2. y 5 2(x 2 1) x 5 0: y 5 2(0 2 1) ; (0, 4) x 5 21: y 5 2(21 2 1) ; (21, 1) 3. f(x) 5 1 } 2 (x 2 3)2 2 4

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1 Graphing Calculator Activit for the lesson Graph Quadratic Functions in Standard Form. 5 6 function is 5 5 and occurs at f () function is f () and occurs at The maimum value of the function is and occurs at function is 5.3 and occurs at h() 5 } 3 function is h() 5.5 and occurs at } The maimum value of the function is 5 9 and occurs at 5 8. Minimum X=3 Y=-5 Minimum X=.5 Maimum X=.5 Minimium X=-0.8 Y=0.75 Y=8.75 Y=-.3 Minimium X=3 Y=-.5 Maimum X=8 Y=9 Lesson. Graph Quadratic Functions in Verte or Intercept Form Guided Practice for the lesson Graph Quadratic Functions in Verte or Intercept Form. 5 ( ) 3 a 5, h 5, k 5 3 Verte: (, 3) Ais of smmetr: 5 5 0: 5 (0 ) 3 5 ; (0, ) 5 : 5 ( ) 3 5 ; (, ) 5 (, 3). 5 ( ) 5 a 5, h 5, k 5 5 Verte: (, 5) Ais of smmetr: 5 5 0: 5 (0 ) 5 5 ; (0, ) 5 : 5 ( ) 5 5 ; (, ) 3. f() 5 } ( 3) a 5 }, h 5 3, k 5 Verte: (3,) Ais of smmetr: : f() 5 } ( 3) 5 ; (, ) 5 : f() 5 } ( 3) 5 ; (, ) (, 5) (3, ). The graphs of both functions open up and have the same verte and ais of smmetr. However, the a values of the functions differ. The graph of the function 5 } 7000 ( 00) 7 is wider than the graph of the function 5 } 6500 ( 00) ( 3)( 7) -intercepts: p 5 3 and q } p q 5 } (5 3)(5 7) 5 Verte: (5, ) Ais of smmetr: f() 5 ( )( ) -intercepts: p 5 and q 5 5 } p q 5 } () 5 } 3 f 3 } 5 3 } 3 } 5 5 } Verte: 3 } 5, } Ais of smmetr: 5 3 } 7. 5 ( )( 5) -intercepts: p 5 and q } p q 5 } ( )( 5) 5 9 Verte: (, 9) Ais of smmetr: 5 (3, 0) 5 5 (5, ) 5 3 (7, 0) (, 0) (, 0) (, 0) 3 (, 5 ) (, 9) 5 (5, 0) Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke

2 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved ( 50) ( 0)( 50) -intercepts: p 5 0 and q } p q 5 } (5)(5 50) ø 5.6 The maimum height of the football is the -coordinate of the verte, or about 5.6 ards ( )( 7) 5 ( 7 ) 5 ( 9 ) ( )( 3) 5 ( 3 3) 5 ( 3) 5 8. f() 5 ( 5)( ) 5 ( 5 0) 5 ( 9 0) ( 6)( ) 5 7( 6 6) 5 7( 5 6) ( 5) 5 3( 5)( 5) 5 3( 5 5 5) 5 3( 0 5) g() 5 6( ) 0 5 6( )( ) 0 5 6( 6) 0 5 6( 8 6) f () 5 ( ) 5 ( )( ) 5 ( ) 5 ( ) ( 3) 9 5 ( 3)( 3) 9 5 ( 3 3 9) 9 5 ( 6 9) Eercises for the lesson Graph Quadratic Functions in Verte or Intercept Form Skill Practice. A quadratic function in the form 5 a( h) k is in verte form.. First identif the -intercepts. Then use the -intercepts to calculate the -coordinate of the verte. Finall, substitute the -coordinate of the verte for into the original function to find the -coordinate of the verte. The -coordinate of the verte is the maimum or minimum value ( 3) a 5, h 5 3, k 5 0 Verte: (3, 0) Ais of smmetr: : 5 ( 3) 5 ; (, ) 5 : 5 ( 3) 5 ; (, ). 5 ( ) a 5, h 5, k 5 0 Verte: (, 0) Ais of smmetr: 5 5 : 5 ( ) 5 ; (, ) 5 3: 5 (3 ) 5 ; (3, ) 5. f() 5 ( 3) 5 a 5, h 5 3, k 5 5 Verte: (3, 5) Ais of smmetr: : f() 5 ( 3) 5 5 ; (, ) (3, 0) (, 0) (3, 5) 5 : f() 5 ( 3) 5 5 ; (, ) ( 7) a 5 3, h 5 7, k 5 Verte: (7, ) Ais of smmetr: : 5 3(6 7) 5 ; (6, ) 5 5: 5 3(5 7) 5 ; (5, ) 7. g() 5 ( ) a 5, h 5, k 5 Verte: (, ) Ais of smmetr: 5 5 3: g() 5 (3 ) 5 0; (3, 0) 5 : g() 5 ( ) 5 ; (, ) 5 7 (7, ) (, ) Algebra Worked-Out Solution Ke 9

