Algebra II. Student Name. C, ÿapter 9 Notes. Teacher,o

Transcription

1 Algebra II C, ÿapter 9 Notes Student Name i Teacher,o

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3 9-4 Simplifying Rational Expressions Do NOT break into addition or subtractionhj Factor, factor, factor!!!!!! State the value or values for which the variable is undefined and simplify. EX 1: X X+I EX 2: 3xZyz3 9xy4z EX 3: x-4 3X-12

4 EX 4: 2x-10 X2-9X+20 EX 5: x2-16 x+4 EX 6: 5-x X-5 EX 7:

5 EX 8: x-4 xs- 64 EX 9: Y y+#xy EX 10: x2-9x + 20 x2-3x- 10

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7 9-4 Multiplying & Dividing Rational Expressions bo NOT break into addition or subtraction!!! Factor, factor, factor!!!!!!!!!!!! Steps for Division: 1) Take the reciprocal of divisor 2) Factor (Look for GCF Ist, then continue to factor) 3) Simplify EX 1: 2a2 3be2! 5b% 8a3 EX 2: 14c2d 35cd3 9m3 n2 24ran

8 EX o o x2 +2x-8 x-2 x2 +4x+3 ' 3x+3 m-1 3m+6 EX 4: i 2m+4 4m-1 EX 5: 2x2-2x-4 2x2+x-6 4-x2 4x2-2x-6

9 EX 6: 4a3 _. 6ab- 18a b2_4' b2_b_6 EX 7: r2+9r+20, r+5 r2_64 r+8 EX 8: 10 x-4 0 x

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11 9-5: Adding and Subtracting RationaJ Expressions i.) Factor denominators 2.) Get common denominators 3.) Make equivalent fractions 4,) Combine numerators, keep denominators 7 2 Ex.1) Ex2) x 2y Ex.3) ÿ-! 4 x+3 x+3

12 Ex.4) 2ÿ+3 5 x+l 3 3 Ex.5) a+2 t 2 a-2 8 x+2 x-2 x2-4 10

13 4 f 2 X.7) 3a2c 6ac2 3 2 Ex.8) ÿ-6 6-ÿ

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15 Complex Fractions Complex Fractions: If the numerator or denominator or both contain fractions, then the expression is called a complex fraction. The fraction bar indicates division. How to simplify: 1. Simplify both numerator and denominator into single fraction expressions. 2. Divide the numerator by the denominator and simplify if possible. Ex. t Q ÿ 8w 3 5

16 5 3 Ex. 3: Ex. 4: X X 2 3 X Ex. 5: 3.2 X ÿ X+4

17 Ex. 6: Ex. 7:?rt -- 7ÿ 2 2 m n Ex. 8: 6 x+l-- X 1 X

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19 Graphing the Reciprocal Parent Function 1,The reciprocal parent function: y ---- ÿ ; X :# 0 Graph: 3-3 y= _ y- m X X.4 aisa a > 0 in Quadrants a < 0 in Quadrants The x and y axes are the asymptotes. the asymptotes: Write the equations of a General form of the reciprocal function: y t- k' X q: h x-h J Vertlcol Asymptote (VA) -Iorizontal Asymptote (HA)

20 Graph: 1. f(x) = -A + 1 VA. HA 3 +4 VA HA 2. Y=x--ÿ 3, y- z+3 z 5 VA HA 6 4, Write the translation of y = - if: X VA:x=2 HA: y=-3 VA:x=O HA:y=5 Shift left 2 and up 5 Shift right 4 and down 3

21 nnll inll i::; i;:;!!!! [111!!!! 1111 ill j i l 'II,il i : I I i ' I I i ' i! i I : : 11:: i1::; i : ] i!! i r ii! :!! ii i i J I1,!!! iiii iiil ilil iiii iiii!!!! IIII IIII i I I II iiii Itl: i II I i'll I I I Ill I I I fill I I llll I I I IIII I t 1 Jill I lill lill I I I IIII I IIII I I IIII I I I IIII I I IIII IIII I I IIII I I IIII IIII I I IIII I I IIII I I lill,' f' Ill I I IHli I f

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23 Rational Functions and Their Graphs Rational Function: quotient of 2 polynomials, 2. Continuous graph: 3, Discontinuous graph: ***For/(x)= ÿ(x )) The graph is discontinuous at values for which h(x) = O, Find the restrictions, The domain is determined before you simprify the function, The restrictions are the DOMAIN!!!! 4, Vertical Asymptotes (VA); 2 A) )'= (x+3)(ÿ;-4) Domain VA x B) Y = x2 +x-6 Doman, VA 'lq oies(rb2rnoiÿable ciiscont lnulti;ÿ)i x2 -I A) y=x2_x_2 Domain VAI Hole at xÿ -3x B) y=x2_x_6 boman VA Hole at

24 6. Horizontal asymptoteÿ (it can have at most! & the graph can cross it 1 time). It is based on the degrees of the numerator and the denominator. A) If the degree of the numerator is larger, no HA, B) If the degree of the denomina'lor is larger, HA at y=o. a C) If degree of numerator = degree of denominator, HA at y = -ÿ, where a & b are the leading coefficients 6x2 + 7x - 2 f(x) = 5x2 _ 3x + 1 HA '-- 2 -ÿ HA. B) y- 3x2 3X2 C) f(x) =ÿ HA, 4x+7 7, X-Intercepts & Y-Znterceptsÿ 2x A) y=x2 9 x-int, y-int 5 x-int. B)ÿV.=:x2..gx y:jn;.... (; + 2) X-i ntl c) y: (x + 1)(x- 4) y-int.

