CALCULUS BASIC SUMMER REVIEW

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Randolf Gibbs
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1 CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept= b Geeral Form of Liear Equatio are ot zero. A By C such that A ad B both What is a Fuctio? A fuctio is a relatio that assigs a sigle elemet of R to each elemet of D. A workig defiitio of a fuctio is that it is a devise that assigs a output to every allowable iput. The iputs make up the domai of the fuctio. The outputs make up the rage. A Fuctio must pass the vertical Lie Test Vertical Lie Test by egative umber Idetifyig the Domai ad Rage: Remember, i the real umber system you ca ot divide zero or fid the eve root of a Eve ad Odd Fuctios A fuctio y = f() is a eve fuctio of if f(-) = f() for every i the fuctio s domai. A fuctio is a odd fuctio of if f(-) = - f() for every i the fuctio s domai. Page

2 Absolute Value Thik of the absolute value fuctio as a piecewise fuctio. The Greatest Iteger Fuctio f ( ) or f ( ) it( ) f ( ) if 0 if <0 The greatest iteger fuctio represets the greatest iteger less tha or equal to Compositio of Fuctios The compositio f g of the fuctios f ad g is defied by ( f g)( ) f ( g( )) The domai of domai of f. ( f g)( ) f ( g( )) cosists of those s for which g() is i the Geometric Trasformatios: Shifts, Reflectios, Stretches, ad Shriks Graph Shiftig Formulas Vertical shifts of the graph of y f () y f ( ) c Shifts graph of y f () dow c uits y f ( ) c Shifts graph of y f () up c uits Horizotal shifts of the graph of y f ( c) Shifts graph of y f () right c uits y f ( c) Shifts graph of y f () left c uits How to stretch or shrik a graph To stretch the parabola each y-coordiate by c. y vertically by a factor of c (c>0), we must multiply If you stretch the graph by a factor of two the ew equatio will be: y How to reflect a graph To reflect the graph of y=f() across the y-ais, we multiply each y coordiate by -. Reflectio Formulas: With respect to the y-ais y f ( ) With respect to the -ais y f () The Parabola y a b c A parabola that opes i the positive y directio if a>0 ad i the egative y directio if a<0. b b b The ais of symmetry is: The verte is at: (, f ( )) a a a POLYNOMIALS Page

3 Polyomial Epressio: a a a... a a0 Polyomial Fuctio: f ( ) a a a... a a0 Polyomial Equatio: a a a... a a0 0 Ratioal Zeros Theorem Suppose all the coefficiets i the polyomial fuctio f ( ) a a a... a a0 are itegers. c If s a ratioal zero of f, where c ad d have o commo factors, the c is a factor d of a 0, ad d is a factor of the leadig coefficiet a. How to Solve f()= 0 usig calculator or your ow brai!!!!. Fid the eact solutio algebraically (ofte by factorig). Draw a complete graph a) Use ZOOM-IN b) Use SOLVE Steps for Solvig a Problem. Fid a algebraic represetatio ivolvig variables.. Draw a complete graph of the fuctio. Fid the domai ad rage 4. Determie the values that make sese i the give situatio 5. Draw a graph of the problem situatio 6. Solve the problem usig appropriate methods For istace: Solve 4 0 Factors of c:,,, 4, 6, Factors of d:, c Possible zeros:,,, 4, 6,,, d Look at the graph to see the zeroes must be betwee - ad - or ad 4. So f ( / ) 0 ( / ) ( ) So ( ) is a factor. By divisio ( )( 4) 0 Use the Quadratic Formula to fid 5 Equatios for Circles i the Plae Circle is the set of poits i a plae whose distace from a fied poit i the plae is a costat. The fied poit is the ceter of the circle. The costat distace is the radius of the circle. Equatio: ( h) ( y k) r Iverse Relatios ad Fuctios: Page

Horizotal Lie Test : The iverse relatio R of the relatio R is a fuctio if ad oly if every horizotal lie itersects the graph of R i at most oe poit.

4 Iverse Relatio: Let R be a relatio. The iverse relatio R of R cosists of all those ordered pairs (b,a) for which (a,b) belogs to R. So the domai of R = the rage of R ad the rage of R = the domai of R. Horizotal Lie Test : The iverse relatio R of the relatio R is a fuctio if ad oly if every horizotal lie itersects the graph of R i at most oe poit. Notice that the iverse of f ( ) 6 is ot a fuctio sice f() fails the horizotal lie test. Oe-to-Oe: The iverse f of a fuctio f is a fuctio if ad oly if f is a oe-to-oe fuctio. Epoetial Fuctios: Defiitio: Let a be a positive real umber other tha. The fuctio f ( ) a whose domai is (, ) ad whose rage is ( 0, ) is the epoetial fuctio with base a. Trigoometric Fuctios: Uit Circle Page 4