3 8. 5 ( ) 3 a 5, h 5, k 5 3 Verte: (, 3) Ais of smmetr: 5 5 0: 5 (0 ) 3 5 ; (0, ) 5 : 5 ( ) 3 5 5; (, 5) 9. f() 5 ( ) 5 a 5, h 5, k 5 5 Verte: (, 5) Ais of smmetr: 5 5 0: f() 5 (0 ) 5 5 : 5 7; (0, 7) f() 5 ( ) 5 5 3; (, 3) 0. 5 } ( ) a 5 }, h 5, k 5 Verte: (, ) Ais of smmetr: : 5 } (0 ) 5 0; (0, 0) 3 (, 3) (, ) 5 5 : 5 } ( ) 5 3; (, 3) 5. 5 } ( 3) 5 3 a 5 }, h 5 3, k 5 Verte: (3, ) Ais of smmetr: : 5 } ( 3) 5 ; (, ) 5 : 5 } ( 3) 5 0; (, 0). B; 5 3( ) 5 (, 5) (3, ) The graph of 5 a( h) k has verte (h, k). The verte of the graph of the function is (, 5) ( 3)( 3) -intercept: p 5 3 and q } p q 5 } (0 3)(0 3) 5 9 Verte: (0, 9) Ais of smmetr: 5 0 (3, 0) 5 0 (3, 0) (0, 9). 5 ( )( 3) -intercept: p 5 and q } p q 5 } ( )( 3) 5 Verte: (, ) Ais of smmetr: ( )( 6) (6, 0) 5 (, 0) (, ) -intercept: p 5 and q } p q (6) 5 } 5 5 3( )( 6) 5 Verte: (, ) Ais of smmetr: 5 6. f() 5 ( 5)( ) -intercept: p 5 5 and q 5 5 } p q 5 } f() 5 (3 5)(3 ) 5 8 Verte: (3, 8) Ais of smmetr: ( )( 6) (, 5) 5 3 (6, 0) (, 0) -intercept: p 5 and q } p q 5 } (6) 5 5 ( )( 6) 5 5 Verte: (, 5) Ais of smmetr: 5 (, 0) 5 (, ) (3, 8) (3, 0) (, 0) (5, 0) 5 3 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0 Algebra Worked-Out Solution Ke

