Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where A polomial is a sum of oe or more moomials. A fuctio give b a is called the leadig coefficiet. a is some real umber (called the coefficiet) other f ( ) a a... a a a is a polomial fuctio of degree. 1 1 1 1 0 The zero fuctio f( ) 0 is a polomial fuctio, ad it has o degree ad o leadig coefficiet. Eamples: Tell whether the give fuctio is a polomial fuctio. If it is, give its degree ad leadig coefficiet. If it is ot, tell wh. f 4 3 ( ) 3 6 1 5 7 3 f ( ) 6 3 3 f ( ) 5 6 4 f ( ) 4 5 9 4 f 3 ( ) 1 f( ) 8-1 -
More Defiitios: f( ) 0 is called the zero fuctio. It has o degree. Its graph is the -ais. f ( ) a, a 0 is called a costat fuctio. Its degree is 0. Its graph is a horizotal lie. f ( ) m b, m 0 is a liear fuctio. Its degree is 1. Its graph is a lie. f ( ) a b c, a 0 is a quadratic fuctio. Its degree is. Its graph is a parabola. Liear Fuctios & Graphs As stated above, f ( ) m b, m 0 is a liear fuctio. Its degree is 1. Its graph is a lie. If the lie is horizotal, the m 0, ad the equatio is reall that of a costat fuctio. If the lie is vertical, m is udefied, ad the lie is ot a fuctio at all (it fails the vertical lie test). A lie that is either horizotal or vertical is called a slat lie. Writig Equatios of Lies Liear equatios ca take o several forms: 1) Slope Itercept Form is the oe above, m b. ) Poit Slope Form is m. 1 1 3) Stadard Form is A B C; A 0; A, B & C are itegers. Eamples: Use the give iformatio to write both the poit-slope ad slope-itercept forms of the liear equatios. A) The lie goes through 3,6 ad 7,14. - -
B) The lie is the liear fuctio g such that g() 4 ad g(5) 13. C) The liear fuctio f has a slope of -4 ad f (5) 8. Where does f cross the horizotal ad vertical aes? D) h is a liear fuctio that crosses the horizotal ais at 8 ad the vertical ais at -4. Write h as a liear fuctio of. Aother Defiitio: The rate of chage of a liear fuctio is the fuctio s slope. Eample: The cost of retig a boat at Bob s Boats o Lake Teoma decreases at a costat rate throughout the moth. O the first of the moth, it costs $80 to ret the boat for the da, but o the 1 th of the moth, it ol costs $58 for the da. A) Write a liear fuctio, C ( ), to represet the cost, C, o da of the moth. B) Use C ( ) to determie the cost of retig a boat o the 5 th da of the moth. - 3 -
Liear Correlatio http://www.stat.uiuc.edu/courses/stat100/java/gcapplet/gcappletframe.html Whe poits i a scatter plot are clustered together alog a lie, we sa there is liear correlatio betwee the quatities represeted b those poits. The correlatio ca be weak, moderatel strog,or strog depedig o how tightl the data poits are clustered. Strog, positive liear relatioship Moderatel strog, positive Weak liear relatioship liear relatioship Estimated r: Estimated r: Estimated r: Strog, egative liear relatioship Moderatel strog, egative liear relatioship Estimated r: Estimated r: - 4 -
REVIEW: Plottig ad Determiig a Liear Regressio Make a scatter plot of this data, fid its liear regressio o our calculator, graph the regressio, the use the regressio to determie f (5). Number of 1 3 4 7 9 1 16 das Amout of Bacteria 11 14 7 35 50 65 USING THE CALCULATOR TO DETERMINE A REGRESSION Algebraic Regressio 1) Press STAT the press Edit (eter data i L1 ad L) ( Xs go i L1, Ys go i L) (Use arrows to chage colums) ) Press Statplot ( d = ) 3) Choose: 1: Plot 1 (highlight it if it is ot alread highlighted) ENTER 4) Highlight: O 5) Arrow dow to the graph that looks like a buch of dots ad highlight it. (This is scatterplot.) 