying Domain and Range from the Graph

From the graph, it can be observed that the x-values (domain) range from -6 to 2 and the y-values (range) range from -4 to 4.

Key Concepts

Center of a Circle

Understanding the center of a circle is key to mastering equations that represent circles. In the equation \(x+2)^2+y^2=16\), the center is easily identifiable through its standard form. Here, we can see that the equation is an altered version of the standard circle equation \(x-h)^2+(y-k)^2=r^2\), where \(h\) and \(k\) represent the center coordinates of the circle on a Cartesian plane.

For our example, the \(h\) value is derived by looking at \(x+2\); since the standard form involves \(x-h\), when we see \(x+2\), it implies that \(h=-2\). There is no \(y-k\) term in our equation, which means \(k=0\). Thus, the center of our circle is at the coordinates \( (-2, 0) \). This concept allows us to locate any circle within the coordinate system.

Radius of a Circle

The radius of a circle is another fundamental concept, representing the distance from the center of the circle to any point on its circumference. In our exercise, \( (x+2)^2+y^2=16 \), the radius can be determined from the constant term on the right-hand side of the equation. In this standard form of a circle's equation, that constant is the square of the radius \(r\).

Therefore, to find the actual length of the radius, we calculate the square root of 16, which is 4. Our circle has a radius of 4 units. The radius helps to understand the size and shape of the circle and is essential when it comes to drawing and measuring the circle on a graph.

Domain and Range

The domain and range of a function or relation are concepts that reflect the set of possible input values (domain) and the corresponding set of output values (range). In the context of a circle, these terms describe the span of x-values and y-values that the circle covers on a graph.

For the given circle equation, the domain is the set of all possible x-values for which the circle has points. This is determined by looking at the horizontal extension of the circle from the center, considering the radius. The range similarly defines the vertical extent. With our center at \( (-2, 0) \) and radius of 4, the domain extends 4 units to the left and right of -2, resulting in a domain from \( -6 \) to \( 2 \) and a range from \( -4 \) to \( 4 \) since the circle extends 4 units above and below the center. These values help graph the circle accurately and understand the space it occupies.

Graphing Circles

Graphing circles requires a clear understanding of the concepts of center and radius, which provide the necessary information to plot the circle accurately. To graph the circle given by \( (x+2)^2+y^2=16 \), we start by plotting the center at \( (-2, 0) \).

We then use the radius of 4 units to measure outward in all directions from the center to mark points on the circle. After plotting several points at this distance from the center, a round shape begins to form. Carefully drawing a smooth curve through these points will result in the graph of the circle. Making sure the circle is even and centered at the right point is crucial for an accurate representation. This visual depiction of the equation not only aids in understanding the relation's domain and range but also solidifies the concepts of center and radius as they relate to the physical shape of the circle.