Problem 6 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$x^{2}+(y-5)^{2}=9$$

Step-by-Step Solution

Answer

The center of the circle is (0, 5) and its radius is 3. To graph the circle, plot the center and draw a circle with radius 3 tangent to the points (3, 5), (-3, 5), (0, 8), and (0, 2).

Step 1: Identify the Center of the Circle

The given equation is \(x^{2}+(y-5)^{2}=9\). By comparing it to the general equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), we can observe that \(h=0\) and \(k=5\). Thus, the center of the circle (h, k) is (0, 5).

Step 2: Identify the Radius of the Circle

From the general equation, we know that \(r^{2}\) is the constant on the right side of the equation. In this case, the given equation has the constant 9, so we have \(r^{2}=9\). To find the radius (r), we take the square root of both sides: \[r=\sqrt{9}\] \[r=3\]

Step 3: Graph the Circle

To graph the circle, start by plotting the center of the circle (0, 5) on the coordinate plane. Then, use the radius (3) to draw the circle around the center. The circle will be tangent to the points (3, 5), (-3, 5), (0, 8), and (0, 2) which are all at a distance of 3 from the center.

Key Concepts

Understanding the Center of a Circle

In the equation of a circle, \(x-h\)^2 + \(y-k\)^2 = r^2, the terms \(h\) and \(k\) represent the x and y coordinates of the center of the circle. \(h\) and \(k\) shift the circle along the x-axis and y-axis, respectively.

By analyzing the given equation \(x^2 + (y-5)^2 = 9\), we can see it's missing some adjustment on the x-term. This means the center isn't moved horizontally, so \(h = 0\).
Meanwhile, \(y\) is compared to 5, indicating the center's position is 5 units up along the y-axis, thus \(k = 5\).

• The center \((h, k)\) is \((0, 5)\).
• This tells us where to start drawing our circle on a graph or coordinate grid.

Understanding these coordinates helps you quickly locate the starting point for any circular graph.

Calculating the Radius of a Circle

The radius is the distance from the center of the circle to any point on its edge. Recognizing this makes it easier to comprehend the circle's size.

In the circle's equation \(x-h\)^2 + \(y-k\)^2 = r^2, the \(r^2\) on the right gives us insight into the radius. The general methodology is straightforward:

• Observe the equation \(x^2 + (y-5)^2 = 9\). The value 9 is equal to \(r^2\).
• Solve for \(r\) by taking the square root of 9, resulting in \(r = \sqrt{9} = 3\).

The circle's radius can thus be visualized or measured out from the center (0, 5) to three units in any direction, making sure we have a perfect round shape. These calculations are essential to graphing, as they outline the boundary of the circle.

Graphing Circles

Graphing a circle involves plotting its center and reaching out the appropriate distance determined by its radius. Here's a step-by-step approach to graphing:

1. **Plot the Center:**
Begin with the center, \(0, 5\). Set this on your coordinate plane. This point is crucial because it dictates the circle's position entirely.

• Use either graph paper or a digital graphing tool to mark it clearly.

2. **Use the Radius:**
From the center, count or measure three units outward in all four cardinal directions (left, right, up, down). These points lie on the circle's circumference.

• The points like \(3, 5\), \(-3, 5\), \(0, 8\), and \(0, 2\) are clear examples of this extension.

3. **Draw the Circle:**
Connect these outer points smoothly with a round line, ensuring the center remains equidistant from all edges.

• Confirm that each point measures precisely three units to guarantee accuracy.

By understanding these steps, you'll find drawing circles on a graph to be a much simpler task, which translates into quick and handy visuals for math problems or geometric projects.