6) Arrow dow to: X list: L1 (make sure it looks like that) Y list: L (make sure it looks like that) 7) Arrow dow to Mark (The little square is easiest to see) 8) Press ZOOM 9 (what do ou see? What tpe of graph do ou thik would fit it best?) 9) Press STAT, the arrow right to CALC. 10) Arrow dow to The regressio ou wat ad ENTER 11) *Operatig Sstem differeces--- As separate etries tpe: L1 ( d 1), (comma) L ( d ), (comma) VARS arrow right to Y- VARS Choose Fuctio ENTER, Choose Y1 ENTER, ENTER or, Table appears ad ou eter L1, L, the for StoreRegEQ select VARS arrow to Y-VARS select Fuctio ad select Y1 1) Graph 13) Set table to ASK Idepedet Variable, ad use the table to predict other outcomes. Directios Fidig the Correlatio Coefficiet (reall ol applicable to liear regressios) 1. Hit d / Zero/ -1. Scroll dow with the arrow util ou fid Diagostics O ad hit eter twice. 3. Now whe ou do our regressio, ou will be give the regressio equatio, its coefficiets, r, ad sometimes r. Hit: r r. Iterpretig r: (the closer to 1 the r is, the better the fit of our chose regressio). See below. Rule of thumb: r <.5 Weak.5 r. 75 Moderatel Strog r. 75 Strog - 5 -
Quadratic Fuctios ad Their Graphs As stated earlier, f ( ) a b c, a 0 Its graph is a parabola. Trasformatios (Review) is a quadratic fuctio. Its degree is. Describe the trasformatios that have take place o f ( ) to obtai the followig. f ( ) 1 f ( ) 3 f ( ) 3 1 Quadratic fuctios are writte i two mai forms: f ( ) a b c is called the stadard quadratic form. f ( ) a h k is called the verte form. Eample: Fid a the write the verte form for the quadratic fuctio with whose graph has verte (, -3) ad additioal poit (4, 6). Sice we kow the basic shape of the graph of a quadratic fuctio (a parabola), it is useful to be able to chage from stadard form to verte form. Doig this requires some algebraic steps. - 6 -
Eamples: Determie the verte ad sketch a graph of the quadratic fuctio f ( ) 1. (Remember, ou ca use the factored form of the quadratic to determie its -itercepts. If factorig is too difficult, ou ca use the quadratic formula, The -coordiate of the verte, h, is foud usig origial fuctio to determie the -coordiate, k. b b 4ac ) a h b a. Oce ou kow h, substitute it ito the Use completig the square to determie the verte of the quadratic fuctio Also determie the fuctios -itercepts ad sketch a graph. f ( ) 6 8. - 7 -
A problem situatio: A cereal compa determies that the umber of boes of cereal, N, is determied b how much the charge per bo. If represets the amout the charge for each bo of cereal, the N( ) 736.50 15358.93. So, for eample, if the charge dollars per bo, the ca epect to sell N() 736.50 15358.93 4904.64 or about 4904 boes of cereal. If N represets the umber of boes sold ad is price per bo, the the compa s icome (or reveue) is determied b N which is 736.50 15358.93 736.50 15358.93. Determie the verte of the graph ad iterpret the meaig of its coordiates. Phsical Applicatios Objects i Vertical Free-Fall 1 You ma have see the equatio s() t gt v0t s0 i a Phsics class. This equatio is used to model the height, s, of a object i free-fall t secods after it leaves its iitial height. s 0 is the iitial height of the object. v is the iitial velocit of the object. 0 g is the acceleratio caused b gravit ad is either 3 ft sec or 9.8m sec. - 8 -
Eample: A object is throw upward from a 10-meter platform with a iitial velocit of 40m. sec Write the fuctio st () to model the object s height, the use the fuctio to determie the maimum height the object will obtai. Also, determie the time at which this maimum height is attaied. Other Questios: Usig the sceario above, determie at what time the object is 5 meters from the groud. Maimizig Area What is the maimum area a rectagle ca attai if the ol coditio set o the legths of its sides is that the perimeter is 140 iches? - 9 -