Problem 28 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The center of the circle is at point (-4,4) and the radius is 2. Plot these on a graph to draw the circle.

Step 1: Identify Center and Radius

The equation given is \((x+4)^{2} +(y-4)^{2}=4\). Comparing to the standard form, it can be seen that \( h = -4 \), \( k = 4 \), and \( r = \sqrt{4} = 2 \). So, the center of the circle is (-4, 4) and the radius is 2.

Step 2: Sketch the Center

Draw a pair of perpendicular axes on a graph. Plot the center of circle, which is at point (-4, 4). The center of the circle will be the reference point for drawing the circle.

Step 3: Plot the Radius

From the center of the circle, measure 2 units in all directions which is the radius of the circle. Plot these points. These points will help in drawing an accurate circle.

Step 4: Draw the Circle

Join all the points plotted in the previous step in a round shape which forms the circle. The points should be an equal distance from the center and this distance is equivalent to the radius of the circle.

Key Concepts

Circle Equation

Understanding circle equations is key to graphing and analyzing circles in mathematics. A circle equation expresses the relationship between the points on a circle in terms of their coordinates.

When you encounter a circle equation like \((x+4)^2 + (y-4)^2 = 4\), it tells you about all the points \((x, y)\) that lie on the circle with a specific center and radius.

• The expression \((x+4)^2\) indicates a horizontal shift from the origin, while \((y-4)^2\) indicates a vertical shift.
• The right side of the equation, \(4\), represents the square of the radius of the circle.

Knowing how to equate these elements to the general form helps in easily determining the circle’s position and size on a graph.

Center and Radius

The center and radius are essential components in defining a circle's shape and size. To extract these from an equation, use the standard form of the circle equation.

Here, the equation \((x+4)^2 + (y-4)^2 = 4\) leads you to identify:

• Center: Represented by the coordinates \((h, k)\), in this case (-4, 4). This means the circle is centered 4 units left and 4 units up from the origin.
• Radius: Found by taking the square root of the radius squared (\(r^2\)). With \(r^2 = 4\), the radius \(r = 2\).

These pieces of information are crucial for sketching the circle on a coordinate plane. Position the center first, then use the radius as a measure to complete the circle.

Standard Form of Circle Equation

The standard form of a circle equation is fundamental to understanding and utilizing circle equations in graphing.

The general format is \((x-h)^2 + (y-k)^2 = r^2\), where:

• \((h, k)\) is the center of a circle.
• \(r\) is the radius.

This format not only provides a clear picture of the circle's dimensions but also simplifies many calculations.

By comparing a given equation to the standard form, such as \((x+4)^2 + (y-4)^2 = 4\):

• Identify \(h = -4\) and \(k = 4\) for the circle's center.
• Calculate the radius \(r = \sqrt{4} = 2\).

Using this form ensures that the graphical representation of a circle is accurate and straightforward.