Lesson 1: Quadratic Equations

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1 Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method Method 1: Fctoring When to Use Fctoring: Steps: 1. Put the qudrtic eqution in.. the epression. 3. Use Property to Solve. Zero Product Property: If b=0, then either or Unit 1 Pge 1 of 4

2 Method : Squre Root Method or Etrction of Roots Method When to Use Squre Root Method: In order to use squre root method, the eqution must be in the formt: =. Notice tht if there is no term in the stndrd form of the qudrtic eqution or if b =, then it is to put in this form. Steps: 1. Put the qudrtic eqution in the form.. Tke the of both sides of eqution nd. 3. When you tke of both sides, you MUST tke the prts. 5 9 (3 1) 9 ( 1) 3 11 Method 3: Completing the Squre Investigtion of Perfect Squres Unit 1 Pge of 4

3 Method 3: Completing the Squre (continued) When to use Completing the Squre Method: This method will work, but I would only use this method if I ws unble to or use. The most equtions to use Completing the Squre Method, hve = nd b is. Steps: 1. From form, mke = by dividing ech term by,. Move the term to the right side of eqution nd dd to ech side. 3. Complete the Squre by the liner term by nd. Put this vlue in the blnks. The right side will now fctor into. 4. Finish Solving by using method Unit 1 Pge 3 of 4

4 Method 4: Qudrtic Formul When to Use Qudrtic Formul: Steps: 1. Put the qudrtic eqution in.. Find the vlues of,, nd. 3. vlues in qudrtic formul, which is: 4. Reduce Choosing the Best Method In summry, when choosing method to solve qudrtic eqution, follow this order. 1. try to first.. If b = or if, then use. 3. If = nd b is, then it is convenient to use. 4. As resort, use, which will solve qudrtic equtions. Unit 1 Pge 4 of 4

5 Lesson : Miscellneous Equtions 1. Higher Order Equtions - Fctoring Steps: 1. Get equtions in generl form, or set. Fctor out, if possible. 3. Fctor the remining epression depending on the number of terms left. Terms: b. 3 Terms: c. 4 Terms: 4. Mke sure ll fctors re. If they re not, then repet step Set ech fctor nd for the vrible. Solve Unit 1 Pge 5 of 4

6 . Rtionl Eponents Review from Intermedite Algebr: Emples: m n n m n m 3 7 = = Solve: Solving Rtionl Eponent Equtions m n Steps: 1. Isolte the with the rtionl eponent. k m n k. Rise both sides to the of the eponent, or.. If of eponent or is, then put sign on vlue. b. If of eponent or is, then DO NOT put sign on vlue. 3. You MUST your solution(s) nd eliminte solutions. m n k If m is even if m is odd n k m n k m Unit 1 Pge 6 of 4

7 More Emples: Solve Solving Equtions of the Qudrtic Form (using Substitution) The following re emples of the qudrtic form. Wht mkes these seemingly different equtions similr? Ech of these 5 similrities to the right 4. the equtions to be in the form. 5. Unit 1 Pge 7 of 4

8 3. Solving Equtions of the Qudrtic Form (using Substitution) continued Steps: 1. Identify the eqution s nd set eqution equl to.. Let some vrible, be equl to the originl eqution s term vrible prt. This eqution is importnt to write down, becuse we will use it in step Find the of the new vrible, which will lwys be the first term s vrible prt. 4. the new vrible into the eqution to get qudrtic eqution nd for the new vrible,. 5. To solve for originl vrible, solution(s) into eqution from step. 6. your solution(s). This is mndtory, if your eqution includes eponents. Solve: Unit 1 Pge 8 of 4

9 Lesson 3: Absolute Vlue Equtions nd Inequlities Remember, the mens distnce from. Often we sy, the bsolute vlue mkes the vlue inside the brs. Emples: Solving Absolute Equtions Solve in your HEAD 3 3 Notice tht the on the vlue determines if the eqution hs. Steps: 1. Isolte the in the. c. If c is, which mens or then split the bsolute vlue eqution into two without ; or 3. If c is, then the nswer is, becuse the bsolute vlue of epression cn be equl to negtive vlue. nd then. Solve: Unit 1 Pge 9 of 4

10 Solve: Intervl Nottion Intervl nottion represents the set of numbers between two. If you would like to the endpoint, then use or symbol. If you would NOT like to include the endpoint, then use or symbol. We lwys write intervl nottion s follows:, Inequlity Nottion Grph Intervl Nottion b b b b All Rel Numbers b b b b No Solution Unit 1 Pge 10 of 4

11 3. Solving Liner Inequlities Remember: the inequlity sign when you multiply or divide by or if you the eqution Solving Inequlities with Absolute Vlue When solving inequlities with, first the bsolute vlue epression. c or c Net, identify if the on the opposite side of the bsolute vlue is, or. A. Let s eplore when c is. Grph of Solutions Inequlity w/ Absolute Vlue Solution in Inequlity form Solution in Intervl Nottion Let be lgebric epression nd let c be positive number. If If c, then. c, then OR And Emples: Or Emples: 3 4 Unit 1 Pge 11 of 4

12 B. Let s eplore when c is. Inequlity w/ Absolute Vlue Grph of Solutions Solution in Intervl Nottion Let be lgebric epression nd let c be negtive number. If If c c, then the solution is., then the solution is. Unit 1 Pge 1 of 4

