Quadratic Functions and Factoring
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1 Chapter Quadratic Functions and Factoring Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Prerequisite Skills for the chapter Quadratic Functions and Factoring. The -intercept of the line shown is.. The -intercept of the line shown is (5) 5 (5)(5) ? ???? ( )( ) ( )( ) (h 7)(h 9) 5 h 7h 7h 5 h h 0. (n 0)(n ) 5 n n 0n 0 5 n n ( ) ( 5) 5 ( ) Lesson. Graph Quadratic Functions in Standard Form Guided Practice for the lesson Graph Quadratic Functions in Standard Form f() f() b () 5 a () 5 verte and ais of smmetr. 5 opens down and is narrower than the graph of 5. ais of smmetr. However, the graph of 5 5 opens down, and its verte is 5 units lower. f() the same ais of smmetr. f () 5 is wider than the graph of f () 5, and its verte is f() 5 units higher. 5 () () 5 Verte: (,) Ais of smmetr: 5 -intercept: ; (0, ) 5 (, ) 5 : 5 () () 5 ; (, ) Algebra Worked-Out Solution Ke
2 b a 5 () (, ) Verte:, 0 Ais of smmetr: 5 -intercept: ; (0, ) 5 : 5 () () 5 ; (, ) f () b a (5) ( 5, 8 ) Verte: 5, 8 Ais of smmetr: 5 5 -intercept: ; (0, ) : 5 () 5() 5 ; (, ) b 5 a () 5 5 () () 5 9 The minimum value is R() 5 (5 ) p (80 0) R() 5, R() ,00 5 b a (0) R(.75) 5 0(.75) 00(.75),00 5 9,80.5 The verte is (.75, 9,80.5), which means the owner should reduce the price per racer b $.75 to increase the weekl revenue to $9, Eercises for the lesson Graph Quadratic Functions in Standard Form Skill Practice 0 0 the same verte and ais of smmetr. However, the graph of 5 is narrower than the graph of the same verte and ais of smmetr. However, the graph of 5 5 is narrower than the graph of verte and ais of smmetr. 5 opens down and is narrower than the graph of Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved.. The graph of a quadratic function is called a parabola.. Look at the value of a in the quadratic function. If a > 0, the function has a minimum value. If a < 0, the function has a maimum value. Algebra Worked-Out Solution Ke
3 f () 5 0 f() 0 verte and ais of smmetr. 5 opens down.. f () f() f () ais of smmetr. However, the graph of f() 5 opens down and its verte is units higher.. g() f() 5 f() 5 f () 0 the same verte and ais of smmetr. However, the graph of f() 5 is wider than the graph of f() 5.. g() 5 f() 5 f() 5 g() g() 5 g() 5 5 g() Both graphs have the same ais of smmetr. g() 5 5 opens down and is narrower than the graph of g() 5. Also, its verte is 5 units lower. 0 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. g () 0 g() g() g() 5 0 the same ais of smmetr. 5 5 is narrower than the graph of 5 and its verte is unit higher the same ais of smmetr. 5 is narrower than the graph of 5 and its verte is unit higher. verte and ais of smmetr. g() 5 opens down and is wider than the graph of g() f () f () f() 5 f() f() g() g () 7 7 g() 5 g() 5 5 Both graphs open up and have the same ais of smmetr. However, the graph of f () 5 5 is wider than the graph of f() 5 and its verte is 5 units lower. Both graphs have the same ais of smmetr. g() 5 5 opens down and is wider than the graph of g() 5. Also, its verte is units lower. 9. The -coordinate of the verte of a parabola is b a, not b. The -coordinate of the verte is: a 5 b a 5 () 5. Algebra Worked-Out Solution Ke