25 Steps to Graphing Rational Functions 1, Factor if possible. Find the domain, Then reduce, if possible, 2, Find the vertical asymptote(s), (This tells you how many sections you will have on your graph). Sketch in the asymptote(s) with a dotted line, 3, Find the holes (if any), 4, Find the x-intercepts (zeros)', 5. Find the y-intercept: 6, Find horizontal asymptote: (it can have I at mast): Graph can only cross one time. It does not have to cross 7, Test points in each section and plot a few points in each section, Graph the following: 3 Ex 1 y:-- X-4 Domain: VA... Holes x-intercepts y-intercept liii!iii...j!!! I I ; : iiii I I : : I I I I J I I : ] I I I ] : iiii! Ill'lit! IIIIIII III!ll Jliiii!!!!!!!!!!!1... i! III'1'...!!! IIJlii IIJIII IIIIII IIIIII!1ii llllllÿ Illlll IIIIJ; ' !!!!!!!!!!!!!! ::::::: ::::::: ::::::: ::::;:: ::lll]l jiiiiii ::::::: iiiiiii :::::::: iiiiiiii HA

26 Ex 2. x+l iiii!! i,i Domain: VA Holes -intercepts y-intercept.!!!!!!!!!11! l : : : : 1, ill::[ II1:::; ::::::: Ji J I I _L i i I I HA..,,,... Ex 3, Domain: VA HoleSl x-intercepts. y-intercept HA. ill ÿl,,t I,l,,,Jl IIIII IIIIIIII I III)1 IIIIIIII III IIIII It111!1t I II IlJll IIIll Ill III II ÿ!ll Illll! II II III III IIIII 111 IIIII IIIIIIII III IIIII IIIIIIII III IIIII IIIIIIII, IIJ" II tjlllll '"'"' IIIII! III IIIII IIIIIII III IIIII IIIIIII III IIIII IIIIIII I II IIIII IIIIIII xz-25 Ex 4. y= x+5 bomain: VA Holes. x-intercepts. y-intercept, HA Illll IIIII!!!! iiii... i+i-iiii: i::: Jiii iiii ::::: iiiii ii1;;

27 _3_ÿx E 5: y - xz_ Domain: VA. Holes <-intercepts. y-intercept, HA. 3 Ex 6 y- ' X2-x-2 i i Domoin;l VA Holes, -intercepts y-intercept. HA

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29 Solving Rational Equations Steps 1o solving rational equations: t 1,Restrict the domain: x ÿ 2. Find the LCD 3. Multiply both sides of the equation by the LCD 4. Reduce/simplify 5. Solve 6.Check for extraneous solutions, Compare the solution(s) to the domain. Examples: = 10 3 x

30 3. x [ X- 5x+8 x+2 x+2 4. x+lo 4 x2-2 x 5x 9x 6. 3 x-'3 x 1 X-3

31 Rational Applications Notes 1 A car traveling 70 miles and a truck traveling 60 miles reach their destinations in the same tlmÿ, The car's rate is 6 mi/hr faster than the truck's rate. Find the rate of the car and the truck. 2, Casey can brick a wall in 6 hours. Kimberly can brick a wall in 5 hours, If they work together how long will it take them to brick a wall? 3, It takes Chris twice as long to clean the house as it takes Erica to clean the house, Together they finish cleaning the whole house in 2 hourÿ, How long does it take each person to separately finish the job?

32 4. Madison can row 4 miles upstream in the same time it takes her to row 6 miles downstream, Her rate of rowing in still water is 2 miles per hour. Find the rate of the current. 5, A new underwater tunnel is being built, One tunne!-boring machine can finish the tunnel in 4 years. A different type of machine can tunnel to the other side in 3 years. If both machines start at opposite ends and work at the same time, when will the tunnel be finished? 6. An oil tank has two inlet pipes and one outlet pipe, One inlet pipe can fill the tank in 12 hours, and the other can fill the tank in 20 hours, The outlet pipe can empty the tank in 10 hours, How long would it take to fill the tank with all three pipes open? 7. Mark was traveling in his car and ran out of gas. By car, he traveled 72 miles... ÿnd thenl;mhad i:0 wÿi-k 4ÿrnore miles ÿreactÿhis ÿdeÿtinÿti6n, 'If he droÿve tlrnes the speed he walked, and it took him 2,5 hours to get there, how fast did he walk?

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