Problem 4 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$(x+1)^{2}+(y+3)^{2}=25$$

Step-by-Step Solution

Answer

The circle with equation \((x+1)^2 + (y+3)^2 = 25\) has center \((-1, -3)\) and radius 5 units. To graph it, plot the center at \((-1, -3)\) on the coordinate plane and draw a circle with a radius of 5 units.

Step 1: Identify a, b, and r from the given equation

Given circle equation: \((x+1)^2 + (y+3)^2 = 25\). In this equation, we can observe that: \(a = -1,\) (since \((x + 1)²\) matches with \((x - a)²\)) \(b = -3,\) (since \((y + 3)²\) matches with \((y - b)²\)) \(r^2 = 25\), so \(r = 5\).

Step 2: Determine the center of the circle

Now that we know the values of a and b, we can determine the center of the circle. Remember, the center is at the point \((a, b)\): Center = \((-1, -3)\).

Step 3: Determine the radius of the circle

Since we found out that \(r = 5\), the radius of the circle is 5 units.

Step 4: Graph the circle

To graph the circle, follow these steps: 1. Plot the center \((-1, -3)\) on the coordinate plane. 2. From the center, draw a circle with a radius of 5 units. And now you have graphed the circle with the equation \((x + 1)^2 + (y + 3)^2 = 25\), which has a center at \((-1, -3)\) and a radius of 5 units.

Key Concepts

Center of a Circle

To find the center of a circle given its equation, you'll typically use the standard equation form: \((x - a)^2 + (y - b)^2 = r^2\). Here, \(a\) and \(b\) represent the x and y coordinates of the circle's center. They are often found by rewriting the given equation in this form and identifying the constants.
For example, if you're given \((x+1)^2 + (y+3)^2 = 25\), compare this with the standard form, \((x-a)^2 + (y-b)^2 = r^2\). You'll notice that:

• \(a\) is 1, but since the form is \((x - a)\), \(a\) should be negated to produce \(-1\).
• \(b\) is 3, but similar to \(a\), \(b\) is \(-3\) in the standard form placement.

This means the center of the circle in this example is \((-1, -3)\). By identifying these values accurately, you set the stage for correct graphing and understanding of the circle's position on a coordinate plane.

Radius of a Circle

The radius of a circle is determined by the value \(r\) in the standard circle equation \((x - a)^2 + (y - b)^2 = r^2\). The equation essentially squares the radius, so to find it, you take the square root of the number on the right side.
In our example, the equation \((x+1)^2 + (y+3)^2 = 25\), the figure 25 is actually \(r^2\). Therefore, to find \(r\), you simply calculate \(\sqrt{25}\), yielding \(r = 5\).
This means the circle has a radius of 5 units, which tells us how far the circle extends from its center point. Remember, the length of the radius is constant from the center to any point on the circle itself.

Graphing Circles

Graphing a circle starts by identifying its center and radius. Once you have these, you can accurately plot the circle on a coordinate grid. Here's a simple guide:
1. **Plot the Center:** Start by marking the center point of the circle on your graph. In our example, this is \((-1, -3)\). Make sure your graph is clearly marked with x and y axes for accuracy.
2. **Draw the Circle:** Use the radius to gauge how wide the circle will be. For a radius of 5 units, measure 5 units in every direction from the center point to create the circle's boundary.

• Go up, down, left, and right from the center, marking these points to help you sketch accurately.
• Ideally, use a compass for precision, or carefully draw by hand if you don't have one.

Graphing a circle in this manner helps visualize its location and size in relation to other figures on the coordinate plane. It also provides a clear geometric representation of the mathematical equation you started with.