Graphs and Equations of Circles in the Rectangular Coordinate System

Graphs, Functions, and Models

Equations and Graphs of Circles

The equation of a circle in the rectangular (Cartesian) coordinate system is a fundamental topic in Precalculus. Understanding how to write, manipulate, and graph these equations is essential for analyzing geometric figures algebraically.

• Standard Form of a Circle: The equation of a circle with center at (h, k) and radius r is given by:

$(x - h)^2 + (y - k)^2 = r^2$

• Center: The point (h, k) is the center of the circle.

• Radius: The value r is the radius of the circle.

• Graphing: To graph a circle, plot the center at (h, k) and draw all points that are exactly r units from the center.

Example 1: Identifying the Center and Radius

• Given the equation $(x + 3)^2 + (y - 2)^2 = 16$, identify the center and radius.

• Solution: Rewrite as $(x - (-3))^2 + (y - 2)^2 = 4^2$.

• Center: (-3, 2)

• Radius: 4

Example 2: Graphing a Circle

• Graph the circle $(x - 1)^2 + (y + 2)^2 = 9$.

• Center: (1, -2)

• Radius: 3

• Plot the center at (1, -2) and draw a circle with radius 3 units.

Converting General Form to Standard Form

Sometimes, the equation of a circle is given in general quadratic form:

$x^2 + y^2 + Dx + Ey + F = 0$

• To convert to standard form, complete the square for both x and y terms.

Example 3: Completing the Square

• Given $x^2 + y^2 - 4x + 6y - 12 = 0$, rewrite in standard form.

• Group variables: $(x^2 - 4x) + (y^2 + 6y) = 12$

• Complete the square:

  • $(x^2 - 4x + 4) + (y^2 + 6y + 9) = 12 + 4 + 9$

  • $(x - 2)^2 + (y + 3)^2 = 25$

• Center: (2, -3)

• Radius: 5

Summary Table: Forms of the Equation of a Circle

Form	Equation	How to Identify Center & Radius
Standard Form	$(x - h)^2 + (y - k)^2 = r^2$	Center: (h, k), Radius: r
General Form	$x^2 + y^2 + Dx + Ey + F = 0$	Complete the square to convert to standard form

Applications

• Circles are used in geometry, physics (e.g., describing orbits), engineering, and computer graphics.

• Understanding the equation allows for analysis of intersections, tangents, and other geometric properties.

Additional info: The notes also include example graphs and step-by-step solutions for finding the center and radius from given equations, as well as practice problems for students to reinforce their understanding.