Question

Graph the equation: \(x^{2}+y^{2}=9\)

Short answer

Graph a circle centered at the origin with radius 3.

Step-by-step solution

Understand the Equation

The given equation is in the form of a circle's equation: \(x^{2} + y^{2} = r^{2}\). Here, \(r^{2} = 9\), so \(r = 3\). This represents a circle with radius 3 and center at the origin (0, 0).

Sketch the Axes

Draw the x-axis and y-axis on a graph. Mark the origin point (0,0) where the axes intersect.

Plot the Radius Points

From the origin (0,0), count 3 units out in all four directions (up, down, left, right) to find points on the circle. Plot the points (3,0), (-3,0), (0,3), and (0,-3).

Draw the Circle

Using the four plotted points as references, draw a smooth, round circle that passes through all of them. Ensure the circle is centered at the origin.

Label Key Points

Label the points where the circle intersects the x-axis and y-axis for clarity. These are (3,0), (-3,0), (0,3), and (0,-3).

Key concepts

equation of a circle

The equation of a circle in a plane is typically written as \(x^2 + y^2 = r^2\). Here, \(r\) represents the radius of the circle. This specific form indicates that the circle is centered at the origin (0, 0). For example, in the equation \(x^2 + y^2 = 9\), we can see that \(r^2 = 9\), so \(r = 3\). This means the circle has a radius of 3 units.
Circles can also be centered at points other than the origin. For instance, the general form \((x-h)^2 + (y-k)^2 = r^2\) describes a circle with center \((h, k)\) and radius \(r\). Understanding these forms is crucial as it allows us to identify and graph circles with various origins and radii.

coordinate geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate plane. It allows us to use algebra to understand geometric concepts. When graphing a circle, we use the coordinate plane to plot points that define the shape and size of the circle.
In a coordinate system, the horizontal axis is called the x-axis and the vertical axis is the y-axis. The point where these axes intersect is known as the origin (0,0).
Coordinate geometry helps in plotting various figures by using equations and coordinates. For our circle equation \(x^2 + y^2 = 9\), we interpret this as a circle around the origin with a radius of 3 units. Using the coordinates, we find and plot the key points that help in drawing the circle accurately on the plane.

graphing steps

Graphing the equation of a circle involves a few clear steps.
First, understand the given equation. In our example, \(x^2 + y^2 = 9\), we identify the radius \(r = 3\) and the center (0,0).
Next, sketch the x and y axes on a graph and mark the origin. This provides a reference frame for plotting the circle.
Then, count 3 units in all four main directions (up, down, left, right) from the origin to determine the circle's furthest points. These points are: (3,0), (-3,0), (0,3), and (0,-3).
Now, draw a smooth, round circle through these points. Make sure it is centered at the origin as dictated by the equation.
Finally, label the intersection points on the axes for clarity and verification. In this example: (3,0), (-3,0), (0,3), and (0,-3).
Following these steps ensures that you precisely graph the circle represented by the given equation.