Problem 34 - Algebra 2

Question

Use the center and the radius to graph each circle. $$ x^{2}+(y+4)^{2}=144 $$

Step-by-Step Solution

Answer

The center of the circle is at \((0,-4)\) and the radius is \(12\) units.

Step 1: Identify the Center

The given equation of the circle is \(x^{2}+(y+4)^{2}=144\), which is in standard form \((x-h)^{2}+(y-k)^{2}=r^{2}\), where the center of the circle is \((h,k)\). Here, \(h\) is \(0\) and \(k\) is \(-4\), so the center is \((0,-4)\).

Step 2: Identify the Radius

The radius \(r\) of the circle is the square root of the right side of the equation. In this case, \(\sqrt{144} = 12\). So, the radius of the circle is \(12\).

Step 3: Graph the Circle

Now plot the center of the circle \((0,-4)\) on the graph. From the center, measure out the radius in all directions by \(12\) units. Draw the circle by connecting these points. This will create a circle centered at \((0,-4)\), with a radius of \(12\) units.

Key Concepts

Circle Graphing

Graphing a circle on a coordinate plane is an enjoyable way to visualize the shape from its algebraic equation. To get started, you need to know both the center and the radius of the circle. These two bits of information will help guide where the circle sits and how big it is.

• Plot the Center: Locate the center of the circle on your graph. For example, if the center is \(0, -4\), find this point on the grid.
• Measure the Radius: Use the radius measurement to determine how far the circle extends from the center point in every direction.
• Draw the Circle: With a compass or by hand, sketch the circle by maintaining a constant distance from the center equal to the radius.

The result should be a perfect circle centered precisely at the point marked, demonstrating how algebra geometry interplay to represent shapes. Adjust your graphing technique for accuracy, extending the radius equally across the four quadrants of the plane.

Standard Form of a Circle

The standard form of a circle's equation is crucial for easily deciphering a circle's geometric attributes. This form is written as \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \(h, k\) represents the coordinates of the center of the circle, and \(r\) is the circle's radius.
Let's consider the equation \(x^{2}+(y+4)^{2}=144\). Notice how it reflects the standard form structure:

• Here, \(h = 0\) and \(k = -4\), indicating that the circle's center is at the point \(0, -4\).
• The expression on the right side, \((y+4)^{2}\), accounts for the vertical shift, effectively translating the circle downward by four units.
• The right side of the equation equals \(r^{2}\), allowing you to calculate the radius by taking the square root of this number.

Using this standardized equation form simplifies identifying and managing key circle traits, ensuring accurate graphing and interpretation.

Radius and Center of a Circle

The radius and center are fundamental components of a circle, acting as the blueprint for both its location and size.

• Center Coordinates: These tell you where the circle is placed on the graph. Expressed as \(h, k\) in the standard form equation, they provide the precise midpoint around which the circle expands. For instance, \(0, -4\) shows this circle's position.
• Radius Length: The radius defines the size of the circle, indicating how far its edge reaches from the center in every direction. In our example, the radius is \sqrt{144} = 12\, suggesting a consistent extension in all dimensions.

Understanding and calculating these values equips you with the tools needed to effectively plot any circle, appreciate its proportions, and engage deeply with the mathematical representation.