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Chapter 12: Problem 27

$$\text {Sketch the graph of each equation.}$$ $$x^{2}+(y-3)^{2}=9$$

Short Answer

Expert verified
The center is (0, 3) and the radius is 3.

Step by step solution

01

Identify the Equation Type

Recognize that the given equation, \(x^2 + (y-3)^2 = 9\), is in the standard form of a circle equation.
02

Determine the Center of the Circle

The general form of a circle is given by \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h=0\) and \(k=3\), so the center of the circle is at (0,3).
03

Calculate the Radius

In the equation \(x^2 + (y-3)^2 = 9\), the value on the right side of the equation is \(r^2\). Thus, \(r^2 = 9\) and \(r = 3\). Therefore, the radius of the circle is 3.
04

Plot the Center of the Circle

On a coordinate plane, plot the center of the circle at the point (0,3).
05

Draw the Circle

Using the center (0,3) and the radius 3, draw a circle that extends from the center 3 units in all directions.

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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
The equation provided, \(x^2 + (y - 3)^2 = 9\), is an example of the standard form of a circle. The standard form of a circle's equation is written as \((x - h)^2 + (y - k)^2 = r^2\). This format makes it easy to identify the circle's center and radius. Here, \(h\) and \(k\) represent the center of the circle, and \(r\) is the radius. By comparing the given equation with the standard form, we can quickly identify the circle's key characteristics.
Center of a Circle
To find the center of a circle using its equation, we look for the values of \(h\) and \(k\) in \((x - h)^2 + (y - k)^2 = r^2\). In the equation \(x^2 + (y - 3)^2 = 9\), we see that \(h = 0\) and \(k = 3\). This tells us that the center of the circle is located at the point (0, 3). The center is a crucial point, as it helps in identifying the location of the circle on a coordinate plane.
Radius of a Circle
The radius of a circle is the distance from the center to any point on the circle's edge. It is calculated using the equation \((x - h)^2 + (y - k)^2 = r^2\). Here, \(r^2\) is the term on the right side of the equation. In our example, \(r^2 = 9\), which means \(r = 3\). Thus, the radius of our circle is 3 units. The radius helps in drawing the circle accurately, as it defines how far the circle extends from its center.
Plotting Points
Plotting points is a fundamental skill in graphing equations. To sketch the circle described by \(x^2 + (y - 3)^2 = 9\), follow these steps:
  • First, plot the center point (0, 3) on the coordinate plane.
  • Next, use the radius of 3 units to mark points that are 3 units away from the center in all directions: up, down, left, and right.
  • These points will include (0, 6), (0, 0), (3, 3), and (-3, 3).
  • Finally, connect these points with a smooth curve to form the circle.
Plotting points helps you to visualize the circle and ensures accuracy when sketching.

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Most popular questions from this chapter

Graph each hyperbola and write the equations of its asymptotes. $$ \frac{y^{2}}{4}-\frac{x^{2}}{25}=1 $$

Solve each problem. Find all points of intersection of the parabolas \(y=x^{2}\) and \(y=(x-3)^{2}\)

Sketch the graph of each ellipse. $$ x^{2}+\frac{y^{2}}{4}=1 $$

Graph both equations of each system on the same coordinate axes. Use elimination of variables to find all points of intersection. $$ \begin{aligned} &\frac{x^{2}}{4}+\frac{y^{2}}{16}=1\\\ &x^{2}+y^{2}=1 \end{aligned} $$

Graph both equations of each system on the same coordinate axes. Use elimination of variables to find all points of intersection. $$ \begin{aligned} &x^{2}+y^{2}=4\\\ &x^{2}-y^{2}=1 \end{aligned} $$
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