91Ó°ÊÓ

When discussing the domain and range of a circle on a graph, we are essentially talking about the smallest and largest values that the circle reaches along the x-axis and y-axis. The **domain** refers to all the possible x-values that are part of the circle. For our circle, the domain is from

• -4 to 8
• Represented as

\([-4, 8]\). This range of x-values occurs because the circle extends 6 units from the center
(2, 0) to the left and 6 units to the right.
The **range** is similar but pertains to the y-values. It tells us the vertical span of the circle on the graph. The range extends from the lowest y-value to the highest:

• -6 to 6
• Expressed as

\([-6, 6]\). This results because the circle stretches 6 units above the center and also 6 below it.In recap, to determine the circle's domain and range,

• Look at how far the circle reaches horizontally and vertically.
• The domain is about the x-axis,
• The range about the y-axis.

Breaking these down into intervals makes it easier to understand and see the limits of the circle's reach on the plane!

Circle equations typically come in the form of

• \((x-h)^2 + (y-k)^2 = r^2\), where:
• \((h, k)\) is the center of the circle.
• \(r\) is the radius.

These variables tell us exactly where the circle sits on a coordinate plane and how large it is.
For example, the equation \((x-2)^2 + y^2 = 36\) lets us know directly that the center of our circle is at point (2, 0). Why? The

• -x term,
• \((x-2)\), instructs us that the circle is shifted 2 units to the right, making the center's x-value 2.
• There is no shift in the y-direction since it's just
• \(y^2\). So, the y-coordinate of the center is 0.

The number on the right of the equation, 36, tells us about the radius because it is equal to the square of the radius. In this case,

• \(r^2 = 36\),
• thus \(r = \sqrt{36} = 6\).

With these pieces of information, we can entirely describe a circle on a graph. Understanding the circle's equation is crucial for graphing it or solving related math problems.

Graphing a circle is straightforward once you've deciphered its equation. Here’s an easy method to follow:

• Plot the center first. For our circle, that’s (2, 0).
• Next, use the radius to mark key points around the center.

The radius is 6, meaning from the center, mark a point 6 units to the right, left, up, and down.

• To the right: (8, 0)
• To the left: (-4, 0)
• Upwards: (2, 6)
• Downwards: (2, -6)

Once these key points are plotted, you can sketch the circle by connecting these points smoothly.
You don't need a compass, but visualizing a smooth, round curve is essential. Some additional tips:

• Ensure your graph's scale is consistent,
• so your circle appears perfectly round.
• Check your key points by ensuring they align with the radius and center.

By following these steps, you can confidently graph any circle given its equation. Graphing helps enhance understanding by providing a visual representation of mathematical concepts.