4 0. It is correct that the -intercept of the graph is the value of c. However, the value of c in 5 7 is b a 5 () 5 5 () () 5 0 Verte: (, 0) Ais of smmetr: 5 -intercept: ; (0, ) 5 : 5 () 5 ; (, ). 5 5 b () 5 a () 5 5 () () 5 Verte: (, ) Ais of smmetr: 5 -intercept: ; (0, ) (, 0) 5 (, ) 5 5 : 5 () () 5 ; (, ) b a 5 8 () 5 5 () 8() 5 Verte: (, ) Ais of smmetr: 5 intercept: ; (0, ) (, ) 5 5 : 5 () 8() 5 0; (, 0). 5 5 b () 5 a () Verte:, 5 Ais of smmetr: 5 -intercept: ; (0, ) (, 5 ) 5 5 : 5 () () 5 5; (, 5) 5. g() 5 5 b () 5 a () 5 g() 5 () () 5 0 Verte: (, 0) Ais of smmetr: 5 -intercept: ; (0, ) 5 (, 0) 5 : g() 5 () () 5 ; (, ). f() b () 5 a () 5 f Verte:, Ais of smmetr: 5 -intercept: 5; (0,5) 5 : f() 5 () () 5 5 5; (,5) b a () Verte: 9, Ais of smmetr: 5 9 -intercept: ; (0, ) (, ) 5 9 (, 8 ) : 5 () () 5 ; (, ) b a () Verte: 8, Ais of smmetr: 5 8 -intercept: ; (0,) 5 8 ( 8, ) 5 : 5 () () 5 ; (, ) 9. g() b a g (, ) 5 5 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke
5 Verte: 5, Ais of smmetr: 5 5 -intercept: ; (0, ) 5 : g() 5 5 () () 5 5 ;, 5. 5 Because a < 0, the function has a maimum value. 5 b a 5 0 () (0) 5 The maimum value is Because a > 0, the function has a minimum value. Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 0. f() 5 5 b a 5 5 f() 5 () () 5 7 Verte:, 7 Ais of smmetr: 5 -intercept: ; (0, ) 5 : f() 5 () 5 ; (, ) b a () Verte: 5, 5 Ais of smmetr: 5 5 -intercept: 5; (0, 5) 5 (, 7 ) 5 (, 5 ) : () () ;, b a () Verte: 77, 0 0 Ais of smmetr: 5 0 -intercept: ; (0, ) : 5 5 () 5 0 ;, 0 (, ) 5 b a 5 0 (9) (0) The minimum value is f() Because a > 0, the function has a minimum value. 5 b a 5 8 () 5 f() 5 () 8() 7 5 The minimum value is f() 5.. g() Because a < 0, the function has a maimum value. 5 b a 5 8 () 5 g() 5 () 8() 5 5 The maimum value is g() f() 5 Because a > 0, the function has a minimum value. 5 b a 5 5 f() 5 () () 5 The minimum value is f() Because a < 0, the function has a maimum value. 5 b a (7) () 7() 5 5 The maimum value is D; Because the -intercept changes from to, the verte moves down the -ais. 0. C; The graph of 5 a b c is wider than the graph of 5 if a < a b 5 c 5 Algebra Worked-Out Solution Ke 5
6 a b c 5. Verte: (, k) b a 5 b a 5 8 Sample answer: C; b () 5 a (0.5) b (.5) 5 a (0.5) ().5() Verte: (, 0.75) Ais of smmetr: 5 -intercept: ; (0, ) 5 : 5 0.5().5() 5 ; (, ) 5. f() 5. 5 b a 5 (.) (, 0.75) 5 0.5() () 5 Verte: (, ) 5. A; b a 5 0 (0.5) (0) 5 Verte: (0, ). B; b () 5 a (0.5) () () 5 Verte: (, ) 7. f() b a 5 0 (0.) 5 0 f(0) 5 0.(0) 5 Verte: (0, ) Ais of smmetr: : f(5) 5 0.(5) 5.5; (5,.5) 8. g() b a 5 0 (0.5) 5 0 g(0) 5 0.5(0) Verte: (0, 5) Ais of smmetr: : g() 5 0.5() 5 5 7; (,7) b a 5 (0.) (5) (5) Verte: (5, 8.5) Ais of smmetr: 5 5 -intercept: ; (0, ) 5 0 (0, ) (0, 5) 5 5 (5, 8.5) 5 : 5 0.() () 5.; (,.) 5 0 f Verte: 5 7, 7 Ais of smmetr: intercept: ; (0, ) ( 5, 7 7 ) : f () 5.() () 5 9.; (, 9.) 5. g() b a 5 0 (.75) 5 0 g(0) 5.75(0) Verte: (0,.5) Ais of smmetr: 5 0 (0,.5) : g() 5.75().5 5.5; (,.5) 5. Because the points (, ) and (, ) have the same -value and lie on the graph of a quadratic function, the are mirror images of each other. The ais of smmetr divides a parabola into mirror images, therefore, the ais of smmetr is halfwa between the -values. The ais of smmetr is a b c The -coordinate of the verte is b a. 5 a a b b a b c 5 ab a b a c 5 ab ab a c 5 a(b b ) b c 5 a a c Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Algebra Worked-Out Solution Ke
