Quadratic Functions. Analyze and describe the characteristics of quadratic functions


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1 Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze sctter plots, nd determine nd pply the equtions for the curves of best fit, using pproprite technology C8 Describe nd trnslte between grphicl, tbulr, written, nd symbolic representtions of qudrtic reltionships B Demonstrte n understnding of the reltionships tht exist between rithmetic opertions nd the opertions used when solving equtions. C9 Trnslte between different forms of qudrtic equtions A7 Describe nd interpret domins nd rnges using set nottion C3 Demonstrte n understnding of how prmeter chnges ffect the grphs of qudrtic functions Assumed Prior Knowledge enerl form of the qudrtic function Effects of the coefficient on the shpe of grph (prbol) of qudrtic function Vertex of prbol Mximum nd minimum yvlues Fctoring nd expnding qudrtic expressions Using lgebr tiles to fctor nd represent qudrtic expressions Using technology to grph qudrtic functions nd to find the eqution of the curve of best fit Pge 
2 Dignostic Worksheet for Selected Assumed Prior Knowledge (Section.3). Which is qudrtic function in generl form? ) y = (x3) + 4 b) y4 = (x3) c) y = x + 3x 5. Which of these sttements is true bout the grph of c(y + ) = (x  ) when compred to the grph of y = x? ) Verticl stretch 8, Trnsltion left, up b) Reflection in yxis, Verticl stretch of c, Trnsltion right, down c) Reflection in xxis, Verticl stretch of 8, Trnsltion right, down 3. Wht is the mpping rule tht describes the trnsformtion of y = x into ¼(y+) = (x  9)? 4. Expnd: ) (x)(x+3) b) (x4)(x+) c) (3x  )(x) 5. Fctor ech of the following completely. Check your work by multiplying your fctors. ) x  9x  36 b) 5x + 5x c) x  6x + 5 d) x + x + e) y + 3y + f) y + 8y + 6 g) x  4x + 40 h) x  6x 7 i) 5  b j) 9n  6 k) y For which vlues of k cn ech of the following be fctored? ) 5x + kx + 4 b) 3x + kx  c) 8x + kx  6 Pge 
3 7. Remove common fctor first, then fctor further if possible. Check your work by multiplying your fctors. ) 3y + 5y + b) 5x  0x + 5 c) 5x 580xy 4 d) 49x 3 + 5xy 8. Fctor completely. Check your work by multiplying your fctors. ) 3x + 3x + 4 b) x + 7x  4 c) 8x  x  0 d) c +9c  4 Answers #. c #. c #3. (x, y) ö (x + 9, 4y  ) #4. ) x + x 6 b) x  3x  4 c) 6x  x + #5. ) (x  )(x + 3) b) 5x(x + 5) c) (x  5)(x  ) d) (x + ) e) (y + )(y + ) f) (y + 4) g) (x  0)(x  4) i) (5  b)(5 + b) j) (3n  4)(3n + 4) k) (y  9)(y + 9) #6. ) ±, ±9, ± b) ±5, ± c) ±49, ±6, ±9, ±6, ±4 #7. ) 3(y + 4)(y + ) b) 5(x  3)(x  ) c) 5x(5x  6y )(5x + 6y ) d) x(49x + 5y ) #8. ) (3x + )(x + 4) b) (x  )(x + 4) c) (8x + 5)(x  ) d)  (4c  )(3c  4) Pge 3
4 Focus C  Forms of Qudrtic Functions LOSSARY TERMS (see text p. 430) Prbol, Trnsformtionl Form of Qudrtic Function, Vertex of Prbol, Axis of Symmetry, Verticl Stretch, Stndrd Form of Qudrtic Function, enerl Form of Qudrtic Function, This Focus is included minly to provide you with n opportunity to review some things you lerned in Mth 04 bout prbols, mpping rules, nd trnsformtions. Some of the bove terminology my be new to you. In prticulr, the Stndrd Form of Qudrtic is new but not difficult to understnd if you re comfortble with trnsformtionl form. Wht you should hve lerned from this Focus: Qudrtic Function A. enerl form y = x +bx+c, 0 L is the verticl stretch, (0,c) is the yintercept B. Stndrd Form y = (xh) +k, 0 L is the verticl stretch, vertex is (h,k) C. Trnsformtionl Form (y k) = (x h), 0 L is the verticl stretch, vertex is (h,k). The mpping rule when compred to y = x is (x,y) º(x + h, y + k). How to write qudrtic equtions in both stndrd nd generl forms How to chnge qudrtic function from Stndrd Form to Trnsformtionl Form nd vice vers. How to find the verticl stretch of grphed prbol by compring with the grph of y = x Pge 4
