# Unit 5: Quadratic Equations & Functions

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1 Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton 5 Comple Numbers Dvson 6 Completng the Square 7 The Quadratc Formula Dscrmnant 8 QUIZ 9 Propertes of Parabolas 10 Translatng Parabolas 11 Graphs of Quadratc Inequaltes and Systems of Quadratc Inequaltes 1 Applcatons of Quadratcs (Applcatons WS) 13 REVIEW

2 Date Perod U5 D1: Modelng Date wth Quadratc Functons The study of quadratc equatons and ther graphs plays an mportant role n many applcatons. For nstance, physcsts can model the heght of an object over tme t wth quadratc equatons. Economsts can model revenue and proft functons wth quadratc equatons. Usng such models to determne mportant concepts such as mamum heght, mamum revenue, or mamum proft, depends on understandng the nature of a parabolc graph. f ( ) = a + b + c a - term b - term c - term Standard Form: f ( ) = a + b + c a postve a negatve Ma or Mn? Verte As of Symmetry y-ntercept Property Eample: y =

3 Sometmes we wll need to determne f a functon s quadratc. Remember, f there s no words, a = 0 ), then the functon wll most lkely be lnear. term (n other When a functon s a quadratc, the graph wll look lke a (sometmes upsde down. When?). We talked a lttle about an as of symmetry what does symmetry mean?! Use symmetry for the followng problems: Warmup: Quck revew of graphng calculator procedures Fnd a quadratc functon to model the values n the table below shown: Step 1: Plug all values nto Step : Solve the of 3 varables. (Favorte solvng method?) Step 3: Wrte the functon *Note: If a = 0

4 Sometmes, modelng the data s a lttle too comple to do by hand Graphng Calc! c. What s the mamum heght? d. When does t ht the ground? The graph of each functon contans the gven pont. Fnd the value of c. 1) y 5 c; (, 14) = + ) y = + c ; 3, Closure: Descrbe the dfference between a lnear and quadratc functon (both algebracally & graphcally). Lst 3 thngs that you learned today. 3

5 Date Perod U5 D: Factorng Quadratc Epresson GCF: Dfference of Squares: 49 ( + 1) Guess & Check: ( ) ( ) Brtsh Method:

6 Factor the followng. You may use the Brtsh method, guess and check method, or any other method necessary to factor completely p w + w 6. ( + 1) + 8( + 1) ( ) ( ) ( 3) 7( 3)

7 y ( ) ( ) ( ) ( 4 49) 7. ( )

8 Date Perod U5 D3: Solvng Quadratc Equatons Objectve: Be able to solve quadratc equatons usng any one of three methods. Factorng Takng Square Roots Graphng + 18 = 9 9 = = 0 Addtonal Notes: Partnered Unfar Game! 7

9 Date Perod U5 D4: Comple Numbers Intro & Operatons (not Dvson) 1. On your home screen, type 9. What answer does the calculator gve you?. Go to MODE and change your calculator from REAL to a + b form (3 rd row from the bottom) 3. On your home screen, type 9 agan. Ths tme what answer does t gve you? 4. Use the calculator to smplfy each of the followng: a. 5 b. 9 4 c. 100 Now look for the on your calculator (t s the nd. near 0), then calculate each of the followng: a. b. ( )( 5 3) + c. ( 4)( 1+ ) 5. From your nvestgaton, what does represent? What knd of number s? 6. What s the meanng of a + b? Imagnary numbers are not nvsble numbers, or made-up numbers. They are numbers that arse naturally from tryng to solve equatons such as + 1= 0 Imagnary numbers : the number whose square s -1. = = Smplfy the followng:

10 Comple number: magnary numbers and real numbers together. a and b are real numbers, ncludng 0. a + b REAL PART IMAGINARY PART Smplfy n the form a + b 5. Wrte the comple number n the form a + b You can use the comple number plane to represent a comple number geometrcally. Locate the real part of the number on the horzontal as and the magnary part on the vertcal as. You graph 3 4 the same way you would graph (3,-4) on the coordnate plane. 1 Imagnary as Real as (3-4) 6. On the graph above, plot the ponts and

11 Absolute value of a comple number s ts dstance from the orgn on the comple number plane. To fnd the absolute value, use the Pythagorean Theorem. Fnd the absolute value of the followng a + b = a + b Addtve Inverse of Comple Numbers Fnd the addtve nverse of the followng: a + b Addng/Subtractng Comple Numbers 14. ( 5+ 7) + ( + 6) 15. ( 8+ 3) ( + 4) 16. ( ) Multplyng Comple Numbers 17. Fnd ( 5)( 4) 18. ( + 3)( 3+ 6) 19. ( 1)( 7 ) 0. ( 6 5)( 4 3) 1. ( 4 9) ( 4 3) ( 3 ) Fndng Comple Solutons. Solve = = = = 0 Closure: What are two comple numbers that have a square of -1? 10

12 Date Perod Warmup: Fll n the table U5 D5: Comple Numbers & Comple Dvson Generalze ths cyclc concept to fnd the followng: 80 =, 133 =, 1044 = = 3 4 = = = Dvde the eponent by 4 and fnd the remander Match the remander the chart on the left. Use that value as your answer. The conjugate of a+b s a-b (note t s NOT the nverse), and the conjugate of a-b s a+b Eamples ; the conjugate s ; the conjugate s ; the conjugate s -5 snce the conjugate of 0+5 s ; the conjugate of 6 s 6 snce 6-0 s the conjugate of 6+0 Comple dvson To dvde comple numbers, multply the numerator and demonnator by the conjugate of the denomnator

13 Date Perod Worksheet U5 D

14 Date Perod U5 D6: Completng the Square Another solvng method for quadratcs s completng the square. The goal s to get the left sde of your equaton to be n the form of ( + #) so that you can take the of both sdes. Quck eample: = 36 Epressons lke because they factor nto ( ) # are called + # + #. + nstead of two dfferent bnomals ( )( ) Unfortunately, sometmes our epresson on the left s not a perfect square. Soluton: the square to make t perfect! Eamples: 1 1) The value that completes the square s always ) 7 + 3) + Now let s apply ths process to solvng an equaton. Eample #1: 5= 0 STEP 1: Get the equaton n the form (move the # s to the rght). STEP : Fnd the amount to be added by takng. STEP 3: Add that amount to both sdes. + = 5 + STEP 4: Factor the left sde and smplfy the rght STEP 5: Take the square root of both sdes. Eample #: = 0 13

15 Notce n the prevous eamples, a = 1. If t does not, we have to t! Eample #3: = Eample #4: = Eample #3: The equaton ht () = t + 3t+ 4models the heght, h n feet, of a ball thrown after t seconds. Complete the square to fnd how many second t wll take for the ball to ht the ground. Classwork Eamples: = 0. 4 = + 3. = The equaton ht () = t + t+ 3models the heght, h n feet, of a ball thrown after t seconds. Complete the square to fnd how many second t wll take for the ball to ht the ground = 0 6. =

16 Date Perod U5 D7: The Quadratc Formula & Dscrmnant When gven an quadratc equaton, we have learned several ways to solve Factor (f applcable), the square, takng square roots, and. Today we wll (re?)learn another method: Everyone s favorte, the formula!!!! If a b c + + = 0, then Dscrmnant Eample 1: 4+ 3= 0 ) = 0 3) 1= 5 Drectons: Just fnd the dscrmnate for each equaton 4) = 0 5) 4 5= 0 6) = 0 The determnant can tell us about the graph and the number of solutons, and even the solvng methods 15

17 On the frst day of the unt, we looked how the values of a quadratc functon effect the graph Look of Graph Dscrmnant Soluton Types Solvng Method WoRdIE: The functon ht ( ) 16t 1t = + models the heght of a bowlng ball thrown nto the ar. Use the quadratc formula to fnd the tme t wll take for the ball to ht the ground. Then, use your calculator to fnd the tme t wll take for the ball to ht the ground (check). Fnally, use your calculator to fnd the tme of the mamum heght, and what that ma heght s More classwork eamples on the net page 16

18 17

19 Date Perod U5 D9: Propertes of Parabolas Forms: ( ) y = 3 y = Quadratcs!!! General Equaton Verte As of Symmetry Intercepts Standard Form Verte Form Today we wll focus more on standard form, and tomorrow we wll cover verte form. Drectons: For each equaton, fnd (a) the verte, (b) the as of symmetry, and (c) the y-ntercept. 1. y = 6 +. y = y =

20 Now we are gong to graph the parabolas of the quadratc functons. 1. y = STEP 1: Fnd the verte. V: STEP : Fnd the as of symmetry AoS: STEP 3: Fnd the y-ntercept. & ts match STEP 4: Fnd one more pont by choosng a value for. Addtonal Informaton: Mn or Ma -ntercepts 19

21 Applcaton: Suppose you are tossng a baseball up to a frend on a thrd-story balcony. After t seconds the heght of the apple n feet s gven by the functon ht ( ) = 16t t Your frend catches the ball just as t reaches ts hghest pont. How long does the ball take to reach your frend, and at what heght does he catch t?! Convertng Forms: Verte Standard ( ) y = Standard Verte y = + 6 (You must complete the square!!!!!!!!!!!!!!!!) Closure: What are the general equatons for standard and verte form of a quadratc? Lst how you can fnd mportant nformaton from each (such as verte, as of symmetry, ntercepts, etc ) 0

22 Date Perod U5 D10: Translatng Parabolas 1. Revew the general equaton for verte form and standard form of a quadratc. Identfy the verte and the y-ntercept from the equatons below a) y = ( 4) + 3 b) y ( ) = + 5 c) y = We wll graph verte form n a smlar way that we dd standard from, ecept now the verte s easy! y = 1 ( ) + 1 STEP 1: Fnd the verte. V: STEP : Fnd the as of symmetry AoS: STEP 3: Fnd another pont. & ts match STEP 4: Repeat step 3 4. Graph each of the followng: a) y = ( + ) 3 b) y ( ) =

23 5. Sometmes we wll need to wrte the equaton of the parabola. Step 1: Locate the Verte Step : Locate another pont Step 3: Plug n to y = a( h) + k and solve for a. 3. verte s ( 3, 6 ) and y-ntercept s 4. verte s ( 3, 6) and pont s ( 1, ) Closure: the equaton of one of the parabolas n the graph at the rght s ( ) y = 4 +. Wrte the equaton of the other parabola. Then, f you have tme, wrte both equatons n standard form, and dentfy the y-ntercepts.

24 Date Perod U5 D11: Graphs of Quadratc Inequaltes & Systems Warm-up: For each nequalty, dentfy above/below and sold/dashed <, >,, Graph the followng: 1. y > 3. y y y > y <

25 Date Perod U5 D1: Applcatons of Quadratcs Worksheet 4

26 Date Perod U5 D13: Revew for Unt 5 Test Problems 1 7 should all be done by hand. The calculator can be used for Answers should be left n smplest radcal form. 1. Wrte the equaton of the parabola n standard form through the ponts (, 7), (-1, 10) and (0, 5).. Wrte the equaton of the parabola wth a verte of (3, 1), through the pont (-1, -15). 3. Wrte each of the followng equatons n verte form by completng the square (f not done already). Sketch the graph by determnng the verte, the lne of symmetry, the y-ntercept, and the -ntercept(s) f they est. a. y = b. y = 1 c. = y d. y = ( + ) Solve each quadratc equaton. Use a varety of methods. a. + 4 = 1 b. 5 5 = 0 c = 5 d. + = 10 e = 0 f. + = 5

27 5. Smplfy each epresson nto a+b form. Show all work. 4 3 a. (8+4)(1-3) b. ( 3 6 ) c. 111 (smplfy- hnt: fnd remander) d f. 5 e. 3 5( 1 + 8) 6. Evaluate the dscrmnant and determne the type and number of solutons. a = 0 b = 0 7. Wrte an equaton n whch the dscrmnant s equal to -9. What type of solutons does your equaton have? 8. Graph the system of quadratc nequaltes. Shade the regon and fnd the ntersecton ponts. y 8 y ( 4) 9. The equaton y = represents the parabolc flght of a certan cannonball shot at an angle of 6, where y s the heght of the cannonball and s the vertcal dstance traveled n meters. Try ths WINDOW [-5, 60, 5, -1, 10, 1], ths follows the order of mn, ma etc. a. What s the mamum heght of the cannonball? How do you know? Eplan your method. b. What s the total horzontal dstance traveled by the cannonball? How do you know? Eplan your method. 10. A rectangular backyard wll be fenced n on 3 sdes. If there s 00ft of fencng, a. Determne the dmensons of the fence for the mamum area. b. Determne the mamum area. 6

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