Problem 43 - 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. $$(x-3)^{2}+(y-1)^{2}=36$$

Step-by-Step Solution

Answer

The center of the circle is at the point (3,1), with a radius of 6. The domain is [-3, 9] and the range is [-5, 7].

Step 1: Identify the Center

The center of the circle can be directly read off from the standard form of the equation, by identifying values of h and k from \((x - h)^2 + (y - k)^2 = r^2\). In the given equation, \((x-3)^{2}+(y-1)^{2}=36\), \(h = 3\) and \(k = 1\) . Hence, the center of the circle is at the point \((3,1)\).

Step 2: Identify the Radius

The radius of the circle can be found from the right side of the standard form equation, by taking the square root of \(r^2\). From the given equation, \(r^2\) is 36, hence the radius \(r\) is \(\sqrt{36}\) which simplifies to 6.

Step 3: Graph the Circle

Sketch the graph of the circle by starting at the center point \((3,1)\) and making a circle with radius 6. This is done by counting 6 units right, left, up, and down from the center and connecting those points to form a circle.

Step 4: Identify the Domain and Range

The domain (possible x-values) of the circle is the interval of all x-values hit by the circle, which in this case goes from the leftmost point of the circle to the rightmost. Since the center is at \(x = 3\) and the radius is 6, this interval ranges from \(3-6\) to \(3+6\), or \([-3, 9]\). Similarly, the range (possible y-values) goes from the bottom of the circle to the top. The center is at \(y = 1\) and radius is 6 so the range is from \(1-6\) to \(1+6\), or \([-5, 7]\).

Key Concepts

Center and Radius of a Circle

When you have the equation of a circle in the standard form \((x - h)^2 + (y - k)^2 = r^2\), the center and radius are easily determined. Here,

• \(h\) and \(k\) represent the x and y coordinates of the circle's center, which means from the equation, \((x-3)^{2}+(y-1)^{2}=36\), the center is at \((3, 1)\).
• \(r^2\) specifies the radius squared, so to get the radius \(r\), you just take the square root of 36, resulting in a radius of 6.

Recognizing the center and radius is crucial because it gives you a fixed point and distance needed to graph your circle.

Standard Form of a Circle Equation

The standard form of a circle equation is a special form that makes understanding circle characteristics straightforward. This form is \((x - h)^2 + (y - k)^2 = r^2\), where:

• \(h\) denotes the x-coordinate of the circle's center,
• \(k\) denotes the y-coordinate of the circle's center,
• and \(r\) is the radius of the circle.

The equation \((x-3)^{2}+(y-1)^{2}=36\) is already in standard form, making it easy to interpret the circle properties, like the center \((3, 1)\) and radius 6.Having the equation in this form is especially beneficial for graphing and further exploring the circle's properties.

Domain and Range of a Circle

The domain and range of a circle help us understand the set of x and y values that the circle covers. For the circle given by \((x-3)^{2}+(y-1)^{2}=36\):

• The **domain** consists of all x-values the circle covers. The center is at \(x = 3\) and the radius is 6. Thus, the domain extends from \((3-6) = -3\) to \((3+6) = 9\). In interval notation, this is \([-3, 9]\).
• The **range** represents all y-values covered by the circle. The center being at \(y = 1\) with a radius of 6 means the range goes from \((1-6) = -5\) to \((1+6) = 7\). This range is denoted by \([-5, 7]\).

Identifying the domain and range is crucial in knowing the extent of your circle on a graph.

Graphing a Circle

Graphing circles can be done systematically by understanding the equation of the circle. First, locate the center of the circle using the form \((x - h)^2 + (y - k)^2 = r^2\). For the equation \((x-3)^{2}+(y-1)^{2}=36\), the center is \((3,1)\). Here are the steps to graph it accurately:

• Place a point at the center \((3, 1)\).
• From this center, count 6 units in all directions (up, down, left, right) which represents the radius.
• Connect these points in a smooth round shape. That’s your circle!

Graphing provides a visual representation and helps you better grasp concepts like domain and range, as it shows the extent of the circle in a visual way.