Mini-Lecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square. 3. Use quadratic equations to solve problems. 4. Ke vocabular: quadratic equation, perfect square trinomial. Eamples: 1. Use the square root propert to solve each equation. a) = 9 b) = 0 c) + 7= 0 d) 4 = 16 e) ( 5) = 5 f) ( + 3) = 11 g) ( 4 + 1) = 36 h) ( ) 5 3 = 48. Solve each equation b completing the square. a) + 4= 3 b) = 35 c) + 0+ 30 = 0 d) 5=3 e) + 11= 1 f) g) 6 + 10+ =0 h) 4 16+ 80= 0 i) + 5 3= 0 + = 1 3. The distance, s() t, in feet traveled b a freel falling object is given b the function s() t = 16t, where t is time in seconds. How long would it take for an object to fall to the ground from 576 feet high? Teaching Notes: Man students forget the +/- when using the square root propert. Most students are confused b completing the square at first and need to see man eamples. Refer students to the Solving a Quadratic Equation in b Completing the Square chart in tet. Answers: 1a) {3,-3}, b) { 5, 5}, c){6i,-6i}, d) {,-}, e) {10,0}, f) { 3 + 11, 3 11}, g) 5 7, 4 4, 3+ 4 3 3 4 3 1 h), ; a) {-3,-1}, b) {7,-5}, c) { 10 + 70, 10 70}, d) 3, 5 5, e) 3, 4, f) 1, 3, g) 5+ 13 5 13 1 3 1 3,, h) {+4i,-4i}, i) + i, i ; 3) t=6 seconds 6 6 Copright 009 Pearson Education, M-46 Inc., publishing as Pearson Prentice Hall
Mini-Lecture 8. Solving Quadratic Equations b the Quadratic Formula Learning Objectives: 1. Solve quadratic equations b using the quadratic formula.. Determine the number and tpe of solutions of a quadratic equation b using the discriminant. 3. Solve geometric problems modeled b quadratic equations. 4. Ke vocabular: discriminant. Eamples: 1. Use the quadratic formula to solve each equation. a) + 5+ 6=0 b) + 4 7= 0 c) 3 9= d) 5 = 10 3 e) 8 5 = f) ( ) 9+ 3 = 8 g) 35 + + = 0 h) ( + 8) ( 9) = ( 1) 7 18 9. Use the discriminant to determine the number and tpes of solutions of each equation. a) 4 8+ 4=0 b) 6 = 5 c) + 8+ 7=0 d) 10 5 = 6+ 5 3. Solve each equation b completing the square. c) Geometr A rectangular sign has an area of 1 square ards. Its length is 6 ards more than its width. Find the dimensions of the sign. d) Geometr The hpotenuse of a right triangle is 7 feet long. One leg of the triangle is 5 feet longer than the other leg. Find the perimeter of the triangle. e) Revenue The revenue for a small compan is given b the quadratic function rt ( ) = 14t + 16t + 860, where t is the number of ears since 1998 and rt ( ) is in thousands of dollars. Find the ear in which the compan s revenue will be $190 thousand. Round to the nearest whole ear. Teaching Notes: Encourage students to memorize the quadratic formula. 4± 5 1± 5 Man students reduce final answers incorrectl. For eample:. 8 Some students prefer to alwas use the quadratic formula because it has no restrictions on when it can be used. Encourage them to master the other methods, which are often quicker and easier to appl. Refer students to the Discriminant chart in the tet. Answers: 1a) {-3,-}, b) { + 11, 11}, c) 3+ 6 3 6 f),, g) { 9 + 11, 9 11}, h) 3 3 9+ 57 9 57 5+ 10 5 10 i 10 i 10,, d),, e),, 6 6 5 5 5 5 1, ; a) one real solution, b) two comple but not real solutions, c) two real solutions, d) two real solutions; 3a) 3+ 30 ds b 3+ 30 ds, b) 7+ 73 ft, c) 003 M-47 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Mini-Lecture 8.3 Solving Equations b Using Quadratic Methods Learning Objectives: 1. Solve various equations that are quadratic in form.. Solve problems that lead to quadratic equations. Eamples: 1. Solve. a) = 8+ b) 8 = 1 1 c) + 11 = + 6 d) 6 1 1 + 3 = 1 6 e) = + 8 f) 4 = 5 4 4 g) + 4=0 h) 13 45= 0 i) d) 1 3 3 + 8=0 j) 1 3 3 3 8=0 k) 5 (3 4) 9(3 4) + 18 = 0 l) + = 1 ( 1). Solve. a) Number The product of a number and 8 less than the number is 15. Find the number. b) Geometr The hpotenuse of a right triangle is 9 feet long. One leg of the triangle is 3 feet longer than the other