Problem 42 - 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. $$x^{2}+y^{2}=49$$

Step-by-Step Solution

Answer

The center of the circle is at the origin (0,0) and the radius is 7. The domain of the relation, set by the x-coordinates covered by the circle, is [-7, 7]. The range, set by the y-coordinates covered by the circle, is also [-7, 7].

Step 1: Identifying Circle's Center and Radius from Equation

Comparing the given equation \(x^{2}+y^{2}=49\) with the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), it can be seen that \(h=0\), \(k=0\) and \(r = \sqrt{49} = 7\). Hence, the circle's center is at the origin (0,0) and radius is 7.

Step 2: Graphing the Circle

Start by placing the center at point (0,0). Then, draw a circle around the center with radius 7, reaching points (7,0), (-7,0), (0,7) and (0,-7).

Step 3: Identifying Domain and Range from Graph

Looking at the graph, the relation includes all x-values between -7 and 7, inclusive. This is our domain. Similarly, this relation includes all y-values between -7 and 7, inclusive. This is our range.

Key Concepts

Domain and Range

Understanding the domain and range of a circle equation is important for solving and graphing problems. The domain of a circle refers to all the possible values that x can take. Similarly, range refers to all the possible values that y can take.
In the equation of a circle, we often see an expression like \(x^2+y^2=r^2\). Given this equation, we determine the domain and range by examining how far the circle extends horizontally and vertically.
For the given equation \(x^2 + y^2 = 49\), the circle is centered at the origin (0,0) with a radius of 7.

• Domain: The x-values range from -7 to 7, as the radius stretches both ways from the center.
• Range: The y-values also range from -7 to 7. The circle is symmetric along the x and y axes.

Graphing Circles

Graphing a circle requires identifying its center and radius from its equation. Let's use the equation \(x^2 + y^2 = 49\) as an example to showcase this process.
First, determine the circle's center, which would be point (0,0) since both x and y terms are not shifted (they have no \(h\) or \(k\) in their expression).
Next, the circle's radius is the square root of 49, which is 7. With these two pieces of information, we can draw the circle:

• Place a point at the center (0,0).
• Measure out 7 units in all four cardinal directions – up, down, left, and right.
• Connect these points with a smooth, round line to illustrate the circle.

This graphing process helps visualize and validate the domain and range as points are included within the radius.

Standard Form of a Circle

The standard form of a circle's equation provides a clear way to understand its geometric properties. This form is \((x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) corresponds to the circle's center, and \(r\) is the radius.
Now, consider our example equation \(x^2+y^2=49\). Notice there are no \(h\) or \(k\) values in this formula, implying a center at the origin \(h=0\) and \(k=0\). The square of the radius, \(r^2\), equals 49 (given by the constant on the equation's right side).
Solving for the radius means taking the square root of 49, resulting in \(r = 7\).

• This form helps quickly identify meaningful properties such as center and radius.
• It also facilitates conversions to other forms, like general form when multiplying out the expressions.

Center and Radius of a Circle

The center and radius are essential parts of understanding and analyzing circles. In the given equation \(x^2 + y^2 = 49\), these values allow you to reconstruct and graph the circle.
To find them, compare the circle equation to the standard form \((x-h)^2 + (y-k)^2 = r^2\).
With the formula above and given that \(x^2 + y^2 = 49\), the absence of \(h\) and \(k\) terms means the center is at the origin (0,0). The term \(r^2=49\) gives a radius of 7:

• The center (0,0) shows how the circle is positioned on the graph with equal extension in all directions.
• The radius of 7 means the circle stretches 7 units from the center, a key detail in graphing.

Understanding these helps ensure accuracy in both placing the circle graphically and determining details like domain and range.