13 Solve Unit 1 Pge 13 of 4

14 Lesson 4: Bsics of Functions nd Their Grphs A reltion is. The set of ll elements in reltion is clled the nd the set of ll elements reltions is clled the. The following is reltion. Student April Bob Crlos Dion Ev Color of their Shirt Let us define the following sets s: A { } B { } The domin of A is : The rnge of A is: A R The domin of B is : The rnge of B is: B R AD B D A is where ech element in the corresponds to element in the. A B Unit 1 Pge 14 of 4

15 Functions cn be epressed severl wys. Functions s Functions s Functions s Functions s Sets Determine if the following reltions re function? Find domin nd Rnge 10,8, 6, 4,,0,, 4 I K 3, 4, 3,5, 8,9, 1,0 J 4,5, 6,8, 8,8, 6,8 Is the reltion Function? Domin: Rnge: Is the reltion Function? Domin: Rnge: Is the reltion Function? Domin: Rnge: Functions s Equtions is nother wy of writing n. Function nottion defines the, or of the function by using ny vlue of the, (). If n eqution is function, then we y with pronounced. y = To find the vlue of function t given, we the into the eqution nd or. Let f 3 Find f 5 = Find f = Find 1 f = Unit 1 Pge 15 of 4

16 Let g 3 5 Find g = Find g = Find 3 g = Let h 1 1 Find h = FInd h 4 = Find h = Functions s Grphs The of is the picture tht represents ll the or for the eqution/function. Remember: If every vlue in the corresponds to only vlue in the, then the grph is. If vlue in the domin corresponds to more thn vlue in the, then the grph is not function. To determine if grph is function, we will use the, which sttes tht if intersects the grph t thn one point, then the grph is function. Are these reltions lso functions? y y y Unit 1 Pge 16 of 4

17 Functions s Grphs: Finding Vlues We cn lso find of function by looking t the. To find function vlue, go to the given on the is. Your is the coordinte t tht input. Find f(-) = Find f(-1) = Find f(0) = Find f(1) = Find f() = Find f(-3) = Find f(-) = Find f(0) = Find f() = Find f(3) = Functions s Grphs: X nd Y Intercepts re where the grph the. Algebriclly, you find it by setting nd solving for. The is where the grph the. Algebriclly, you find it by setting nd solving for. Find the nd y intercepts of the following grphs: X Intercept(s): X Intercept(s): X Intercept(s): Y Intercept: Y Intercept: Y Intercept: Unit 1 Pge 17 of 4

18 Functions s Grph: Domin nd Rnge We will write the domin nd rnge using. Remember: The of reltion is ll the or vlues tht reltion includes. In order to find the of the grph, look t the end points of the reltion grphed from to. The of reltion is ll the or vlues tht reltion includes. In order to find the of the grph, look t the end points of the reltion grphed from to. Domin: Domin: Domin: Rnge: Rnge: Rnge: Domin: Domin: Domin: Rnge: Rnge: Rnge: Unit 1 Pge 18 of 4

19 Functions s Grphs Summry Zeros: The vlue of when f() =, or the coordinte of the intercept. 1.. Domin Rnge X Intercept(s) Y intercept Zeros Find f() = Find f(3) = Domin Rnge X Intercept(s) Y intercept Zeros Find f(-3) = Find f(-6) = Domin Rnge X Intercept(s) Y intercept Zeros Find f(-1) = Find f(0) = Domin Rnge X Intercept(s) Y intercept Zeros Find f(-) = Find f(-5) = Unit 1 Pge 19 of 4

20 Lesson 5: More on Bsics of Functions nd their Grphs Even nd Odd functions nd their Symmetry TYPE GRAPH SYMMETRY ALGEBRAIC DETERMINATION If f(-) = for ll in the domin, then the functions is. If f(-) = for ll in the domin, then the functions is. Algebriclly determine if the following functions re even, odd or neither. 3 5 f 3 1 f 4 f 1 Use possible symmetry to determine whether the following grphs re even, odd or neither. Unit 1 Pge 0 of 4

21 Piecewise Defined Functions: A function is tht is defined by more thn eqution over specified. Emple: For Cell Phone Pln, you will py $0 for the first 60 minutes, nd then $0.40 per dditionl minute. Ct Find C(30)= Find C(10)= To find function vlue with piecewise defined function you must look for the the function vlue belongs in. Then you substitute tht vlue in to the corresponding. Emple: 6 1 if 0 Let f 7 3 if 0 Find f(-3)= Find f(0)= Find f(4)= if 5 0 Let g 4 if 0 Find g(-3)= Find g(0)= Find g(3)= Unit 1 Pge 1 of 4

22 Grphing Piecewise Functions To piecewise defined function, choose vlues for, including the of ech domin, whether or not tht the endpoint is in the domin. Lbel ech endpoint s or not. Sketch the of the function. Remember our Cell Phone Pln Function: C t 0 if 0 t t 60 if t 60 t C(t) Grph the following piecewise defined functions. Unit 1 Pge of 4

23 Unit 1 Pge 3 of 4

24 Difference Quotient The difference quotient is used to understnd the rte t which functions chnge, which is covered hevily in future courses. For this College Algebr course, we will need to understnd how to evlute this rtio. Definition: f h f h where h 0 Emples: Find the difference quotient for the following functions 6 1 f f 5 f 1 Unit 1 Pge 4 of 4