leg. Find the perimeter of the triangle. c) Combined Rate Two pipes can be used to fill a pool. Working together, the two pipes can fill the pool in 6 hrs. The larger pipe can fill the pool in 3 hours less than the smaller pipe can alone. Find the time to the nearest tenth of an hour it takes for the smaller pipe working alone to fill the pool. Teaching Notes: Most students find this section difficult due to the various number of solutions that are possible. Remind students to check for etraneous solutions. Encourage students to draw a diagram or make a chart when solving applied problems. Refer students to the Solving a Quadratic Equation chart in the tet. 1 Answers: 1a) {16}, b), c) {5}; d) {5 + 7,5 7}, e) {1 + 17,1 17} ; f) { 5, 5, 5, 5} 16 i i, g) {,, i 6, i 6}, 10 10 64 h) 3, 3, i, i, i) {8,-64},j) 8, 7, k) 7 10, 3 3, l) 1 1, ; a) 3 or 5, b) 1.37 ft, c) 13.7 hrs 4 Copright 009 Pearson Education, M-48 Inc., publishing as Pearson Prentice Hall
Mini-Lecture 8.4 Nonlinear Inequalities in One Variable Learning Objectives: 1. Solve polnomial inequalities of degree or greater.. Solve inequalities that contain rational epressions with variables in the denominator. 3. Ke vocabular: standard form, related equation, test point. Eamples: 1. Solve. a) ( + ) ( + 3) > 0 b) ( + )( + 3) 0 c) 9+ 18 0 d) 5 4 9 e) ( + 6) ( ) < 0 f) ( + 1) ( 3) ( 6) > 0 g) ( 36)( 4) 0 h) 3 16 + 48 5 75 > 0. Solve. + 5 a) < 0 3 b) 7 > 0 c) 4 0 3 d) 3 3 + 4 e) ( + 5)( 5) (3 ) ( 1) < 0 f) 0 ( ) g) 4 < + 6 h) ( 3) > 0 5 Teaching Notes: Man students understand the concepts of this section better if the are shown a graph of the quadratic function in 1a) and b) and can see where the parabola is above or below the -ais. Some students are confused b how to pick test points. Remind them that the can pick an convenient point ecept for the critical points that define regions. Encourage students to make a region chart as in the tetbook eamples. Refer students to the Solving a Polnomial Inequalit of Degree or Higher and Solving a Rational Inequalit charts in the tet. Answers: 1a) (, 3) (, ), b) [-3,-], c) (,3] [6, ), d) M-49 9 (, 1],, e) (, 6) (0,), f) ( 1,3) (6, ), 5 5 5 g) [ 6, ] [,6], h) 3,, ; a) (-5,3), b) (,) (7, ), c) (,3), d) [ 5, 4) 4 4, e) (, 5) (0,5), f) [1, g) ( 6, ) (0, ), h) (, 5) (5, ), ) (,3] Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Mini-Lecture 8.5 Quadratic Functions and Their Graphs Learning Objectives: 1. Graph quadratic functions of the form. Graph quadratic functions of the form 3. Graph quadratic functions of the form 4. Graph quadratic functions of the form f ( ) = + k. f ( ) = ( h). f ( ) = ( h) + k. f ( ) = a. 5. Graph quadratic functions of the form f ( ) = a( h) + k. 6. Ke vocabular: verte, ais of smmetr. Eamples: 1. Graph each quadratic function. Label the verte and sketch and label the ais of smmetr. a) f ( ) = b) f( ) = + c) f( ) = 3. Graph each quadratic function. Label the verte and sketch and label the ais of smmetr. a) f( ) = ( ) b) f( ) = ( + 3) 3. Graph each quadratic function. Label the verte and sketch and label the ais of smmetr. a) f( ) = ( ) + 1 b) f( ) = ( + 1) 3 4. Graph each quadratic function. Label the verte and sketch and label the ais of smmetr. a) c) f ( ) = b) f ( ) = d) 1 f ( ) = f ( ) = 3 5. Graph each quadratic function. Label the verte and sketch and label the ais of smmetr. a) f( ) = ( + 1) b) c) f( ) = ( 3) + 4 d) 1 f( ) = ( ) + 1 1 f( ) = ( + 3) 3 Teaching Notes: Most students find vertical shifts eas to understand. Some students are confused b the direction of a horizontal shift. Man students are uncertain of how to quickl determine the verte until the have seen the graphs in objective 5 and can visualize how the verte is ( hk, ) Refer students to the man graphing charts in the tet. Answers: (graph answers at end of mini-lectures) 1a) (0,0), =0, b) (0,), =0, c) (0,-3), =0; a) (,0), =, b) (-3,0), =-3; 3a) (,1), =, b) (-1,-3), =-1; 4a) (0,0), =0, b) (0,0), =0, c) (0,0), =0, d) (0,0), =0; 5a) (-1,0), =-1, b) (,1), =, c) (3,4), =3, d) (-3,-), =-3 Copright 009 Pearson Education, M-50 Inc., publishing as Pearson Prentice Hall