7 Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. Problem Solving 55. R() 5 ( 0.05) p (000 80) R() R() b 0 5 a () R(5) 5 (5) 0(5) Price: (5) 5.75 The store should increase the price per song to $.75 to increase the dail revenue to $900. Revenue (dollars) 5 Price (dollars/camera) p Sales (cameras) R() 5 (0 0) p (70 5) R() 5, R() ,00 5 b a 5 00 (00) 5 R() 5 00() 00(),00 5,500 Price: () 5 00 The store should decrease the price per digital camera to $00 to increase the monthl revenue to $, b 5 a (00) 7 (00) The height above the road of a cable at its lowest point is 0 feet b a 5. (0.) (.5).(.5) ø. No, the mouse cannot jump over a fence that is feet high because the maimum height it can jump is about. feet. 59. a. Profit (dollars) 5 Price (dollars/ticket) p Sales (tickets) Epenses (dollars) P() 5 (0 ) p (50 0) b P() c. Profits (dollars) P() P() 5.5 (.5, 5.5) 0 0 Price decrease The theater should reduce the price per ticket b $.50 to increase the weekl profit to $5.50. g ,000 a. e 5 0, b. m , m e The golf ball travels.5 feet on Earth. The golf ball travels 88.8 feet on the moon. c. A golf ball travels 88.8 or about times further on.5 the moon than on Earth. Smaller values of g produce longer distances.. P 5 w l P w 5 l A 5 lw 5 (P w)w 5 Pw w 5 w Pw w 5 b a 5 P () 5 P A 5 P P P 5 8 P P 5 8 P In terms of P, the maimum area that the swimming section can have is 8 P ft. Algebra Worked-Out Solution Ke 7
8 Graphing Calculator Activit for the lesson Graph Quadratic Functions in Standard Form. 5 The minimum value of the function is 5 5 and occurs at 5.. f () 5 The minimum value of the function is f () and occurs at The maimum value of the function is and occurs at The minimum value of the function is 5. and occurs at h() 5 The minimum value of the function is h() 5.5 and occurs at The maimum value of the function is 5 9 and occurs at 5 8. Minimum X= Y=-5 Minimum X=.5 Maimum X=.5 Minimium X=-0.8 Y=0.75 Y=8.75 Y=-. Minimium X= 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 ( ) a 5, h 5, k 5 Verte: (, ) Ais of smmetr: 5 5 0: 5 (0 ) 5 ; (0, ) 5 : 5 ( ) 5 ; (, ) 5 (, ). 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 ; (, ). f() 5 ( ) a 5, h 5, k 5 Verte: (,) Ais of smmetr: 5 5 : f() 5 ( ) 5 ; (, ) 5 : f() 5 ( ) 5 ; (, ) (, 5) 5 5 (, ). 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 ( 00) 7 is wider than the graph of the function ( 00) ( )( 7) -intercepts: p 5 and q p q (5 )(5 7) 5 Verte: (5, ) Ais of smmetr: 5 5. f() 5 ( )( ) -intercepts: p 5 and q 5 5 p q 5 () 5 f Verte: 5, Ais of smmetr: ( )( 5) -intercepts: p 5 and q p q ( )( 5) 5 9 Verte: (, 9) Ais of smmetr: 5 (, 0) 5 5 (5, ) 5 (7, 0) (, 0) (, 0) (, 0) (, 5 ) (, 9) 5 (5, 0) Copright Houghton Mifflin Harcourt Publishing Compan. All rights reserved. 8 Algebra Worked-Out Solution Ke
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