Etra Eample. Graph.. 6. 7. (, ) (, ) REVIEW KEY VOCABULARY quadratic function, p. 6 standard form of a quadratic function, p. 6 parabola, p. 6 verte, p. 6 ais of smmetr, p. 6 minimum, maimum value, p. 8 verte form, p. intercept form, p. 6 monomial, binomial, trinomial, p. quadratic equation, p. VOCABULARY standard form of a quadratic equation, p. root of an equation, p. zero of a function, p. square root, p. 66 radical, radicand, p. 66 rationalizing the denominator, p. 67 conjugates, p. 67 imaginar unit i, p. 7 comple number, p. 76 standard form of a comple number, p. 76 Multi-Language Glossar Vocabular practice imaginar number, p. 76 pure imaginar number, p. 76 comple conjugates, p. 78 comple plane, p. 78 absolute value of a comple number, p. 79 completing the square, p. 8 quadratic formula, p. 9 discriminant, p. 9 quadratic inequalit in two variables, p. 00 quadratic inequalit in one variable, p. 0 best-fitting quadratic model, p.. WRITING Given a quadratic function in standard form, eplain how to determine whether the function has a maimum value or a minimum value. If a < 0, the function has a maimum value and if a > 0, then the function has a minimum value.. Cop and complete: A(n)? is a comple number a bi where a 0 and b Þ 0. pure imaginar number. Cop and complete: A function of the form a( h) k is written in?. verte form. Give an eample of a quadratic equation that has a negative discriminant. Sample answer: (, ) REVIEW AND Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of Chapter.. Graph Quadratic Functions in Standard Form pp. 6 Graph. Because a < 0, the parabola opens down. Find and plot the verte (, ). Draw the ais of smmetr. Plot the -intercept at (0, ), and plot its reflection (, ) in the ais of smmetr. Plot two other points: (, ) and its reflection (, ) in the ais of smmetr. Draw a parabola through the plotted points. (, ) EXAMPLE on p. 8 for Es. 7 Graph the function. Label the verte and ais of smmetr. 7. See margin.. 6. 7 7. f() 6 8 Chapter Quadratic Functions and Factoring 8
,, and on pp. 7 for Es. 8. Graph Quadratic Functions in Verte or Intercept Form pp. Graph ( )( ). Identif the -intercepts. The quadratic function is in intercept form a( p)( q) where a, p, and q. Plot the -intercepts at (, 0) and (, 0). Find the coordinates of the verte. p q () ( )( ) 9 Plot the verte at (, 9). Draw a parabola through the plotted points as shown. Graph the function. Label the verte and ais of smmetr. Chapter Review Practice (, 0) (, 0) 6 (, 9) 8. ( )( ) 9. g() ( )( ) 0. ( )( 6). ( ). f() ( 6) 8. ( 8). BIOLOGY A flea s jump can be modeled b the function 0.07( ) where is the horizontal distance (in centimeters) and is the corresponding height (in centimeters). How far did the flea jump? What was the flea s maimum height? cm; about 0 cm 8. See margin. Etra Eample. Graph ( ) Etra Eample. Solve 0.,.. (, ). Solve b c 0 b Factoring pp. 8 (6, 8) Solve 8 0. Use factoring to solve for. 8 0 ( 6)( ) 0 Factor.. 8 6 6 0 or 0 Zero product propert (8, ) 6 or Solve for. EXAMPLE on p. for Es.. 0, 0 6. z 6z 0, 6 7. s 6s 7 0, 9 8. k k 0, 9. 8 8 9 0. n n 8,. URBAN PLANNING A cit wants to double the area of a rectangular plaground that is 7 feet b 8 feet b adding the same distance to the length and the width. Write and solve an equation to find the value of. (7 )(8 ) (7)(8); ft Chapter Review 9 8. 9. 0. (, 6 ) (, 9) (, 6 ) 9