Mini-Lecture 8.6 Further Graphing of Quadratic Functions Learning Objectives: 1. Write quadratic functions in the form f ( ) = a( h) + k.. Derive a formula for finding the verte of a parabola. 3. Graph quadratic functions b graphing the verte and all intercepts. 4. Find the minimum or maimum values of quadratic functions. Eamples: 1. Find the verte of the graph of each quadratic function b completing the square. a) f( ) = + 6+ 9 b) f( ) = + 4+ 6 c) f( ) = + + 6. Find the verte of the graph of each quadratic function. Determine whether the graph opens upward or downward, find an intercepts, and graph the function. a) f( ) = + 5+ 4 b) f( ) = + + 8 c) e) f ( ) = + 8 d) 5 f( ) = + 4+ f) f( ) = 4+ 4 1 9 f( ) = + + 4 4 3. Solve. a) Number Find two numbers whose sum is 44 and whose product is as large as possible. b) Geometr The length and width of a rectangle must have a sum of 94 feet. Find the dimensions of the rectangle whose area is as large as possible. c) Projectile An arrow is fired into the air with an initial velocit of 96 feet per second. The height in feet of the arrow t seconds after it was shot into the air is given b the function h ( ) = 16t + 96t. Find the maimum height of the arrow. Teaching Notes: Most students need to be reminded of the completing the square procedure. Most students are comfortable using the verte formula but some are confused at first b wh the calculated value must be substituted back into the quadratic function. Remind students to alwas check that their graph opens in the epected direction. 1 3 Answers: (graph answers at end of mini-lectures) 1a) (-3,0), b) (1,8), c), 4 ; a) 5 9,, opens up, -int (- 4 4,0), (-1,0), -int (0,4), b) (1,9), opens down, -int (-,0), (4,0), -int (0,8), c) (,8), opens down, -int (0,0), (4,0), -int 1 (0,0), d) (,0), opens up, -int (,0), -int (0,4), e) 1,, opens up, no -int, -int 5 0,, f) 7 4,, opens up, -int 4 ( 4+ 7,0 ),( 4 7,0), -int 9 0, ; 3a) and, b) 47 ft b 47 ft, c) 144 ft 4 M-51 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.1 Form I Name Date Use the square root propert to solve each equation. 1.. 3. 4. 5. 6. 5 16 81 3 4 3z 7 0 11 0 1.. 3. 4. 5. 6. 7. 3 4 7. Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. 8. 9. 10. 11. a z 16 1 10a z 8. 9. 10. 11. Solve each equation b completing the square. 1. 6 9 1. 13. 4 1 14. 6 16 13. 14. 15. 8 15 15. E-175 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.1 Form II Name Date Use the square root propert to solve each equation. 1.. 3. 144 7 1 0 1.. 3. 4. z 5 9 5. z 3 5 4. 5. 6. 7. 9 0 4 0 6. 7. Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. 8. 9. 10. 11. z r 14 18z 0r 8. 9. 10. 11. Solve each equation b completing the square. 1. 13. 14. 15. 6 8 3 4 9 5 0 1. 13. 14. 15. E-176 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.1 Form III Name Date Use the square root propert to solve each equation. 1.. 145 6 5 1.. 3. 3 18 4. 10 11 3. 4. 5. z 16 0 5. 6. 1 16 7. z 4 18 6. 7. Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. Then factor the trinomial. 8. 9. 10. 11. p r 9 5p 7r 8. 9. 10. 11. Solve each equation b completing the square. 1. 13. 14. 15. 7 0 3 4 0 3p 1 p 0 4 6 15 0 1. 13. 14. 15. E-177 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8. Form I Name Date Use the quadratic formula to solve each equation. 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. r 8 15 0 5r 6 0 6 9 0 10 5 0 1 3 0 5 5 5 4 1 5 4 0 0 8 4 6 1 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. 1. 1. m m 6 1 Use the discriminant to determine the number and tpes of solutions of each equation. 13. 14. 15. 5 0 1 0 4 1 9 13. 14. 