Graphing Circles

To graph a circle, we use the standard form of the equation of a circle, which is given by:

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

where $(h, k)$ is the center of the circle and $r$ is the radius.

Example:

Graph the circle with the equation:

$$(x + 2)^{2} + (y - 3)^{2} = 16$$

Step 1: Identify the center and radius.

• The center $(h, k)$ is $(-2, 3)$.
• The radius $r$ is $\\sqrt{16} = 4$.

Step 2: Plot the center on the graph at $(-2, 3)$.

Step 3: From the center, use the radius to find points on the circle:

• Move 4 units up to $( -2, 7)$.
• Move 4 units down to $( -2, -1)$.
• Move 4 units left to $(-6, 3)$.
• Move 4 units right to $(0, 3)$.

Step 4: Draw the circle through these points.

Key Questions:

• What is the center of the circle given by the equation $(x - 1)^{2} + (y + 2)^{2} = 25$?
• How do you find the radius from the equation $(x + 5)^{2} + (y - 4)^{2} = 9$?
• If the center of the circle is at $(3, -1)$ and the radius is 5, what is the equation of the circle?
• How can you determine if a point lies inside, outside, or on the circle?

Keywords:

• Circle
• Radius
• Center
• Equation
• Graphing