Etra Eample. Solve z 7z 0., Etra Eample. Solve ( 8) 08. 8 Ï 6, 8 Ï 6 Etra Eample.6 Write i as a comple number i 7 in standard form. 7 i. Solve a b c 0 b Factoring pp. 9 6 Solve 0 9 0. 0 9 0 0 0 Divide each side b. ( )( ) 0 Factor. 0 or 0 Zero product propert or Solve for. EXAMPLE on p. 6 for Es... 6 8r r. 8 0. 0a a 0, 6 Ï, Solve Quadratic Equations b Finding Square Roots pp. 66 7 Solve ( 7) 80. ( 7) 80 ( 7) 0 Divide each side b. 7 6 Ï 0 7 6 Ï Take square roots of each side. Add 7 to each side and simplif. and on pp. 67 68 for Es. 8. 08 66 6. 6 Ï 7. (p ) 8 6 Ï 8. GEOGRAPHY The total surface area of Earth is 0,000,000 square kilometers. Use the formula S πr, which gives the surface area of a sphere with radius r, to find the radius of Earth. about 67 km.6 Perform Operations with Comple Numbers pp. 7 8 Write (6 i)( i) as a comple number in standard form. (6 i)( i) 6 8i i i Multipl using FOIL. 6 i () Simplif and use i. 6 i Write in standard form. 0 Chapter Quadratic Functions and Factoring 0
,, and on pp. 76 78 for Es. 9.7 Write the epression as a comple number in standard form. 9. 9i( i) 9 8i 0. ( i)( i) 6i. ( i)( i) 9. (8 6i) (7 i). ( i) (6 i) i i i. 8 6i i Complete the Square pp. 8 9 Solve 8 0 b completing the square. 8 0 8 Write left side in the form b. 8 6 6 Add 8 () 6 to each side. ( ) Write left side as a binomial squared. Chapter Review Practice Etra Eample.7 Solve 0 7 0 b completing the square. Ï, Ï Etra Eample.8 Solve 8 0. i Ï 9, i Ï 9 6 Ï Take square roots of each side. 6 Ï Solve for. and on pp. 8 86 for Es. 7.8 Solve the equation b completing the square.. 6 0 6 Ï 6 6. 0 6 Ï 7. 0 6 Ï Use the Quadratic Formula and the Discriminant pp. 9 99 Solve 6. 6 6 0 6 6 Ï 6 ()() () 6 Ï Write in standard form. Use a, b 6, and c in quadratic formula. Simplif.,,, and on pp. 9 9 for Es. 8 Use the quadratic formula to solve the equation. 6 i Ï 8. 0 6 Ï 7 9. 9 6 0. 6 6 8. VOLLEYBALL A person spikes a volleball over a net when the ball is 9 feet above the ground. The volleball has an initial vertical velocit of 0 feet per second. The volleball is allowed to fall to the ground. How long is the ball in the air after it is spiked? about 0. sec Chapter Review
Etra Eample.9 Solve 6 < 0. approimatel.7 < <.7 Etra Eample.0 Write a quadratic function for the parabola shown. verte (, ) (6, ) ( ) 8. 7 ( ) 7.9 EXAMPLE on p. 0 for Es..0 Graph and Solve Quadratic Inequalities pp. 00 07 Solve 0. The solution consists of the -values for which the graph of lies on or below the -ais. Find the graph s -intercepts b letting 0 and using the quadratic formula to solve for. 6 Ï ()() () 6 Ï 6 Ï ø.6 or ø.6 Sketch a parabola that opens down and has.6 and.6 as -intercepts. The solution of the inequalit is approimatel.6 or.6..6.6 Solve the inequalit b graphing.. < 0. 0. 6 > 0 0. < < 0.6.6 < 7.6 or >.6 Write Quadratic Functions and Models pp. 09 Write a quadratic function for the parabola shown. Because ou are given the -intercepts p and q, use the intercept form a( p)( q) a( )( ). Use the other given point, (, ), to find a. a( )( ) Substitute for and for. (, ) a Simplif coefficient of a. a Solve for a. c A quadratic function for the parabola is ( )( ). and on p. 09 for Es. 8 Write a quadratic function whose graph has the given characteristics.. -intercepts:, 6. passes through: 7. verte: (, 7) passes through: (, ) (, ), (0, ), (8, 6) passes through: (, ) ( )( ) ( )( 6) 8. SOCCER The parabolic path of a soccer ball that is kicked from the ground ( ) 7 passes through the point (0, 0) and has verte (, 7) where the coordinates are in feet. Write a quadratic function that models the soccer ball s path. See margin. Chapter Quadratic Functions and Factoring