5 Focus D  Creting the Trnsformtionl Form of Qudrtic As you hve lredy lerned, there re three forms of the Qudrtic function: enerl form y = x +bx+c, 0 Stndrd Form y = (xh) +k, 0 Trnsformtionl Form (y k) = (x h), 0 Ech of these forms re useful becuse we cn esily red certin informtion tht is different from tht using nother form. It will therefore be helpful to be ble to convert from one form to nother, nd in Focus C you lredy lerned how to chnge from Stndrd Form to Trnsformtionl Form nd vice vers. In this Focus you will lern how to convert from enerl Form to either of the other forms. First, however, we need to look t specil types of trinomils which cn be fctored s perfect squres Wht is Perfect Squre Trinomil? Recll tht perfect squre number is one tht hs two fctors which re EXACTLY the sme e.g. 4 = =, 5 = 5 5 = 5. In similr mnner, some trinomils cn be fctored s the product of two fctors which re EXACTLY the sme e.g. x + 0x + 5 = (x+5)(x+5) = (x+5) Prctice  Fctor ech of the following perfect squre trinomils: ) x + x + 36 b) x  0x + 5 c) x  8x + 8 In ech of the bove perfect squre trinomils, there is specil reltionship between the middle nd lst coefficients (tht is, the vlues of b nd c): Tke hlf of the middle coefficient, then squre it nd you should obtin the lst coefficient. For exmple, in question b), 0 is the middle coefficient; hlf of tht is 5; (5) = 5, which is the lst coefficient. In generl, for perfect squre trinomil of the form x + bx + c, the reltionship is (b/) = c. Pge 5
6 Prctice  Find the vlue tht should be plced in ech blnk in order to crete perfect squre trinomil. ) x + 4x + b) x  8x + c) x  4x + d) x  x + e) x  3x + f) d + 9d + g) h) q  q + i) p p + 49 Chnging from enerl Form to Trnsformtionl Form If we look t the Trnsformtionl or Stndrd Forms of Qudrtic Function, we cn see the result of fctoring perfect squre trinomil e.g. EXAMPLE: Chnge to Trnsformtionl Form: y = x  8x + 5 Step y  5 = x  8x Bring the constnt term (the c vlue) to the other side of the eqution Step y  5 = (x  8x) The coefficient of the qudrtic term (the vlue) must be removed s common fctor Step 3 y ( 6) = (x  8x + 6) Chnge the expression in the brckets to perfect squre trinomil. Blnce the eqution by dding equivlent mounts to both sides; in this cse, 6 hs been dded to both sides. Step 4 y + 7 = (x  4) Fctor the perfect squre trinomil Step 5 ½(y + 7) = (x  4) OR y = (x  4)  7 Move the cross the = sign using its reciprocl to crete Trnsformtionl Form Move the 7 cross the = sign to crete the Stndrd Form. Pge 6
7 Wht you should hve lerned from this Focus: How to identify nd crete perfect squre trinomil written in the form x +bx+ c How to convert from enerl Form to either Trnsformtionl or Stndrd Form How to use the Trnsformtionl or Stndrd Form to solve problems, including those involving mxim nd minim (See text p. 33, #8, 9,, ) Extr Prctice  Revisit the Extr Prctice problems from Investigtion 4 nd solve them by creting the Trnsformtionl or Stndrd Form of the qudrtic function insted of using the grphing clcultor s you did previously. Pge 7
8 Focus E  Determining Qudrtic Functions from Prbols Red p Another exmple using both methods is provided below. Fill in the spces to complete the explntions nd/or the steps Method (y k) = (x h) y = x, Vertex is (0,0) From vertex, Over up Over up 4 Over 3 up is the trnsformtionl form. Method For the grph, Vertex is (3,4). From vertex, Over down Verticl stretch is therefore Since the grph is opening down nd the grph of y = x opens up, then there is reflection cross the. The qudrtic function is ½(y  ) = (x  ) Vertex (h,k) = (3,4). Therefore, we know (y ) = (x _) The grph psses through the point (4,). Substitute these coordintes for x nd y, then solve for : ( _) = ( _ ) ( ) = () = = Therefore, the eqution is Pge 8
9 Extr Prctice Find the Trnsformtionl nd the Stndrd Form of the function for ech grph shown: Pge 9
10 Answers Trnsformtionl Form. (y) = x. (y+) = x 3. 3y = (x+) 4. 8(y) = (x+5) 5. /4. (y5.8) = (x.5) or 5/(y5.8) = (x.5) 6. /. (y + 0.3) = (x.4) or 0.8 (y + 0.3) = (x.4). y = x +. y = 3x  3. y = (x+) 4. y = /8(x+5) + 5. y = 4.(x.5) y=.(x.4) 0.3 Stndrd Form Wht you should hve lerned from this Focus: How to find the verticl stretch of Qudrtic Function from its grph (review from Focus C) How to find Qudrtic Function in either Trnsformtionl or Stndrd Form using the grph How to find Qudrtic Function in either Trnsformtionl or Stndrd Form given the vertex (or informtion to help you find it) nd nother point (e.g. see text p.38 #39, 4) How to find Qudrtic Function in either Trnsformtionl or Stndrd Form from problem sitution (e.g. see text, p.38, #38, 45) Pge 0
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