Graph of the Sie Curve: y = si() Graph of

Taget Curve: Coic Sectios y = ta() Circle

Defiitio: A coic sectio is the itersectio of

Ellipse (v) Parabola (v) Hyperbola (v) By

5 Graph of the Sie Curve: y = si() Graph of the Cosie Curve: y = cos() Graph of the Taget Curve: Coic Sectios y = ta() Circle Ellipse (h) Parabola (h) Hyperbola (h) Defiitio: A coic sectio is the itersectio of a plae ad a coe. Ellipse (v) Parabola (v) Hyperbola (v) By chagig the agle ad locatio of itersectio, we ca produce a circle, ellipse, parabola or Page 5

... 7 (9 7 5) = 5 7 = (4 5) = 8. (4i ) (i ) = 9. (4i)(i ) = 0. 5[4( y ) ( y )] =. = 4. 0 0 (5 ) a = 5. (7 y) ( 5 y) = 6. (7 y)( 5 y) = 7.

6 hyperbola; or i the special case whe the plae touches the verte: a poit, lie or itersectig lies. Poit Lie Double Lie The Geeral Equatio for a Coic Sectio: A + By + Cy + D + Ey + F = 0 psimplify the followig algebraic ad umeric epressios (9 7 5) = 5 7 = (4 5) = 8. (4i ) (i ) = 9. (4i)(i ) = 0. 5[4( y ) ( y )] =. = (5 ) a = 5. (7 y) ( 5 y) = 6. (7 y)( 5 y) = 7. (4i ) (i ) = Simplify without a calculator, givig aswer i eact form (ot decimal). I your aswer, epress all epoets as positive values ad covert ay fractioal powers to radical form y z = z y t t 5 5t 8 5t Page 6

7 y z ( ) 6 y z ( ) (6 ) ( ) 7 98 Fid the eact value without your calculator (o decimal aswers) Solve each equatio algebraically; verify your solutio by substitutig i the origial equatio. 5. ( 7) y solve for i terms of ( ) ( 7)( ) 6. c, where a b a 0, b 0, b a, solve for.. Solve by completig the square: Fid the solutio to the system of equatios. y y 9 a 7. r s, solve for r. 9. y 98y 4 Page 7

8 4. Determie the slope betwee the poits (4, -) ad (-6, 4). 40. Determie the slope of the lie -y = Write i slope-itercept form the equatio of the lie cotaiig the poit (-, ) ad parallel to the give lie y = Write i slope-itercept form the equatio of the lie cotaiig the poit (4, 5) ad perpedicular to the give lie y = 6. You should kow how to quickly sketch the graphs of these five basic paret fuctios: a) y b) y c) y d) y e) y 4. From the paret graph of y ( ) 5 ad graph the fuctio. y describe the shift to obtai the ew graph of 44. From the paret graph of y describe the shift to obtai the ew graph of y ad graph the fuctio. 45. State whether the give set of poits is a relatio or a fuctio{(,),(,0),(,0),(,)}. 46. For the poits give F= {(,),(,0),(,0),(,)}., state the domai ad rage. 47. State the domai ad rage of the fuctio h ( ) ad vertical ad horizotal asymptotes if ay eist. 48. Fid the domai, rage, zero(s), ad y-itercept of f ( ) ad verify by graphig. 49. Fid the domai, rage, zero(s), ad y-itercept of g( ) 4 ad verify by graphig. 50. Give 5. Give f ( ) fid its iverse f ( ) ad the sketch the graph of both. g( ) 4 fid its iverse g ( ) ad the sketch the graph of both. For #5 ad 5 use the followig: 5. Fid f ( g( )) f g 4 ( ) ; ( ) Fid g( f ( )) Factor the followig: 58. 9y y 4z y y Page 8

9 Summer Readig AP Calculus: Complete the idicated operatio to simplify the polyomials. Ratioal aswers should have a commo deomiator What are the legths of the missig sides of a triagle if the loger leg of the triagle is 8 cetimeters? 66. The hypoteuse of a triagle is iches. Fid the measures of the other two sides. 68. What is the legth of the hypoteuse of a triagle if oe leg measures 9 cetimeters? 69. The leg of a triagle is 4 cetimeters. Fid the measures of the other two sides. 70. If the radius of a circle is 6 cetimeters, what is its eact circumferece? 7. If the radius of a circle is 6 cetimeters, what is its area? 7. What is the area of a triagle with base of 7 cm ad altitude to the base of 4 cm? 7. If the base of a parallelogram is 5 iches ad altitude to the base is oe third of the base, what is the area of this parallelogram? 9

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.