15. E-178 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8. Form II Name Date Use the quadratic formula to solve each equation. 1. 6 10. 8 1 4 3. 3 1 4. 1 1.. 3. 4. 5. 6. 7. 8. 9. 10. z 5z 3 1 z 4z 4 9 1 6( r 1) 5 r 4 4 5. 6. 7. 8. 9. 10. 11. ( 9) 5 11. 1. 1 1 1. Use the discriminant to determine the number and tpes of solutions of each equation. 13. 14. 15. 9r 1 6 r 3 5 8 4 5 13. 14. 15. E-179 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8. Form III Name Date Use the quadratic formula to solve each equation. 1.. 3. 3 7 3 0 8 9 0 1 3 1 0 1.. 3. 4. r r 5 5 4. 5. 6. 7. 15 15 5 0 1 5 0 4 1 1 1 4 4 5. 6. 7. 8. 6 3 9. s 5s 3 4 10. 8. 9. 10. 11. 1. 1 a a 4 5 0 11. 1. Use the discriminant to determine the number and tpes of solutions of each equation. 13. 14. 15. 9 8 3 5 9 0 4 3 1 0 13. 14. 15. E-180 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.3 Form I Solve. 1. 6 13 Name Date 1.. 3. 4 5 0 4 3 0 4. 3. 4. 4 5 5. 7 1 0 6. 1 3 4. 5. 6. 7. 8. 9. 10. 3 1 1 1 3 4 1 a 1a 3 0 4 4 4 8 48 0 7. 8. 9. 10. 11. 16 64 11. 1. 13. 14. 3 3 4 1 0 7 6 0 3 1 3 15 0 3 1 3 1. 13. 14. 15. 4 3 15. E-181 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.3 Form II Name Date Solve. 4 1. 10 9 0 1. 4 0 1.. 3. 3 1 3. 4. 5 8 3 5. 4 3 4. 5. 6. 1 1 6. 7. z 4 z 7. 8. 1 1 8. 1 1 5 3 4 9. 9. 10. 11. 1. 13. 1 1 1 r 1 r 3 /3 1/3 6 8 4 3 1 10. 11. 1. 13. 14. 15. 3 3 0 4 9 5 4 0 14. 15. E-18 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.3 Form III Name Date Solve. 1. 5 4 1.. 3 1 1 3. ( 3) ( 3) 6 4. 3 1. 3. 4. 5. 6. d 14d 40 0 4 3 4 3 3 5. 6. 7. 5 6 5 1 5 8. 7 4 3 0 5 4 7. 8. 9. 10. 4 4 11 3 4 5 6 8 0 9. 10. 3 1 4(3 1) 5 0 11. 11. 1. 13. 14. 15. 3 1 3 9 5 3 1 3 3 1 14 3 1 3 15 7 0 0 3 4 4 b 3 b 5 1. 13. 14. 15. E-183 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.4 Form I Name Date Solve each quadratic inequalit. Write the solution set in interval notation. 1. 1 3 0. 0 3. 4 0 4. 5 3 0 5. 4 6 0 1.. 3. 4. 5. 6. 7. 8. 9. 4 3 0 7 1 0 1 3 5 6. 7. 8. 9. 10. 1 3 5 0 11. 3 4 0 1. 3 5 0 13. 4 9 0 14. 16 4 0 10. 11. 1. 13. 14. 15. 3 0 4 15. E-184 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.4 Form II Solve each quadratic inequalit. Write the solution set in interval notation. 1. ( 5)( ) 0. ( 7) 4 3. 4 3 0 4. 3 10 8 Name Date 1.. 3. 4. 5. 6. 7. m 1 m 5 3 4 7 5u 3u 5. 6. 7. 8. ( )( 4) 0 8. 9. 10. 11. ( 4)( 5) 0 3 0 4 9. 10. 11. 1. 1 5 1. 13. 3 0 7 13. 14. 3 1 14. 15. 1 4 15. E-185 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.4 Form III Solve each quadratic inequalit. Write the solution set in interval notation. 1.. 3. 3 5 4 1 4 3 9 18 0 Name Date 1.. 3. 4. 5. 6. 7. ( 8)( 3) 0 1 6 3 5 3 0 3 5 8 4. 5. 6. 7. 8. 3 5 3 0 8. 9. 10. 11. 1. 13. 14. 15. 3 4 4 5 3 5 3 4 7 3 3 4 1 0 5 144 0 4 5 6 0 9. 10. 11. 1. 13. 14. 15. E-186 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 Form I Name Date Sketch the graph of each quadratic function. Label the verte, and sketch and label the ais of smmetr. 1. f ( ) 1 1.. g ( ) 3. 3. h ( ) 5 3. 4. h ( ) 4 4. E-187 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) f ( ) 4 5. Name 5. g( ) 4 6. 6. f ( ) 7. 7. h( ) 1 8. 8. E-188 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) Name 9. f ( ) 3 9. 10. 1 g( ) 10. 11. h( ) 11. 1. 1 g( ) 4 1. E-189 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 Form II Name Date Sketch the graph of each quadratic function. Label the verte, and sketch and label the ais of smmetr. 1. f ( ) 7 1. g( ) 6.. h( ) 3. 3. h( ) 6 1 4. 4. E-190 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) f ( ) 3 4 5. Name 5. g( ) 3 6. 6. 7. f ( ) 4 7. 8. 1 h( ) 5 8. E-191 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) f ( ) 3 1 1 9. Name 9. g( ) 3 10. 10. h( ) 3 1 4 11. 11. g( ) 1 4 5 1. 1. E-19 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 Form III Name Date Sketch the graph of each quadratic function. Label the verte, and sketch and label the ais of smmetr. 1. f ( ) ( 4) 15 1.. f ) ( ( 3) 9. 3. 