4 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8. g() 5 ( 3)( 7) (5, 6) 5 5 (7, 0) (3, 0) -intercept: p 5 3 and q } p q 3 (7) 5 } 5 5 g() 5 (5 3)(5 7) 5 6 Verte: (5, 6) Ais of smmetr: ( )( ) -intercept: p 5 and q 5 5 } p q () 5 } 5 } } 3 } 5 } Verte: 3 }, } Ais of smmetr: 5 3 } 0. f() 5 ( 3)( ) (, 9 ) 5 3 (, 0) (3, 0) -intercept: p 5 3 and q 5 5 } p q 5 } 3 () 5 } 5 3 (, 0) (, 0) 3 ( 3, ) f () 5 } 3 } 5 9 } Verte: }, 9 } Ais of smmetr: 5 }. 5 ( 7)( ) -intercept: p 5 7 and q 5 5 } p q 5 } 7 () 5 } } 7 5 } 5 8 Verte: 5 }, 8 Ais of smmetr: 5 5 }. A; 5 ( 6)( ) -intercepts: p 5 6 and q 5 5 } p q 5 } 6 () 5 5 ( 6)( ) 5 5 Verte: (, 5) 0 (, 0) (7, 0) (, 8 ) 3. The -intercepts of the graph of 5 a( p)( q) are p and q. Therefore, the -intercepts of the graph of 5 5( )( (3)) are and ( )( 3) 5. 5 ( 5)( 3) h() 5 ( )( 6) 5 ( 6 6) 5 ( 5 6) ( )( ) 5 3( 8) 5 3( 6 8) f() 5 ( 5) 5 ( 5)( 5) 5 ( 5 5 5) ( 3) 6 5 ( 3)( 3) 6 5 ( 3 3 9) g() 5 ( 6) 0 5 ( 6)( 6) 0 5 ( ) 0 5 ( 36) Algebra Worked-Out Solution Ke

5 3. 5 5( 3) 5 5( 3)( 3) 5 5( 3 3 9) 5 5( 6 9) f() 5 ( ) 5 ( )( ) 5 ( ) 5 ( ) ( 3) Because a > 0, the function has a minimum value. The minimum value is g() 5 ( 6) Because a < 0, the function has a maimum value. The maimum value is ( 5) 30 Because a > 0, the function has a minimum value. The minimum value is f() 5 3( 0)( 8) Because a > 0, the function has a minimum value. 5 } p q 5 } f() 5 3( 0)( 8) 5 3 The minimum value is f() ( 36)( 8) Because a < 0, the function has a maimum value. 5 } p q 36 (8) 5 } (9 36)(9 8) 5 79 The maimum value is ( 9) 5 ( 0)( 9) Because a < 0, the function has a maimum value. 5 } p q 5 } } } 5 3 The maimum value is ( 5) 5 8( 0)( 5) Because a > 0, the function has a minimum value. 5 } p q 5 } 0 (5) 5 } } 5 } The minimum value is ( 3)( 6) Because a > 0, the function has a minimum value. 5 } p q 5 } } } 3 9 } } The minimum value is 5 9 }.. g() 5 5( 9)( ) Because a < 0, the function has a maimum value. 5 } p q 5 } 9 5 } 5 g 5 } } 9 5 } 5 } 85 The maimum value is g() 5 85 }.. 5 a( h) k 5 ( 3) a. If a changes to 3, a < 0 so the graph will open down instead of up. Also because a >, the graph will be narrower than the original graph. b. If h changes to, the graph will be translated horizontall units to the left. c. If k changes to, the graph will be translated verticall units up. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke

6 3. 5 5(.5) (.5,.75) a 5 5, h 5.5, k 5.75 Verte: (.5,.75) Ais of smmetr: : 5 5(.5).75 ø 5.06; (, 5.06) 5 : 5 5(.5).75 ø.; (,.). g() 5 8( 3.) 6. (3., 6.) 6. 5 } 3 } } 5 (, 5) a 5 } 3, h 5 }, k 5 } 5 Verte: }, } 5 Ais of smmetr: 5 } 5 5 : 5 } 3 } } 5 5 } 7 0 ;, } : 5 } 30 } } 5 5 } 9 30 ; 0, } f() 5 } 3 ( 5)( 8) 3 7 (, 6) 3 5 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved a 5 8, h 5 3., k 5 6. Verte: (3., 6.) Ais of smmetr: : g() 5 8(3 3.) ; (3, 6.08) 5 : g() 5 8( 3.) ; (, 5.) ( 5.) 8.5 (5., 8.5) 5 5. a 5 0.5, h 5 5., k Verte: (5., 8.5) Ais of smmetr: : 5 0.5(0 5.) ; (0,.7) 5 : 5 0.5( 5.) ; (,.09) -intercepts: p 5 5 and q } p q 5 (8) 5 } 5 } 3 f() 5 } 3 } 3 5 } 3 Verte: 3 }, 7 } 6 Ais of smmetr: 5 3 } 8. g() 5 } 5 } 3 } 5 -intercepts: p 5 } 3 and q 5 } 5 5 } p q } 5 3 } 5 } 5 } } 7 6 g() 5 } 5 } 3 5 } 3 } 3 5 } 5 5 } 9 90 Verte: 3 } 5, 9 } 50 Ais of smmetr: 5 3 } (, ) 9. The graph of f() 5 ( )( 5) is a parabola that opens down, because the leading coefficient is negative. The graph crosses the -ais at its -intercepts, and 5; it lies above the -ais between 5 and 5 5; and it lies below the -ais to the left of 5 and to the right of 5 5. So, the function values are positive on the interval (, 5) and negative on the intervals (`, ) and (5, `) a( h) k 5 a( h)( h) k 5 a( h h h ) k 5 a( h h ) k 5 a ah ah k Algebra Worked-Out Solution Ke 3

7 a 5 a, b 5 ah, c 5 ah k 5 } b (ah) 5 } 5 h a (a) 5 a( p)( q) 5 a( q p pq) 5 a ap aq apq 5 a (ap aq) apq a 5 a, b 5 ap aq, c 5 apq 5 } b (ap aq) (a)(p q) 5 } 5 } 5 } p q a (a) a Problem Solving ( ) 6 The verte is (, 6). The maimum height of the kangaroo is 6 feet. () 5 8 The kangaroo s jump is 8 feet long ( 5.5) 5 The verte is (5.5, 5). (5.5) 5 05 The width of the arch is 05 meters. 53. a ( 60) ( 0)( 60) -intercepts: p 5 0 and q 5 60 The width of the field is 60 feet. b. 5 } p q 5 } (80)(80 60) ø.5 The maimum height of the field s surface is about.5 feet ( 6) 8 The maimum height of the jump with a conventional spring is 8 inches. 5.7( 6) The maimum height of the jump with a bow spring is inches. The jump on the pogo stick with a bow spring is inches higher than the jump on the pogo stick with a conventional spring. The constant k affects the maimum heights of the jumps, while the constants a and h do not. 55. a ( 5.5)(.6) p 5 5.5, q } p q } ( )(.06.6) ø 55.5 For hot-air popping, a.06% moisture content maimizes popping volume. The maimum popping volume is 55.5 cubic centimeters per gram. b ( 5.35)(.8) p , q } p q } ø ( )(3.58.8) ø. For hot-oil popping, a 3.58% moisture content maimizes popping volume. The maimum popping volume is. cubic centimeters per gram. c ( 5.5)(.6) ( 5.35)(.8) hot-air popping: domain: range: hot-oil popping: domain: range: 0. The -intercepts of the graph of each function determined the domain. The -coordinate of the verte of the graph of each function determined the range. Also, the range did not include an negative values because it does not make sense to have a negative popping volume a( h) k h 5 33 k a( 33) 5 At (0, 0): 0 5 a(0 33) a(089) a 5 } a 5 5 } 089 ( 33) 5 Changing the value of a affects the width of the flight path. Changing the value of h affects the horizontal position of the flight path. Changing the value of k affects the height of the flight path. Lesson.3 Solve b c 5 0 b Factoring Guided Practice for the lesson Solve b c 5 0 b Factoring. Factors of 8: m, n, 8, 8, 9 Sum of factors: m n Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Factors of 8: m, n, 9 3, 6 3, 6 Sum of factors: m n ( 3)( 6) Algebra Worked-Out Solution Ke