Now that you know how to find the equation of a circle if you're given its graph, what about the other way around? Can you find the graph of a circle if you are given its equation? Let's return to the director's challenge of using geometry to block the main characters' positions on stage.
One of your classmates has determined that the following equation describes Jasbo's position and the projection of the spotlight around him:
\(\mathsf{(x+3)^2 + (y+4)^2 = 9}\)
Can you represent Jasbo and the spotlight encircling him using a graph drawn on the coordinate plane?
From earlier lessons, you may remember that to graph a circle, you need to know the coordinates of the circle's center as well as the measure of the circle's radius. If you know a circle's equation, then you already have the information you need for a graph. You can follow these steps to draw the circle.
| Step 1 | Using the general equation of a circle as a reference, identify the center coordinates (h,k). |
| Step 2 | Plot the center on the coordinate plane. |
| Step 3 | Plot at least four points that are r units from the center of the circle. |
| Step 4 | Connect the points that you have plotted, to make the shape of a circle. |
Generally, the more points you plot representing the circle's radius, the more accurate your sketch will be.
Question
What geometry tool can you use to make your circle even more accurate?
A compass
Try applying the steps above. Using a coordinate plane to represent the stage, graph a circle that shows the location of Jasbo and the spotlight encircling him on the stage, according to your classmate's equation:
\(\mathsf{(x+3)^2 + (y+4)^2 = 9}\)
Jasbo is the center of the circle. From the equation, you know that h = -3 and k =-4. The spotlight projected on Jasbo encircles him with a radius of 3.