1 3 g( ) 4 3. 4. H ( ( ) 1) 1 4. E-193 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) Name 5. 3 13 G( ) 5. 6. 1 3 f( ) ( 1) 6. 7. f 1 ( ) 9 7. 8. h 3 ( ) 5 8. E-194 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.5 (cont.) Name 9. 1 f ( ) 4 9. 10. 1 g( ) 4 3 10. h( ) 5 4 11. 3 11. g( ) 5 6 1. 1 1. E-195 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 Form I Name Date Find the verte of the graph of each quadratic function 1.. 3. 4. 5. g ( ) 4 3 f ( ) 6 9 h ( ) 6 5 f ( ) 6 8 g ( ) 3 1.. 3. 4. 5. Find the verte of the graph of each quadratic function. Determine whether the graph opens upward or downward, find an intercepts, and sketch the graph. 6. f ( ) 5 4 6. 7. f ( ) 1 7. 8. f ( ) 9 8. E-196 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 (cont.) Name 9. f ( ) 16 9. 10. f ( ) 4 10. 11. f ( ) 6 5 11. 1. f ( ) 4 3 1. E-197 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 Form II Name Date Find the verte of the graph of each quadratic function 1. f ( ). g( ) 4 1 3. H( ) 8 4. G( ) 3 5. h( ) 4 4 1.. 3. 4. 5. Find the verte of the graph of each quadratic function. Determine whether the graph opens upward or downward, find an intercepts, and sketch the graph. 6. f ( ) 7 8 6. 7. f ( ) 7. 8. f ( ) 4 8. E-198 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 (cont.) 9. f ( ) 4 1 Name 9. 10. f ( ) 5 3 10. 11. f () 3 4 11. 1. f ( ) 3 1. E-199 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 Form III Name Date Find the verte of the graph of each quadratic function 4 1. f ( ) 1 3 3 1.. 3. 4. 5. f ( ) 1 1 f ( ) 8 f ( ) 3 f ( ) 3 17 0. 3. 4. 5. Find the verte of the graph of each quadratic function. Determine whether the graph opens upward or downward, find an intercepts, and sketch the graph. 6. f ( ) 4 6. 7. f ( ) 3 15 7. 8. 1 4 f ( ) 3 8. E-00 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Additional Eercises 8.6 (cont.) Name 9. 1 1 4 f ( ) 9. 10. f ( ) 3 4 9 10. 11. f ( ) 3 1 11. 1. f ( ) 3 13 10 1. E-01 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Section 8. Solving Quadratic Equations b the Quadratic Formula Objective: Derive the quadratic formula. Suggested Format: Small group format. Time: 15 minutes Stud pages 489 490. Then close the book and our notes and use the completing the square method on the equation a + b + c = 0 to derive the quadratic formula. A- 19 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Section 8.5 Quadratic Functions and Their Graphs Objective: Use the a quadratic equation with real life data. Suggested Format: Small Group Time: 0 minutes A manufacturing compan found the that the equation f ( ) = 3. + 68.+ 57 could be used to approimate their profit per da for items produced. 1. Find the profit if 40 items are made.. Find the profit if 50 items are made. 3. How man objects would need to be made to get a profit of $543? 4. How man objects should be sold to maimize profit? 5. What is the maimum profit? A- 0 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form A Solve b completing the square. 1.. + 3 = 0 1. + =. 3 6 15 0 Solve using an appropriate technique. 3. ( 3) = 7 3. 4. 5. 6. 7. 8. 9. 10. + 7 60 = 0 4. = 5. 3 8 16 0 14 + 50 = 0 6. = 7. 10 11 6 0 6 1 = 8. 3 = 9. 8 0 10 + 4 = 0 10. 4 1 3 11. = + 1 + 1 1 11. 1. 1 = + 1 1. 3 13. ( ) ( ) + 1 = 0 13. Solve the inequalit and write the solution in interval notation. 14. 3 10 0 14. 15. 3 < 0 5 15. 16. 3 > 0 16. + 5 T- 159 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form A cont d 17. + 1 > + 3 17. 18. A square picture frame will hold a poster with 18. an area of 196 square inches. Find the dimensions of the frame. 19. A whole number increased b its square is 4 19. times itself. Find the number. 