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

2 STANDARD 4: 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 ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 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 22 2 What would be the equation for this circle? (3,2)(3,2) h = 3 k = 2 r = x y 6 (x – ) + (y – ) = ( ) (x – ) + (y – ) = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 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). r = ( - ) + ( - ) 22 = ( 3 ) + ( -5 ) 22 = r = (x –x ) + (y –y ) y 1 x 1 y 2 x 2 =(2,3) =(5,-2) r= x y (5,-2) (2,3)(2,3) 34 (x – h) + (y – k) = r 22 2 (x – ) + (y – ) = ( ) (x – ) + (y + ) = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 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). r = ( - ) + ( - ) 22 = ( 3 ) + ( -8 ) 22 = r = (x –x ) + (y –y ) y 1 x 1 y 2 x 2 =(3,4) =(6,-4) r= x y (6,-4) (3,4)(3,4) 73 (x – h) + (y – k) = r 22 2 (x – ) + (y – ) = ( ) (x – ) + (y + ) = PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 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 x + y + 8x +2y -32=0 22 x + 8x + + y + 2y = (4) 2 2 (1) 2 2 x + 8x + + y + 2y = = 17 + (x + 4) 2 (y + 1) (x + ) + (y + ) = Changing the form: Rewriting the equation to graph it: (x –( )) + (y –( )) = ( ) h= -4 k= -1 r= 7 (x – h) + (y – k) = r x y 7 (-4,-1) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 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 x + y + 6x +4y -23=0 22 x + 6x + + y + 4y = (3) 2 2 (2) 2 2 x + 6x + + y + 4y = = 13 + (x + 3) 2 (y + 2) (x + ) + (y + ) = Changing the form: Rewriting the equation to graph it: (x –( )) + (y –( )) = ( ) h= -3 k= -2 r= 6 (x – h) + (y – k) = r x y 6 (-3,-2) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 Standard 4, 9, 17 Write an equation of a circle if the endpoints of a diameter are at (-1,-3), and (7,5) x y First we find the midpoint or center of the circle: y x, =, x 1 7 x 2 y 1 5 y 2 -3 = y x, , = y 1 3,3, = x 1 x 2, 2 + y 1 y Using: y x, Now we calculate the radius’ length using: y 1 x 1 y 2 x 2 =(7,5) =(3,1) r = ( - ) + ( - ) 22 = ( -4) + (-4) 22 = r = (x –x ) + (y –y ) r= = = =2 2 2 = (3,1)(3,1) Center of the circle PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 Standard 4, 9, 17 (x – h) + (y – k) = r 22 2 h = 3 k = 1 (x – ) + (y – ) = ( ) (x – ) + (y – ) = x y 4 2 (3,1)(3,1) Now using the coordinates found for the center and the radius to find the equation for the circle: r = 4 2 r= 32 2 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved