Name Date PRECALCULUS SUMMER PACKET

Transcription

1 Name Date PRECALCULUS SUMMER PACKET This packet covers some of the cocepts that you eed to e familiar with i order to e successful i Precalculus. This summer packet is due o the first day of school! Make sure to show all work! Some study tips to help you with the packet: If you are ufamiliar with ay of the cocepts elow, you should start y doig a search for the topic title or the cocepts i Google. This will take you to a variety of wesites that ca help explai the cocept. Kha Academy is great for videos o the cocepts! YouTue is awesome for videos as well. INTERVAL NOTATION Iterval otatio is aother way to represet iequalities. For example, x > ca e writte as (, ) sice all of the values of x that satisfy this iequality are i the iterval from to ifiity. Aother example, x ca e writte as (, ] sice all the values of x that satisfy this iequality are i the iterval from to. The iterval otatio for this iequality uses the racket o the sice x ca actually equal. Write the iterval otatio for each iequality. 1) x 7 ) x < 18 ) 4 < x < 4 4) 5 < x 5

2 The Laws ad Properties of Expoets Operatio Actio Law/Property Example(s) Multiplicatio Add expoets of the same a m a a m+ x x 4 x 7 ase Divisio Sutract expoets of the a m 7 x same ase a m a x x 4 Power to a Power Multiply expoets Power to a Power of a Product Power to a Power of a Quotiet Zero Expoet Negative Expoets Multiply expoets Multiply expoets Take reciprocal of the ase ad chage the sig of expoet to positive Simplify with oly positive expoets. ( a ) m a m x 4 ( a) a x a a y x x 1 x 6 7x 6 y a 0 1, a , x y a a a 1 a a a x 1 x x y 4 y 4 x x y x 0 1 y x y x 1. ( x )( 4x 5 ). x 5 y x y. ( x y ) 4 4. ( x y ) 4 5. x y 6. ( 5x 0 y ) 7. x 5 y 4 w 8. x y 9. ( x 5 )( x )( 4x 7 ) 10. ( x 4 y ) 11. ( y ) y 1. ( y 4 ) x 5 y 4 x 0 y

3 The Laws ad Properties of Radicals I a, is the idex. Operatio Law Example Law Example Product a a a a a a 6 Same idex Quotiet a a a a Same idex Additioal Examples Simplify each radical. Assume all variales represet positive real umers. 8 Example 1a: Usig Perfect Squares Example 1: Usig Prime Factorizatio Example a: Usig Perfect Cues Example : Usig Prime Factorizatio Example : x y 6x 5 y x y 6x 5 y 18x 7 y 4 x y x Simplify, remove radicals if at all possile. Assume all variales represet positive real umers x x 5 y xy x y

4 Multiplicatio of Polyomials Example 1: Distriutive Property 5x x 10x 15x 5x x Example : Multiplyig iomials 4 x + 5 Multiply, use product formulas whe they apply. 1. x( x + 5x ). ( x + 5) ( x + 7) ( x ) ( 4x + 5) ( x ) 1x 8x +15x 10 1x + 7x 10. ( x + 4) ( x 7) 4. ( x ) 5. ( x 1) ( x + x 10) 6. ( x + ) ( 4x 7x +1)

5 Factorig Polyomials Example 1: Greatest Commo Factor 15x 5x 15x 5x 5x x 5 Example : Triomial Factorig of ax + x + c where a 1 Factorig asic triomials x x 15 Fid two umers that multiply to 15 ad add to x x 15 x 5 x + Examples : Triomial Factorig of ax + x + c where a 1, a 0 Factors of Same Sigs 6x +19x +10 Factors of a c with sum of , Rewrite triomial as 6x +19x +10 6x + 4x +15x +10 Group as two iomials 6x +19x +10 6x + 4x Factor Greatest Commo Factor 6x +19x +10 x x x ( x + ) Factor Greatest Commo Factor 6x +19x +10 ( x + ) x + 5 Factorig Formulas Name Formula Example x y ( x + y) ( x y) 9x 16 ( x) 4 Differece of Two Squares Differece of Two Cues Sum of Two Cues x y ( x y) x + xy + y x + y ( x + y) x xy + y Prolems for Factorig Polyomials Factor each completely. 8x 7 x 8x + 7 x 1. 1x +14. x +1x + 5 ( x + 4) ( x 4) ( x ) ( x) + ( x) + ( x ) 4x + 6x ( x + ) ( x) ( x) + ( x + ) 4x 6x + 9. x 10x x 4x 1

6 5. x x x +1x x x 49y 9. x x +1x x 11x x + x 6 1. x x x x +15

7 Fractioal Expressios A fractioal expressio is a quotiet of two algeraic expressios. As i ay fractio, the deomiator may ot equal zero, divisio y zero is ot permitted. A special case of fractioal expressios is the quotiet of two polyomials, which is referred to as a ratioal expressio. Follow these three steps i simplifyig fractioal expressios ad ratioal expressios. Step 1 Factor the umerator ad the deomiator if possile Step Determie ay values that would make the deomiator equal to zero. Step Remove ay factors of oe, factors that are the same i the umerator ad deomiator. Example 1: Simplify the fractioal expressio ad determie ay restricted values. x + x 10 x 5 Step 1 Factor x + x 10 ( x + 5 )( x ) x 5 ( x + 5) ( x 5) Step Determie Restricted Values x ±5 Step Remove Factors of Oe x + x 10 ( x + 5 )( x ) x 5 x + 5 x 5 ( x 5) ( x ) Example : Simplify the fractioal expressio ad determie ay restricted values. x 5x x + x 10 x 5x ( x +1) ( x ) Step 1 Factor x + x 10 ( x + 5) ( x ) Step Determie Restricted Values x 5, x Step Remove Factors of Oe Prolems for Fractioal Expressios Simplify each if possile ad state ay restrictios. x 5x x + x 10 ( x ) ( x ) x +1 x + 5 x +1 x ( x + ) x 4 x + ( x + 5). 5x + 0 x +10x + 4. x +1x + 0 x 9 4. x +1x + 5 x x 40

8 Multiple Choice 1) If f x 4x ad g( x) 5x, what is? g f x a. 100x. 80x c. 0x d. 10x e. 16x ) What is the value of: a. 1/6. 5/15 c. 10 d. 15 e. 0 ) 6y(y ) + (y ) 5!! ( )? a. (y + ) (6y). (y + )(6y 1) c. (y )(6y + 1) d. (6y 1)(y ) e. (6y)(y)(y )( 1) 4) A certai lie i the coordiate plae has the equatio 5a + 7, where a represets the idepedet variale, ad represets the depedet variale. Which of the followig is the equatio of a lie that is perpedicular to 5a + 7? a. 5a +. 5a 5 c. 1/5a d. 1/5a + 1 e. 7/5a 5) A cady store sells a ag of cotto cady for $0.70 ad chocolate ars for $0.50. Sheila ought some cotto cady ad several chocolate ars from the store ad spet a total of $6.0. How may ags of cotto cady ad ars of chocolate did Sheila purchase? a c. 11 d. 1 e. 17 6) If - is oe solutio of the equatio x + x + z 1, where z is a costat, what is the other solutio? a. 0. c. 6 d. 9 e. 1 7) A equatio of the lie that cotais the origi ad the poit (-4,7) is a. y 7/4x. y 4/7x c. y 7/4x d. y 7/4x + e. y 7/4x 8) If x p ad y p what is the value of x y + xy? a c. p 5 d. p 5 e. p 10

9 9) I the arithmetic sequece x 1, x, x,, x, x 1 4, ad x x -1 + for each >1. What is the value of whe x 60? a c. 18 d. 19 e. 0 10) Which value of x satisfies the equatio: log (18) log () log 4 (x)? a c. 0 d. 4 e. 11) For which of the followig values of x is 1 x ot a real umer? a. 1. c. d. 4 e. 5 1) If si x ad x > π, what is cos x? a. 1. ½ c. 0 d. ½ e. 1 1) A root of x 5x 1 0 is a.. c. d. e ) The sequece a is defied y a 0 1 ad a +1 a + for 0, 1,, What is the value of a? a c. 16 d. 0 e. 15) A apartmet uildig cotais 1 uits cosistig of oe- ad two- edroom apartmets that ret for $60 ad $450 per moth, respectively. Whe all uits are reted, the total mothly retal is $4,950. What is the umer of two-edroom apartmets? a.. 4 c. 5 d. 6 e. 7