DEFINITION OF A CIRCLE and example Standard 4, 9, 17 CIRCLES DEFINITION OF A CIRCLE and example PROBLEM 1a PROBLEM 1b PROBLEM 2a PROBLEM 2b PROBLEM 3 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

ALGEBRA II STANDARDS THIS LESSON AIMS: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes ESTÁNDAR 4: Los estudiantes factorizan polinomios representando diferencia de cuadrados, trinomios cuadrados perfectos, y la suma de diferencia de cubos. STANDARD 9: Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) + c. ESTÁNDAR 9: Los estudiantes demuestran y explican los efectos que tiene el cambiar coeficientes en la gráfica de funciones cuadráticas; esto es, los estudiantes determinan como la gráfica de una parabola cambia con a, b, y c variando en la ecuación y=a(x-b) + c STANDARD 17: Given a quadratic equation of the form ax + by + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. Estándar 17: Dada una equación cuadrática de la forma ax +by + cx + dy + e=0, los estudiantes pueden usar el método de completar al cuadrado para poner la ecuación en forma estándar y pueden reconocer si la gráfica es un círculo, elipse, parábola o hiperbola. Los estudiantes pueden graficar la ecuación 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Definition of a Circle: Standard 4, 9, 17 CIRCLES Definition of a Circle: A circle is the set of all points in a plane that are equidistant from a given point in the plane, called the center. Any segment whose endpoints are the center and a point on the circle is a radius of the circle. Equation of a Circle: The equation of a circle with center at (h,k) and radius r units is (x – h) + (y – k) = r 2 4 2 6 -2 -4 -6 8 10 -8 -10 x y What would be the equation for this circle? h = 3 k = 2 (3,2) 6 r = 6 (x – ) + (y – ) = ( ) 2 3 2 6 (x – ) + (y – ) = 36 2 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standard 4, 9, 17 Write the equation of a circle whose center is at (5, -2) and the circle passes through the point (2, 3). 4 2 6 -2 -4 -6 8 10 -8 -10 x y y 1 x =(2,3) (2,3) y 2 x =(5,-2) 34 r = (x –x ) + (y –y ) 2 1 (5,-2) r = ( - ) + ( - ) 2 5 2 -2 3 = ( 3 ) + ( -5 ) 2 (x – h) + (y – k) = r 2 = 9 + 25 (x – ) + (y – ) = ( ) 2 5 -2 34 r= 34 (x – ) + (y + ) = 34 2 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standard 4, 9, 17 Write the equation of a circle whose center is at (6, -4) and the circle passes through the point (3, 4). 4 2 6 -2 -4 -6 8 10 -8 -10 x y y 1 x =(3,4) (3,4) y 2 x =(6,-4) 73 r = (x –x ) + (y –y ) 2 1 (6,-4) r = ( - ) + ( - ) 2 6 3 -4 4 = ( 3 ) + ( -8 ) 2 (x – h) + (y – k) = r 2 = 9 + 64 (x – ) + (y – ) = ( ) 2 6 -4 73 r= 73 (x – ) + (y + ) = 73 2 6 4 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

and then graph the corresponding circle. (x – h) + (y – k) = r Standard 4, 9, 17 Given that x + y + 8x +2y -32=0 is the equation of a circle. Put it in the form and then graph the corresponding circle. (x – h) + (y – k) = r 2 Changing the form: Rewriting the equation to graph it: (x –( )) + (y –( )) = ( ) 2 -4 -1 7 x + y + 8x +2y -32=0 2 (x – h) + (y – k) = r 2 8 2 2 x + 8x + + y + 2y + -32 = + 2 h= -4 k= -1 r= 7 (4) 2 (1) 2 4 2 6 -2 -4 -6 8 10 -8 -10 x y x + 8x + + y + 2y + -32 = + 2 x + 8x + + y + 2y + -32 = + 2 16 1 7 (x + 4) 2 (y + 1) 2 + - 32 = 17 +32 +32 (-4,-1) (x + ) + (y + ) = 49 2 4 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

and then graph the corresponding circle. (x – h) + (y – k) = r Standard 4, 9, 17 Given that x + y + 6x +4y -23=0 is the equation of a circle. Put it in the form and then graph the corresponding circle. (x – h) + (y – k) = r 2 Changing the form: Rewriting the equation to graph it: (x –( )) + (y –( )) = ( ) 2 -3 -2 6 x + y + 6x +4y -23=0 2 (x – h) + (y – k) = r 2 6 2 4 2 x + 6x + + y + 4y + -23 = + 2 h= -3 k= -2 r= 6 (3) 2 (2) 2 4 2 6 -2 -4 -6 8 10 -8 -10 x y x + 6x + + y + 4y + -23 = + 2 x + 6x + + y + 4y + -23 = + 2 9 4 (x + 3) 2 (y + 2) 2 6 + - 23 = 13 +23 +23 (-3,-2) (x + ) + (y + ) = 36 2 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

First we find the midpoint or center of the circle: Standard 4, 9, 17 Write an equation of a circle if the endpoints of a diameter are at (-1,-3), and (7,5). x 2 -1 y 2 -3 4 2 6 -2 -4 -6 8 10 -8 -10 x y x 1 7 y 1 5 First we find the midpoint or center of the circle: x 1 2 , + y 1 2 + = Using: x, y (3,1) 4 2 = , 2 + x, y = y x, 2 6 , = y x, 1 3, Center of the circle r = (x –x ) + (y –y ) 2 1 Now we calculate the radius’ length using: 32 2 2 r = ( - ) + ( - ) 2 3 7 1 5 y 1 x =(7,5) 16 2 8 2 2 4 2 = ( -4) + (-4) 2 y 2 x =(3,1) 2 2 1 = 16 + 16 = 2 2 2 2 32 = =2 2 2 =4 2 r= 32 =4 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Standard 4, 9, 17 Now using the coordinates found for the center and the radius to find the equation for the circle: 4 2 6 -2 -4 -6 8 10 -8 -10 x y 4 2 (3,1) h = 3 k = 1 r = 4 2 r= 32 r = 32 2 (x – h) + (y – k) = r 2 (x – ) + (y – ) = ( ) 2 3 1 4 2 (x – ) + (y – ) = 32 2 3 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved