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Kty Name Class Date Writing Quadratic Functions in Vertex F rrn : COMMON CORE ••.......* CC.9-12ASSE.3b*, CC.9-12ACED.2*, CC.9-121.1F.2, CC.9-12.EIE7*. CC.9-121.IE7a*, CC.9-12.1.1F.8a Essential question: How do you convert quadratic functions to vertex form f(x) = a(x — h) 2 + k? Writing Quadratic Functions in Different Forms Write the quadratic function in the form described. A Write the functionf(x) = 2(x — 4)2 + 3 in the form f(x) = ax2 + bx + c. fix) = 2(x — 4) 2 + 3 f(x) = 2(2 - - 1 )( + Iç + 3 Multiply to expand (x — 4) 2 . f(x) Distribute 2. 2(x2) — .f(x) = 212 - ( 8x) + (16) + 3 K + 3) + 3 Multiply. ..f(x) = 2? - 16x + '35 Combine like terms. - f So, fix) = 2(x — 4)2 ± °C 3 is equivalent to /61X C Write the function f(x) = 12 + 6x + 4in vertex form. Recall that the vertex form of a quadratic function is f(x) = a(x h)2 4 k. Write the given function in vertex form by completing the square. Ci Houg hto n Mifflin Harcourt Pu blishing Co mpany — f(x) = x2 + 6x +4 Set up for completing the square. f(x) = (x2 + 6x + I )+4—Li Ax) = (12 + 6x + 9) + 4 — 9 Add a constant so the expression inside the parentheses is a perfect square trinomial. Subtract the constant to keep the equation balanced. 2 f(x) =(x + Write (x2 + 6x + 9) as a binomial squared. fix). (x + 3)2 - 5 Combine like terms. z Sodix) = x2 6x + 4 is equivalent to — 5 REFLECT \ 1 a. In part A, how does the value of a of the function in vertex form compare with the value of a when the function is in the form fix) = ax2 + bx c? 14 voAu Unit 2 n kaik -r)oi(); 47 s tke- `1 "-e- ) Lesson 4 lb. In part B, how could you check that you found the vertex form of the quadratic equation correctly? \&(4€)( -co cm 01/45-e_ -40 W1,-1C- 110,r, (6 c.1 4 liorkTL yok s D-C ke VA c,b1 A4 t fix); ^ 4`e• -Cr rn rno4LL 41nosx- ¥ in - lika e 1 ■• 4'1 o lc. Describe how to complete the square for the quadratic expression x2 +8x+ 1 1:1;114,2. kka coe-(41“2,"A o-C )<. 121 Ad -ChP- (e 5 ,1-k 1 L ), ainC) 5(kutca ne u o+e: fly ;5 6 . ott(oA\c- -.2)(6e55c0 'n: The standard form of a quadratic equation isf(x) = ax2 + bx + c. Any quadratic function in standard form can be written in vertex form, and any quadratic function in vertex form can be written in standard form. EXAMPLE\ Graphing by Completing the Square Graph the function by first writing it in vertex form. Then give the maximum or minimum of the function and identify its zeros. A • Write the function in vertex form. Set up for completing the square. Bo )4-12- IL = (X - 9 0 H ou ghton Miffli n H arcourt P ubli shi n gCom pa ny /Cr) Add a constant to complete the square. Subtract the constant to keep the equation balanced. 2 Write the expression in parentheses as a binomial squared. fix) (x - 4)2 - tj Combine like terms. • Sketch a graph of the function. The vertex is 1 Li 1.6 Two points to the left of the vertex are (2, D ) and (3, - 3 ). Two points to the right of the vertex are (5, - ) and (6, Q)• • Describe the function's properties. The minimum is _ _ The zeros are Unit 2 and G 48 Lesson 4 B f(x) = -2x2 - 12x - 16 • Write the function in vertex form. f(x) = (x2 + 6x) - 16 Factor the variable terms so that the coefficient of x2 is 1. f(4= Set up for completing the square. 16 - 9)— 16 — (-2) 9 Complete the square. Since the constant is multiplied by -2, subtract the product of -2 and the constant to keep the equation balanced. 5 Write the expression in parentheses as a binomial squared. f(x) = -2(x + 3) 2 - 16- ( - Simplify (-2)9. Combine like terms. • Sketch a graph of the function. The vertex is Two points to the left of the vertex are (-5, -ç ) and (-4, C 0 Houg hton Mi fflin HarcourtPu b lishi ng Comp any Two points to the right of the vertex are (-2: ) and (-1, - )• • Describe the function's properties. The maximum is The zeros are — [REFLECT and — N 2a. How do you keep the equation of a quadratic function balanced when completing the square? Sub-\ fo,(2` ickt canUr 1/4 cja 4 CaYn_fAelc k stvue. TI 41-1- r)1\ kkt_v, jvt_3 04 -0, olvAkv?Vict) by. nvm6c 1 5¼4o(? casA cAnic ayS - L, nuirydgc. 2b. In part B, why do you factor out -2 from the variable terms before completing the square? ■ Unit 2 C J-2.4-(“A 'nit arl) 49 1.1 5 Ot5S Lum1k4 41,4 scragt. Lesson 4 2c. Why might the vertex form of a quadratic equation be more useful in some situations than the standard form? In gzeice-x c0 (r4, can 1j 5t -UN2 'job Vcava5 A Ck sil i a \c-?, A'ko_ aX , 5 0C 5 ,/rnmeAri / 04 4 k. k- c.o_ n,) k 4,, e40') t'd ■Q n 4 S nnoorimAnn ciC 1 rn;n;niurn b-C - 1 v,nt.AS or, an A ;As XAMPLE Modeling Quadratic Functions in Standard Form The function h(t) = —16t2 + 64t gives the height h in feet of a golf ball t seconds after it is hit. The ball has a height of 48 feet after 1 second. Use the symmetry of the function's graph to determine the other time at which the ball will have a height of 48 feet. A Write the function in vertex form. — 4t) Factor so that the coefficient of t2 is 1. — -16) LI kt) —16(1 2 — 4t + Complete the square and keep the equation balanced. 2 /KO= - 1+ — Write the expression in parentheses as a binomial squared. h(t) = —16(t— 2) 2 + 4 Simplify. This point is 2 units to the left of the vertex. Based on symmetry, there is a point 2 units to the right of the vertex with the same y-coordinate at ( L.I KY :E "0 The point (0, 0 is on the graph. ak 8 0 , o). VA MN i Vt) Lvic The vertex is "7 a • MR NM D, 3 1-1- me (5) The point (1, 48) is on the graph. Based on symmetry, the point( 3 , 48) is also on th e graph. So, the ball will have a height of 48 feet after 1 second and again after 3 seconds. RE'iLECT 3a. How can you check your answer to the problem? \,(0Aki.orke 0 Unit 2 -( 44 -Cr, , r 0,N 9 1h. -Cyr 1:: 3 otnd civic 41,10, 1, -14€ IvQ_ LIS 50 Lesson 4 0 H ou ghton Miffl in H arcourt P ubli shi ng Com pany Use symmetry to sketch a graph of the function and solve the problem. 3b. What is the maximum height that the ball reaches? How do you know? ce, 4(m ver ,-c -16 -co, -,n6LA-65 jloa-v i-L veA-ex ()162-), fincs ■ mokx,r, hp4) ,,v1 o G 1-1 . ko_ ball re aLL2) . 3c. If you know the coordinates of a point to the left of the vertex of the graph of a quadratic function, how can you use symmetry to find the coordinates of another point on the graph? means -k.ko1/41 ,5 A cc-°n(i` AL Q'S i"`isic v4\ \9"C /4- S° V"- 1 51 0;04 51OnU oi 1-4 AO 1-t sal,. 1- coof();n6,--'m o a Se co 45 a-Przc o-S 4)•Q_ ver-l-e x d NRctex2 11, A S [PGIngVng 7-1Zkrr,•Jr—vle---ar+r.owr".. Graph each function by first writing it in vertex form. Then give the maximum or minimum of the function and identify its zeros. 1. f(x) = x,21 — fix + 9 -7 I ; I I- — — ME 1 - - TAL L 'T +1 _L I - - PIT H Houg hto n Mi fflin Harcou rt Pu blishing Company fix) = x2 — 2x — 3 Icy .— k(x-3 3 3. f(x) = 2. 7x2 - m ft-kin:- o:L_ 14x 4. fix) arc) 3 = 3x2 — 12x + 9 SIM “Y"): -.1 (1 A" \/* -1 YY10Vi \ Unit 2 2.404 0 51 'ant 3(2('' M n: 3, I atql 3 Lesson 4 5. A company is marketing a new toy. The function s(p) = — 50/92 + 3000p models how the total sales $ of the toy, in dollars, depend on the price p of the toy, in dollars. a. Complete the square to write the function in vertex form and then graph the function. ODO 35, 000 b. What is the vertex of the graph of the function? What does the vertex represent in this situation? Rfejsc-\ea Ao fesv a5, 000 ko.-1 ■.5 - 15,000 - 5000 n 0 t . •• • I HI 20 HO Co 0 (15, OOD) 10L\ 5 4 \t5 ( C. (1130) ?ir ‘ 1,0_ 4 C3_0_2_ 1-15 f L ii 5 CO - CO (f - 30T LHH U. 1.15, 00 go 100 ywk 14) The model predicts that total sales will be $40,000 when the toy price is $20. At what other price does the model predict that total sales will be $40,000? Use the symmetry of the graph to support your answer. S PTII'Qkf 1 c\e() On -1-1‘. (1 Vv.\ 01 L 9rpLiflverc wit 05. bt ;Alf, iL-c -16 vede o c -11,0 vcr-{e x 114. 11-1()_:1k it,;,,A 6.0 ) HO1 o0o) a 10 von ,1 s y - capcti1/4ir,,,Ac n-1 1 40,000. "TI\ Vpi unjj 46 p4. -',5 6. A circus performer throws a ball from a height of 32 feet. The model h(0 = —16? + 16t + 32 gives the height of the ball in feet 1 seconds after it is thrown. a. Complete the square to write the function in vertex form and then graph the function. 0 H ou gh ton Miffl i n H arcourt P ubl i sh in g Com pany ( G (-{ - h( 1 b. What is the maximum height that the ball reaches? 4- n-a 3[o 4 C. What is a reasonable domain of the function? Explain. I0 0,4 k Mv•S\ rsy 05 2 3 oni./ nen ivyth-L vacts , so A" yaks JC less -L si R ir‘ DA 4/ea4r€ 1no,r5 5D -t er0 1. 3. d. What is the y-intercept of the function's graph? What does it represent in this situation? What do you notice about they-intercept and the value of c when the function is written in standard form? LibAIi-Ccscr) 2",dnAL 6,t1 b Al2/0,4 n 1 1‘e_t_:Acruz t\ nJ 4k vakvq of Unit 2 52 C CA( 4t 5ArKt_ Lesson 4