0. A ball is thrown upward from a 100-foot tall 0. building with an initial velocit of 14 feet per second. Its height s( t) is given b the function ( ) s t = 16t + 14t + 100. Find the interval of time for which the ball is greater than 103 feet. 1. Write the equation of a parabola with the shape 1. as f ( ) = but with the verte ( 3, ).. Find the verte of the graph of. f = 3 1 + 5 and determine whether ( ) the graph opens upward or downward. T- 160 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form A cont d Give the verte, determine whether the graph opens upward or downward, and graph. 3. = + 8 4. h( ) = 3( + 1) 3 f = + 3 + 5. Give the verte and -intercepts and graph. ( ) T- 161 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form B Solve b completing the square. 1.. + 4 = 0 1. + =. 3 1 0 Solve using an appropriate technique. 3. ( 3) = 18 3. 4. 5. 6. 7. 5 50 = 0 4. = 5. 3 1 0 = 6. 4 5 3 + = 7. 7 0 8. 3 6 + 1 = 8. 9. 10. 4 + = 3 9. 8 4 1 3 3 5 + 6 = 0 10. 11. + 1 = 4 11. 1. 13. = 1. 3 7 3 4 + 4 = 0 13. Solve the inequalit and write the solution in interval form. 14. ( )( ) 3 + 5 > 0 14. 15. 0 + 3 15. T- 16 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form B cont d 16. 4 > 0 9 16. 17. + 6 17. 18. Find the verte of the graph of 18. f = 4 + 1 + 7 and determine whether ( ) the graph opens upward or downward. 19. Find the verte and -intercepts of the graph 19. h = 9 6 3. ( ) Graph the parabolas. Label the verte. 0. f ( ) ( ) = 3 4 1. = 4 4 Find the -intercepts. T- 163 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form B cont d. Give the verte f ( ) ( ) = 1 1 + 1 and graph. t 3. Use the formula A P( 1 r) = + to find the 3. r to make $1600 grow to $1764 in ears. 4. A rectangular window needs to have an area 4. 1 of 6 square feet and its height must be 1 times its width. Find the dimensions of the window. 5. A trailer compan has found that the revenue 5. from sales of heav-dut truck trailers is a function of the price, p, it charges. If the 3 revenue R is given b R = p + 700 p, 4 what unit price p should be charged to maimize revenue? What is the maimum revenue? T- 164 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form C Solve b completing the square. 1.. + 6 = 0 1. + =. 4 16 40 0 Solve using an appropriate technique. 3. ( + 3) = 3 3. 4. 5. 6. 7. 3 8 = 0 4. = 5. 6 17 1 + = 6. 3 6 0 1 3 3 + 1 = 0 7. 4 + 4 = 9 + 1 8. 8. ( ) 9. 10. 11. 1. 13. z z = 9. 4 3 5 0 1 54 + = 10. 7 7 1 + 4 = 1 11. 5 + 4 = 0 1. 4 = 13. 6 1 0 Solve the inequalit and write the solution in interval notation. 14. ( )( ) 6 3 + 5 > 0 14. 15. 16. 7 0 + 3 < 0 + 1 15. 16. T- 165 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form C cont d 17. 7 0 + 17. 18. 5 3 19. z z z 6 18. 19. 0. Find the verte of the graph of 0. f = 8 11 +. ( ) Graph and find the verte. f = 5 + 6 1. ( ) ( 3) g = + + 4. ( ) 3. Trone and Elias are painting a house. 3. Together it will take them 4 hours to paint the house. Trone can paint the house in 6 hours less than Elias working alone. How long to the nearest hour would it take each of them working alone to paint the house? 4. The hpotenuse of a right triangle is 3 meters and 4. legs are equal in length. Find the length of a leg in the triangle. 5. Find two positive numbers whose sum is 4 and 5. whose product is as large as possible. T- 166 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form D Circle the correct answer. 1. The perfect square trinomial that results from completing the square in + 9 is a. 81 + 9 + b. + 9 + 9 c. + 9 + d. 9 81 + 9 + 4. Solve b completing the square. + 4 = 6 a. ± 6 b. ± 6 c. ± 10 d. ± 6 3. The equation 6 4 = has a. two real solutions. b. one real solution. c. two comple solutions d. no solution 4. The equation ( )( ) + 7 1 = 16 has a. two real solutions. b. one real solution. c. two comple solutions d. no solution Use the appropriate method to solve each of the following equations. 5. ( 3 + 6) = 18 a. 3, + 3 b. 3, + 3 c., + d., + 6. + = 64 0 a. 8 i,8i b. 8, 8 c. 4, 4 d. 4 i, 4i 7. 