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Chapter 2: Problem 27

Sketch the graph of the circle or semicircle. $$(x+3)^{2}+(y-2)^{2}=9$$

Short Answer

Expert verified
The graph is a circle centered at (-3, 2) with radius 3.

Step by step solution

01

Identify the Equation Type

The equation \((x+3)^{2}+(y-2)^{2}=9\) is in the standard form for a circle. The general form of a circle鈥檚 equation is \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h, k)\) is the center and \(r\) is the radius.
02

Determine the Center

From the equation \((x+3)^{2}+(y-2)^{2}=9\), identify \(h = -3\) and \(k = 2\). So the center of the circle is at \((-3, 2)\).
03

Calculate the Radius

The equation \((x+3)^{2}+(y-2)^{2}=9\) shows that \(r^{2} = 9\). Taking the square root, we find the radius \(r = 3\).
04

Sketch the Circle

To sketch the circle, plot the center at \((-3, 2)\) on a coordinate plane. Then, use the radius \(r = 3\) to draw a circle by placing points 3 units away from the center in all directions. Connect these points to form the circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Circle
The equation of a circle can be thought of as the blueprint for drawing a perfect round shape on a coordinate plane. It tells us everything we need about the circle: its position and size. In general, the equation of a circle is given in the format
  • \( (x-h)^2 + (y-k)^2 = r^2 \)
This form helps identify essential features directly:
  • \((h, k)\): The center of the circle.
  • \(r\): The radius of the circle.
By recognizing this format, anyone can quickly determine where a circle sits on the coordinate plane and how big it is. In our example, the equation is \((x+3)^2 + (y-2)^2 = 9\), and from looking at it, we can pick out the center and radius immediately.
Radius of a Circle
The radius is one of the most critical features of a circle. It's the constant distance from the center to any point on the edge of the circle. In the equation of a circle
  • \((x-h)^2 + (y-k)^2 = r^2\)
the \(r^2\) term represents the square of the radius. To find the radius, simply take the square root of this number. For the equation \((x+3)^2 + (y-2)^2 = 9\), we find that the square of the radius \(r^2\) is 9. By taking the square root of 9, we get the radius \(r = 3\). The radius helps us understand how far the circle's edge stretches from its center.
Circle Graph Sketching
Sketching the graph of a circle involves a few reliable steps that make it easier. Let's break down the process:First, locate the center of the circle on your graph by finding the point \((h, k)\) from the equation. In our exercise, this point is \((-3, 2)\).Next, use the radius to draw the circle. For our example, the radius \(r\) is 3. Place a mark 3 units away from the center in all cardinal directions (up, down, left, right). Connect these marks with a smooth curve, making sure all parts of the circle are the same distance from the center. If done correctly, these steps will give a perfect circle on your graph!
Standard Form of a Circle
The standard form of a circle's equation is an incredibly helpful tool that simplifies understanding and working with circles. It appears as
  • \((x-h)^2 + (y-k)^2 = r^2\)
and contains all necessary information in an organized way. From this form, it's straightforward to identify both the center and the radius at a glance. Here's how it breaks down:
  • \(h\) and \(k\) are shifts from the origin, locating the circle's center.
  • \(r^2\) represents the square of the radius of the circle.
Using this form makes sketching and graphing circles more manageable because you do not need to mess around with more complex algebra or guesswork. Knowing the center and radius directly serves as a roadmap to drawing and understanding circles effectively.

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Most popular questions from this chapter

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