1. CCGPS GEOMETRY UNIT QUESTION: How are the equations of circles and parabolas derived? Standard: MCC9-12..A.REI.7, G.GPE.1,2 and 4 Today’s Question: How is the equation of a circle derived? Standard: MCC9-12..G.GPE.1 UNIT QUESTION: How are the equations of circles and parabolas derived? Standard: MCC9-12..A.REI.7, G.GPE.1,2 and 4 Today’s Question: How is the equation of a circle derived? Standard: MCC9-12..G.GPE.1

2. OPENER

3. EOCT PRACTICE QUESTION

4. LET’S GRAPH AN EQUATION!

5. xY(1)Y(2) 0 6 8 10 -6 -8 -10 3 -3 Find the y-values associated with each of the listed x-values and plot them. Try x=12 and see what solutions you get for y! After plotting all the points, connect each point with the point nearest it on each side in “best fit” fashion.

6. Now that we have connected our points, we can answer some questions: What shape have we plotted? What are the coordinates of its center? What is its radius?

7. TRY ANOTHER!!!!

8. xY(1)Y(2) 2 8 10 12 -4 -6 -8 Find the y-values associated with each of the listed x-values and plot them. Why did we choose x=2 to start with? After plotting all the points, connect each point with the point nearest it on each side in “best fit” fashion.

9. What are the coordinates of this circle’s center? What is its radius? How does it differ from our first circle?

10. LAST ONE!!!!

11. xY(1)Y(2) 2 5 6 7 -2 -3 Find the y-values associated with each of the listed x-values and plot them. After plotting all the points, connect each point with the point nearest it on each side in “best fit” fashion.

12. What are the coordinates of this circle’s center? What is its radius? How does it differ from the other circles?

13. WHAT ARE YOUR CONJECTURES??? How did the radius of our circle change when we changed the right side of our equation from 100 to 25? What is the relationship between the radius and the sum of the squared terms (the right side of our equation)? Can you come up with a generalized equation for a circle using the x and y coordinates of the circle’s center and the radius of the circle? What does this equation look like? How did the radius of our circle change when we changed the right side of our equation from 100 to 25? What is the relationship between the radius and the sum of the squared terms (the right side of our equation)? Can you come up with a generalized equation for a circle using the x and y coordinates of the circle’s center and the radius of the circle? What does this equation look like?

14. LET’S MATCH CIRCLES TO THEIR EQUATIONS! Use your newly derived formula to match the circles around the room to their equations.

15. EQUATIONS OF CIRCLES

16. Standard Form of a Circle Circle with center at the origin (0,0) Standard form of a circle that is translated **Center: (h, k) Radius: r **

17. FINDING THE EQUATION OF A CIRCLE Write the standard form of the equation for the circle that has a center at the origin and has the given radius. 1.r = 92.r = 143.

18. WRITING EQUATIONS OF CIRCLES Write the standard equation of the circle: Center (4, 7) Radius of 5 (x – 4) 2 + (y – 7) 2 = 25

19. Write the standard equation of the circle: Center (-3, 8) Radius of 6.2 (x + 3) 2 + (y – 8) 2 = 38.44 Writing Equations of Circles

20. WRITING EQUATIONS OF CIRCLES Write the standard equation of the circle: Center (2, -9) Radius of (x – 2) 2 + (y + 9) 2 = 11

21. EQUATION OF A CIRCLE The center of a circle is given by (h, k) The radius of a circle is given by r The equation of a circle in standard form is (x – h) 2 + (y – k) 2 = r 2

22. Circle B The center is (4, 20) The radius is 10 The equation is (x – 4) 2 + (y – 20) 2 = 100

23. Circle O The center is (0, 0) The radius is 12 The equation is x 2 + y 2 = 144

24. GRAPHING CIRCLES (x – 3) 2 + (y – 2) 2 = 9 Center (3, 2) Radius of 3

25. GRAPHING CIRCLES (x + 4) 2 + (y – 1) 2 = 25 Center (-4, 1) Radius of 5

26. GRAPHING CIRCLES (x – 5) 2 + y 2 = 36 Center (5, 0) Radius of 6

27. APPLYING GRAPHS OF CIRCLES A bank of lights is arranged over a stage. Each light illuminates a circular area on the stage. A coordinate plane is used to arrange the lights, using the corner of the stage as the origin. The equation (x – 13) 2 + (y - 4) 2 = 16 represents one of the disks of light. A. Graph the disk of light. B. Three actors are located as follows: Henry is at (11, 4), Jolene is at (8, 5), and Martin is at (15, 5). Which actors are in the disk of light?

28. APPLYING GRAPHS OF CIRCLES 1.Rewrite the equation to find the center and radius.  (x – h) 2 + (y – k) 2 = r 2  (x - 13) 2 + (y - 4) 2 = 16  (x – 13) 2 + (y – 4) 2 = 4 2  The center is at (13, 4) and the radius is 4. The circle is shown on the next slide.

29. APPLYING GRAPHS OF CIRCLES 1.Graph the disk of light  The graph shows that Henry and Martin are both in the disk of light.

30. Graphing a circle in Standard Form!! Example: Center: (4, 0) r: 3 To write the standard equation of a translated circle, you may need to complete the square. To write the standard equation of a translated circle, you may need to complete the square. Standard Form!!

31. WRITE THE STANDARD EQUATION FOR THE CIRCLE. STATE THE COORDINATES OF ITS CENTER AND GIVE ITS RADIUS. THEN SKETCH THE GRAPH. Another one you ask!?! Ok, here it is!!

32. LAST ONE!!! Write the standard equation for the circle. State the center and radius.