1 3 3 6 + 8 = 0 a., 4 b. 4, 16 c.,8 d. 8, 64 T- 167 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form D cont d 8. + 5 1 = 0 a. c. 5 73 5 73, + 5 i 3 5 + i 3, b. d. 5 73 5 73, + 5 i 3 5 i 3, 9. 3 16 = 0 a. b., 1 i 3, 1+ i 3 c. d., 1 i 3, 1+ i 3 10. 6 7 = 0 4 a. 7, 7, i, i b. 1,1, i 7, i 7 c. i 7, i 7, i, i d. 7, 7, 1,1 11. 3 8 = 0 a.,4 b. 3 4, c. 3 4, d. 3 4, 3 1. + 4 = 0 a. 6, + 6 b. 6, + 6 c. 3, + 3 d. 4 6, 4 + 6 13. 3 = 15 0 a. 5 5 i 3, 5 + 5 i 3,5 14. ( ) ( ) 3 + 3 3 18 = 0 b. 5 c. 5, 5 d. 5 5i 3 5 + 5i 3,,5 a. 6, 1 b. 7, 4 c. 6, 3 d. 3, 6 T- 168 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form D cont d 15. 3 = 7 + a. c. 13 57 13+ 57, 14 14 14 57 14 57, + 13 13 b. d. 57 13 57 13, + 14 14 13 57 13 57, + 14 14 16. If a square room has a diagonal of 38 feet, the sides of the room have length a. 1444 feet b. 19 feet c. 38 feet d. 38 feet 17. Solve. + 1 a. [ 3, ] b. [, 3] c. (, 3] [, ) d. (, ] [3, ) 18. Find the equation that has the solution (, ] (0, ) a. 3( + ) > 0 b. ( ) 3 + 0 c. + 3 0 d. 0 3 + 19. Solve. m m 3 + 4 < 1 a. c. 1 1, 3 1 1, 3 1, 1, 3 1, 1, 3 b. ( ) d. ( ) 0. The graph of ( ) f = 4 7 has its verte at and opens. a. (, 3); downward b. (, 3); downward c. (, 3); upward d. (, 19); downward 1. The ais of smmetr of the graph of ( ) f = 4 + 8 0 is a. = b. = 1 c. = d. = 1 T- 169 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form D cont d. Graph f ( ) ( ) = 3. a. b. c. d. T- 170 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form D cont d 3. Graph = + 3. a. b. c. d. 4. The difference between the square of a negative number and five times the number is 40 more than the number. The number is a. 5 b. 10 c. 4 d. 8 5. A large pipe and a small pipe can fill a tank in 6 hours. The small pipe alone takes two more hours than the large pipe alone. Find the time it would take the large pipe alone to fill the tank. a. 1.1 hours b. 3.1 hours c. 13.1 hours d. 11.1 hours T- 171 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form E Circle the correct answer. 1. The possible missing terms to make + + a perfect square trinomial are 100 a. 4,5 b. 10,10 c. 0,0 d. 4, 5. The perfect square trinomial that results from completing the square 16 is a. 16 + 64 b. 16 64 c. 16 + 4 d. 16 8 3. The equation = has 5 10 a. two comple solutions b. one real solution c. two real solutions d. no solution 4. The equation ( 8) = 16 has a. two comple solutions b. one real solution c. two real solutions d. no solution Solve each equation using an appropriate method. 5. ( 4) = 3 a. 4, + 4 b. 4, + 4 c., + d., + 6. + = 5 0 a. 5 i,5i b. 5 c. 5, 4 d. no solution 7. 3 = 64 a. 4 b. 4, i 3, + i 3 c. 4, 4 i,4i d. 4, 4i 3, + 4i 3 8. ( ) ( ) + 4 5 + 4 1 = 0 a. 3,4 b. 11,0 c. 4 d. 4 T- 17 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form E cont d 9. 3 1 + 0 = 0 a. 6 i 6, 6 + i 6 b. 1 4 i, 1 + 4i 3 3 c. 4i 6, + 4i 6 d. 4i 6, + 4i 6 10. 8 = 0 1 a., 4 b. 4, 3 c. 1 1, d. no solution 4 11. + 6 < 0 a. [ 3,] b. (, 3) (, ) c. ( 3, ) d. (, 3) 1. ( + 3) ( )( + 4) 0 a. ( 4,3] [0,) b. [ 4, 3] [0,] d. (, 4) [ 3,0] (, ) c. ( 4, 3] (0,) 13. 5 0 + 8 a. (, 5] (8, ) b. [ 5, ) c. ( ), 8 [5, ) d. ( 8, 5] 14. The equation of the parabola with the same shape as ( ) at ( 3, ) is a. f ( ) = 9( + 3) b. f ( ) ( ) f = 9 but with the verte = 9 + 3 + c. f ( ) = 9( ) 3 d. f ( ) ( ) = 9 3 + 15. What is the ais of smmetr of 5 =? a. 5 = b. = c. 5 5 = d. = 5 T- 173 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form E cont d 16. Graph f ( ) ( ) = + 3. a. b. c. d. T- 174 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form E cont d 17. Graph = + 3. a. b. c. d. 18. The graph of f ( ) ( ) opens. = 9 + 16 is a parabola with verte which a. (9, 16); upward b. (9, 16); upward c. (9, 16); downward d. (9, 16); downward 19. The minimum value of ( ) f = 1.3 6.5 + 10.35 occurs when = a..5 b..5 c..5 d. 0. The equation = 3 + 4 + 1 has a maimum value of a. 5 b. 7 3 c. 7 1 d. 9 T- 175 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form