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Chapter 7: Problem 148

Graph each circle. Identify the center and the radius. \((x-1)^{2}+(y+3)^{2}=16\)

Short Answer

Expert verified
Center: \((1, -3)\), Radius: \4\.

Step by step solution

01

Identify the standard form of a circle equation

Compare the given equation \((x - 1)^2 + (y + 3)^2 = 16\) with the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) is the center and \(r\) is the radius.
02

Determine the center of the circle

From the equation \((x-1)^2 + (y+3)^2 = 16\), identify \((h, k) = (1, -3)\). Thus, the center of the circle is \((1, -3)\).
03

Determine the radius of the circle

Compare the given radius term with \((r^2)\). Here, \(r^2 = 16\) which means \(r = \sqrt{16} = 4\). The radius of the circle is \4\.
04

Graph the circle

Plot the center of the circle at \((1, -3)\). Draw a circle with a radius of \4\ units around this center point.

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Key Concepts

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

standard form of a circle
Understanding the standard form of a circle's equation is crucial for identifying key features such as the center and the radius. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\).
This compact form tells us everything we need to know about the circle:
  • The terms \(h\) and \(k\) represent the x and y coordinates of the center of the circle, respectively.
  • The term \(r\) represents the radius of the circle, which is the distance from the center to any point on the circle.
If an equation is given in this form, it simplifies the process of graphing the circle and identifying its properties. Compare this with any provided circle equation to extract the necessary information.
center of a circle
To find the center of a circle, we need to look at the terms inside the squared parentheses in the standard form equation \((x - h)^2 + (y - k)^2 = r^2\).
The center of the circle is represented by the coordinates \( (h, k) \).
  • In the given equation \((x - 1)^2 + (y + 3)^2 = 16\), we compare and find that \( h = 1 \) and \( k = -3 \).
  • So, the center of the circle is at the point \((1, -3)\).
Always ensure to adjust for the signs; the standard form uses \((x - h)\) and \((y - k)\). A negative sign inside the parentheses indicates a positive coordinate value.
radius of a circle
Calculating the radius of a circle involves understanding the term \(r^2\) in the standard form equation \((x - h)^2 + (y - k)^2 = r^2\).
The radius \(r\) is simply the square root of the given constant term.
  • For the equation \((x - 1)^2 + (y + 3)^2 = 16\), we recognize that \(r^2 = 16\).
  • Taking the square root of both sides, we get \(r = \sqrt{16}\ = 4\).
So, the radius of this circle is 4 units. The radius is the measure from the center to any point on the circle's circumference.
graphing circles
Graphing a circle becomes straightforward once we identify the center and the radius. To plot the circle given by \((x - 1)^2 + (y + 3)^2 = 16\) on a graph:
  • Start by plotting the center at the point \( (1, -3) \) on the coordinate plane.
  • Next, use a compass or a consistent scale to draw a circle with a radius of 4 units around this center point.
The result is a perfect circle whose properties match the given equation. Always double-check the center and radius to avoid any graphing errors. This visual representation makes understanding circles in coordinate geometry much easier.

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