Problem 52 - 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. $$ (x+2)^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

The center of the circle is (-2,0), the radius of the circle is 4. The domain of this relationship is [-6, 2] and the range is [-4,4].

Step 1: Identify the Center

The circle's equation is in the form \((x-h)^2+(y-k)^2=r^2\), which means the center of the circle is (h,k). Comparing this with the given equation \((x+2)^{2}+y^{2}=16\), it can be inferred that the center of the circle is (-2,0).

Step 2: Calculate the Radius

The radius can be found from the same standard form. In this case, the radius squared \(r^2\) is equal to 16. Taking the square root of both sides, we find that the radius \(r\) is equal to 4.

Step 3: Identify the Domain and Range

From the equation and graph of the circle, the domain (all possible x-values) is from -2 - r to -2 + r, which is from -6 to 2. Similarly, the range (all possible y-values) is from 0 - r to 0 + r, which is from -4 to 4.

Key Concepts

Center of a Circle

In the realm of geometry, understanding the center of a circle is crucial. A circle's equation often appears in the standard form: \((x-h)^2 + (y-k)^2 = r^2\). Here, the values \(h\) and \(k\) represent the coordinates of the circle's center. For our given circle equation \((x+2)^2 + y^2 = 16\), we can re-organize it to easily identify the center. Comparing it with the standard form, we see that \(h = -2\) and \(k = 0\), which tells us the center of this circle is

• (-2, 0)

The center gives a circle its definite position on the coordinate plane. It acts as an anchor point, from which all the points of the circle are equidistant. This simple feature plays a vital role in drawing and interpreting circles.

Radius of a Circle

The radius is another essential component when dealing with circles. It is the distance from the center of the circle to any point on its circumference. Again, let’s refer to the standard form equation: \((x-h)^2 + (y-k)^2 = r^2\). The term \(r^2\) indicates the square of the radius.For our specific equation, \((x+2)^2 + y^2 = 16\), the radius squared is 16. To find the actual radius, take the square root of 16, giving us a radius of 4. Remember:

• The radius is always positive.
• It is consistent throughout the circle.

Understanding the radius is key for determining the size of the circle and helps in sketching its proper scale.

Domain and Range

The terms domain and range might sound complex, but they are simply the extents of the x-values and y-values, respectively, that a circle covers. Domain refers to all possible x-values within the circle, and range refers to all possible y-values.For the circle given by our equation \((x+2)^2 + y^2 = 16\) with its center at (-2, 0) and radius 4:- The domain is calculated by considering the farthest left and right x-values. - Center x-value: -2 - Reach: -4 and +4 based on radius - Thus, the domain is from -6 to 2.- The range is found by stretching the circle vertically from the center - Center y-value: 0 - Reach: -4 and +4 based on radius - Therefore, the range is from -4 to 4.

• Domain: [-6, 2]
• Range: [-4, 4]

This understanding helps you know the full extent of a circle's spread across the x and y axes.

Graphing Circles

Graphing a circle accurately involves plotting its path through all quadrants of the coordinate plane. The basic equation guide is: \((x-h)^2 + (y-k)^2 = r^2\) which translates graphically to a circle with a center (h, k) and radius \(r\).To graph - Start by marking the center of the circle, at (-2, 0).- Use the radius of 4 to determine the circle's boundary.

• From the center, count 4 units in all directions: up, down, left, and right
• Draw a smooth curve connecting these points to form the circle.

Graphing is a visual representation, making understanding geometric relationships easier. It helps solidify comprehension and is a handy tool to check your algebraic solutions.