E cont d 1. If the hpotenuse of an isosceles right triangle is 3 inches longer than either leg, find the length of the hpotenuse. a. 3+ 3 inches b. 6 + 3 inches c. 6 3 inches d. 3 inches. A rocket fired straight up from the ground with an initial velocit of 96 feet per second is at height s( t) = 16t + 96t at an time t. The height of the rocket is greater than 18 feet for t between a. 1 and 4 seconds b. and 4 seconds c. 1 and seconds d. 4 and 6 seconds 3. What is the maimum product of two numbers if their sum is 7? a. 160 b. 36 c. 115 d. 196 A compan kept track of its sales for the 4 months of a ear and found that the equation f = 3.4 9.1 + 98 gives the monthl sales in thousands of dollars, where represents ( ) the month of the ear. Use this equation for problems 4 and 5. 4. What would be the sales for the sith month? a. $18,400 b. $165,800 c. 159,600 d. $173,000 5. What month would give $4,800 in sales? a. 10 th b. 9 th c. 5 th d. 8 th T- 176 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form F Circle the correct answer. 1. The binomial that results from completing the square in 1 is a. 1 + 144 b. 1 + 36 c. 1 + 6 d. 1 + 1. The possible missing terms to make + + as perfect square trinomial are 81 a. 9, 9 b. 3,3 c. 9,9 d. 18,18 3. The equation + = has 49 0 a. two comple solutions b. two real solution c. one real solutions d. no solution Solve. 4. ( 5 + 10) = 40 a. 10 10 10 10, + 5 5 b., + c. 10, + 10 d. 10, + 10 5. 4 + 5 = 0 a. c. 1 i 19 1+ i 19, 1 i 19 1+ i 19, 4 4 1 i 1 1+ i 1 b., 4 4 d. no solution 6. 3 50 = 0 a. 5 b. c. 5 d. 5 5i 3 5 + 5i 3 5,, 5 5i 3 5 + 5i 3 5,, T- 177 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form F cont d 7. 8 9 = 0 4 a. 1, 1, 3, 3 b. 1,, 3i, 3i c. i, i, 3, 3 d. i, i, 3i, 3i 8. 3 4 = + 4 + 4 a. c. 11 3 17 11 3 17, + 4 4 5 3i 15 5 + 3i 15, 4 4 b. d. 5,4 11 3 17 11 3 17, + 9. p 1 3 3 + 8 = 11p a., 7 b. 4, 7 c. 8,7 7 d. 64, 343 10. + 3 0 = 0 a. 5 4, b. 5,4 c. 1 4, d. 1,4 11. ( ) ( ) 5 5 1 = 0 a. 3, 4 b., 9 c. 4, 3 d. 1, 7 1. = 3 a., b. 8, 8 c. 4, 4 d., 13. < 4 0 a. (, ) (, ) b. (, ) c. (, ) d. (, ) 14. 6 0 a. [3, ) b. [, 6] c. (,] [ 6, ) d. (,) [ 6, ) T- 178 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form F cont d 15. The equation of the parabola that has the same shape as ( ) opposite direction and has verte (8, 1) is a. f ( ) = 4( 8) + 1 b. f ( ) ( ) = 4 + 8 1 c. f ( ) = 4( + 8) + 1 d. f ( ) ( ) = 4 8 1 f = 4 but opens in the 16. The equation of the parabola that opens downward with the verte (1, 5) is a. f ( ) = ( 1) 5 b. f ( ) ( ) = 1 + 5 c. f ( ) = ( 1) + 5 d. f ( ) ( ) 17. The ais of smmetr of the graph ( ) = + 5 + 1 f = 3 + 9 + 1 is a. 9 = b. = 3 c. 9 = d. 3 = 18. The minimum value of ( ) f = 34 + 307 a. 17 b. 307 c. 1 d. 18 19. What is the equation of the following graph? a. 3 6 = + b. 3 6 = + c. 6 3 = = d. = + 6 T- 179 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form F cont d 0. What is the equation of the following graph? a. f ( ) = ( 1) + b. f ( ) ( ) = 1 c. f ( ) = ( + 1) d. f ( ) ( ) 1. Graph ( ) f = 4 + 7. = + 1 + a. b. c. d. T- 180 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall
Name: Instructor: Date: Section: Chapter 8 Test Form F cont d. Graph ( ) = + 1. a. b. c. d. 3. A rectangle is 4 times a long as it is wide. If the length of the diagonal is 51 centimeters, the length of the rectangle is a. 3 17 cm b. 1 17 cm c. 17 cm d. 3 cm 4. The diagonal of a square is 6 meters more than one of its sides. What is the length of a side in meters? a. 3 3 m b. 6 6 m c. 3+ 3 m d. 6 + 6 m 5. George and Cind together can stock the shelves of the local market in 5 hours. When she works alone, it takes Cind 1 hour longer to stock than it takes George working alone. About how long does it take George to stock the shelves? a. 9.5 hrs b. 11.54 hrs c. 10.5 hrs d. 10.54 hrs T- 181 Copright 009 Pearson Education, Inc., publishing as Pearson Prentice Hall