( )! 1! 3 = x + 1. ( ) =! x + 2

Transcription

1 7.5 Graphing Parabolas 1. First complete the square: y = x 2 + 2x! 3 = x 2 + 2x + 1 ( )! 1! 3 = x + 1 ( ) 2! 4 The x-intercepts are 3,1 and the vertex is ( 1, 4). Graphing the parabola: 3. First complete the square: y =!x 2! 4x + 5 =! x 2 + 4x + 4 ( ) =! x + 2 The x-intercepts are 5,1 and the vertex is ( 2,9). Graphing the parabola: ( ) The x-intercepts are 1,1 and the vertex is (0, 1). Graphing the parabola:

2 7. The x-intercepts are 3,3 and the vertex is (0,9). Graphing the parabola: 9. First complete the square: y = 2x 2! 4x! 6 = 2 x 2! 2x + 1 ( )! 2! 6 = 2 x! 1 The x-intercepts are 1,3 and the vertex is (1, 8). Graphing the parabola: ( ) 2! First complete the square: y = x 2! 2x! 4 = ( x 2! 2x + 1)! 1! 4 = ( x! 1) 2! 5 The x-intercepts are 1 ± 5 and the vertex is (1, 5). Graphing the parabola:

3 13. The vertex is (1,3) and there are no x-intercepts. Graphing the parabola: 15. The vertex is ( 2,4) and the x-intercepts are 4,0. Graphing the parabola: 17. The vertex is (2, 4) and the x-intercepts are 2 ± 2 2. Graphing the parabola:

4 19. The vertex is (4, 1) and there are no x-intercepts. Graphing the parabola: 21. First complete the square: y = x 2! 4x! 4 = x 2! 4x + 4 The vertex is (2, 8). Graphing the parabola: ( )! 4! 4 = x! 2 ( ) 2! First complete the square: y =!x 2 + 2x! 5 =! x 2! 2x + 1 The vertex is (1, 4). Graphing the parabola: ( ) + 1! 5 =! x! 1 ( ) 2! 4

5 25. The vertex is (0,1). Graphing the parabola: 27. The vertex is (0, 3). Graphing the parabola: 29. First complete the square: g(x) = 3x 2! + 4x + 1 = 3 x x + 4 $ " # 9 % & ' = 3! x + 2 $ " # 3 % & " The vertex is! 2 3,! 1 % # $ 3& '. Graphing the parabola: 2 ' Completing the square: y = x 2! 6x + 5 = x 2! 6x + 9 ( )! = x! 3 The vertex is (3, 4), which is the lowest point on the graph. 33. Completing the square: y =!x 2 + 2x + 8 =! x 2! 2x + 1 The vertex is (1,9), which is the highest point on the graph. 35. Completing the square: y =!x 2 + 4x + 12 =! x 2! 4x + 4 ( ) 2! 4 ( ) =! x! 1 The vertex is (2,16), which is the highest point on the graph. 37. Completing the square: y =!x 2! 8x =! x 2 + 8x + 16 ( ) ( ) =! x! 2 ( ) + 16 =! x + 4 The vertex is ( 4,16), which is the highest point on the graph. ( ) ( )

6 39. First complete the square: P(x) =!0.002x x! 800 =!0.002 x 2! 1750x + 765,625 =!0.002 x! 875 ( ) + 1,531.25! 800 ( ) It must sell 875 patterns to obtain a maximum profit of $ The ball is in her hand at times 0 sec and 2 sec. Completing the square: h(t ) =!16t t =!16 t 2! 2t + 1 The maximum height of the ball is 16 feet. 43. Completing the square: R = xp = 1200 p! 100 p 2 =!100 p 2! 12 p + 36 ( ) + 16 =!16 t! 1 ( ) =!100 p! 6 The price is $6.00 and the maximum revenue is $3,600. Sketching the graph: ( ) ( ) Completing the square: R = xp = 1700 p! 100 p 2 =!100 p 2! 17 p ( ) =!100 p! 8.5 The price is $8.50 and the maximum revenue is $7,225. Sketching the graph: ( )

7 47. The equation is given on the graph: 49. Solving the equation: x 2! 2x! 8 = 0 ( x! 4) ( x + 2) = 0 x =!2, Solving the equation: 6x 2! x = 2 6x 2! x! 2 = 0 2x + 1 ( )( 3x! 2) = Solving the equation: x 2! 6x + 9 = 0 ( x! 3) 2 = 0 x = 3 x =! 1 2, 2 3