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Chapter 7: Problem 150

Graph each circle. Identify the center and the radius. \((x+4)^{2}+(y+1)^{2}=25\)

Short Answer

Expert verified
Center: (-4, -1). Radius: 5.

Step by step solution

01

- Identify the equation form

The given equation \( (x+4)^{2}+(y+1)^{2}=25 \) is in the standard form of a circle equation \( (x-h)^2 + (y-k)^2 = r^2 \).
02

- Determine the center

Compare the given equation \( (x+4)^{2}+(y+1)^{2}=25 \) with the standard form. We see that \( h = -4 \) and \( k = -1 \). So, the center of the circle is \( (-4, -1) \).
03

- Find the radius

The radius \( r \) is given by the square root of the number on the right side of the equation. Here, \( r^2 = 25 \), thus \( r = \sqrt{25} = 5 \).
04

- Graph the circle

Plot the center of the circle at \( (-4, -1) \). Then, use the radius of 5 units to draw a circle around this center.

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Key Concepts

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

Graphing Circles
Graphing a circle might seem tricky, but with the right steps, it becomes simple. A circle's equation in standard form looks like this: \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h, k\) are the coordinates of the center and \(r\) is the radius. To graph a circle:

\1) Identify the center and radius from the equation.
\2) Plot the center point on a coordinate plane.
\3) Measure out the radius in all directions from the center and mark the points.
\4) Draw a smooth curve through all the points to complete the circle's outline.

Remember, the circle should be perfectly round, and all points on the circle are equidistant from the center.
Circle Center
The circle's center is a crucial part of its graph. It is the point from which all distances (radius) are measured. Let's use the equation \((x+4)^{2}+(y+1)^{2}=25\) as an example.

To find the center:
\1) Compare the given equation with the standard form \((x-h)^2 + (y-k)^2 = r^2\). Notice how \(h\) and \(k\) change the signs when plugging in the center.
\2) For the equation \((x+4)^{2}+(y+1)^{2}=25\), center is \((-4, -1)\).

The center tells us exactly where the middle of the circle sits on the graph.
Circle Radius
The radius is the distance from the center of the circle to any point on its edge. It helps us know how 'big' the circle is. From the equation \((x+4)^{2}+(y+1)^{2}=25\), we derive the radius like this:

\1) Identify the number on the right side of the equation, which is \25\.
\2) Take the square root of this number to find the radius: \sqrt{25} = 5\.

The radius can also tell us more:
  • It's always positive.
  • It is consistent for every point on the circle.
  • Helps in plotting the exact shape by marking points at a uniform distance from the center.

    So, in our example, the radius is 5 units.

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