VERTEX FORM (OF A QUADRATIC FUNCTION) STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q. f(x) = ax 2 + bx + c
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2 VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c
3 Rewrite the equation in vertex form by completing the square. y = x 2 + 6x + 5 y = (x 2 + 6x) + 5 y = (x 2 + 6x + 9-9) + 5 y = (x 2 + 6x + 9) y = (x + 3) y = (x + 3) 2-4 (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2-2ab + b 2 Group the first two terms. Add and subtract the square of half the coefficient of the x-term. Group the perfect square trinomial. Rewrite as the square of a binomial. Simplify.
4 How can you solve quadratic equations of these form? VERTEX FORM (OF A QUADRATIC FUNCTION) f(x) = a(x - p) 2 + q 0 = a(x - p) 2 + q a(x - p) 2 + q = 0 STANDARD FORM (OF A QUADRATIC FUNCTION) f(x) = ax 2 + bx + c 0 = ax 2 + bx + c ax 2 + bx + c = 0
5 Solve x 2 9 = 0 x 2 = 9 x 2 = 9 x = ±3 x = 3 and x = 3 ax 2 + bx + c = 0 The roots are 3 and -3.
6 Solve (x 1) 2 49 = 0 (x 1) 2 = 49 a(x - p) 2 + q = 0 (x 1) 2 = 49 (x 1) = ±7 x 1 = ±7 x = ±7 + 1 x = = 8 and x = = -6 The roots are 8 and -6.
7 A wide-screen television has a diagonal measure of 42 in. The width of the screen is 16 in. more than the height. Determine the dimensions of the screen, to the nearest tenth of an inch.
8 A wide-screen television has a diagonal measure of 42 in. The width of the screen is 16 in. more than the height. Determine the dimensions of the screen, to the nearest tenth of an inch. h 2 + (h + 16) 2 = 42 2 h 2 + (h h + 256) = h h = h h = 1508 h h = 754 h h + 64 = (h + 8) 2 = 818 (h + 8) 2 = 818 h + 8 = ± 818 Draw a diagram. Let h represent the height of the screen. Then, h + 16 represents the width of the screen. Use the Pythagorean Theorem. Solve by Completing the Square
9 A wide-screen television has a diagonal measure of 42 in. The width of the screen is 16 in. more than the height. Determine the dimensions of the screen, to the nearest tenth of an inch. h 2 + (h + 16) 2 = 42 2 h 2 + (h h + 256) = h h = h h = 1508 h h = 754 Since the height of the screen cannot be negative, h = is an extraneous root. (A root that does not satisfy the initial restrictions on the variable) h h + 64 = (h + 8) 2 = 818 h = ± (h + 8) 2 = 818 h = and h = h + 8 = ± 818 h 20.6 in and h 36.6 in
10 A wide-screen television has a diagonal measure of 42 in. The width of the screen is 16 in. more than the height. Determine the dimensions of the screen, to the nearest tenth of an inch. h = ± h = and h = h 20.6 and h 36.6 Since the height of the screen cannot be negative, h = is an extraneous root. (A root that does not satisfy the initial restrictions on the variable) height 20.6 in and width = 36.6 in
11 The circular Canadian two-dollar coin consists of an aluminum and bronze core and a nickel outer ring. If the radius of the inner core is 0.84 cm and the area of the circular face of the coin is 1.96π cm 2, what is the width of the outer ring?
12 The circular Canadian two-dollar coin consists of an aluminum and bronze core and a nickel outer ring. If the radius of the inner core is 0.84 cm and the area of the circular face of the coin is 1.96π cm 2, what is the width of the outer ring?
13 Solve a Quadratic Equation by Completing the Square when a = 1. Confirm your answer(s) graphically. Express your answers to the nearest tenth. x 2 21 = 10x x x = 21 (x x ) = 21 (x x + 25) 25 = 21 (x + 5) 2 25 = 21 (x + 5) 2 = (x + 5) 2 = 46 (x + 5) 2 = 46 x + 5 = ± 46 x = ± 46 5 The roots are and x = x = and x =
14 Solve a Quadratic Equation by Completing the Square when a = 1. Confirm your answer(s) graphically. Express your answers to the nearest tenth. The roots are and x =
15 Solve a Quadratic Equation p 2-4p = 11 by Completing the Square when a = 1. Confirm your answer(s) graphically. Express your answers to the nearest tenth.
16 Solve a Quadratic Equation -2x 2 3x + 7 = 0 by Completing the Square when a 1. Confirm your answer(s) graphically. Express your answers to the nearest hundredth.
17 Solve a Quadratic Equation -2x 2 3x + 7 = 0 by Completing the Square when a 1. Confirm your answer(s) graphically. Express your answers to the nearest hundredth.
18 Solve a Quadratic Equation -2x 2 5x + 2 = 0 by Completing the Square when a 1. Confirm your answer(s) graphically. Express your answers to the nearest hundredth.
19 A defender kicks a soccer ball away from her own half. The path of the kicked soccer ball can be approximated by the quadratic function h(x) = -0.06x x , where x is the horizontal distance travelled, in metres, from the goal line and h is the height, in metres. a) You can determine the distance the soccer ball is from the goal line by solving the corresponding equation, -0.06x x = 0. How far is the soccer ball from the goal line when it is kicked? Express your answer to the nearest tenth of a metre. b) How far does the soccer ball travel before it hits the ground?
20 A defender kicks a soccer ball away from her own half. The path of the kicked soccer ball can be approximated by the quadratic function h(x) = -0.06x x , where x is the horizontal distance travelled, in metres, from the goal line and h is the height, in metres. a) You can determine the distance the soccer ball is from the goal line by solving the corresponding equation, -0.06x x = 0. How far is the soccer ball from the goal line when it is kicked? Express your answer to the nearest tenth of a metre.
21 A defender kicks a soccer ball away from her own half. The path of the kicked soccer ball can be approximated by the quadratic function h(x) = -0.06x x , where x is the horizontal distance travelled, in metres, from the goal line and h is the height, in metres. a) You can determine the distance the soccer ball is from the goal line by solving the corresponding equation, -0.06x x = 0. How far is the soccer ball from the goal line when it is kicked? Express your answer to the nearest tenth of a metre.
22 A defender kicks a soccer ball away from her own half. The path of the kicked soccer ball can be approximated by the quadratic function h(x) = -0.06x x , where x is the horizontal distance travelled, in metres, from the goal line and h is the height, in metres. b) How far does the soccer ball travel before it hits the ground?
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