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# Col Alg for Math,Science,Engin MTH 111C

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This 5 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 111C at Portland Community College taught by Staff in Fall. Since its upload, it has received 40 views. For similar materials see /class/224641/mth-111c-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

Haberman King MTH 1110 Section IV Power Polynomial and Rational Functions Module 1 Power Functions DEFINITION A power function is a function of the form fx kxP where k and p are constants EXAMPLE Which of the following functions are power functions For each power function state the value of the constants k and p in the formula y kxP a bx 5x 34 b mx 74 c lx32x cl sx Z5 SOLUTIONS a The function bx 5x 34 is not a power function because we cannot write it in the form y kxP bThe function mx 74 is a power function because we can rewrite its formula as mx7x14 Sok7 and p c The function 1x 32x is not a power function because the power is not constant In fact 1x 32x is an exponential function J7 xSZ 7x752 cl Since we see that sx can be written in the form y kxP where k J7 and x p so s is a power function a primer functmn we can nd tne tune un s tunnu1a are untne graph uf afu ctm t 5 EXAMPLE Suppuset att e pumts 1 81 and 3729 Fwd an a1gebra1e nuetur assummgthau 1 a ah earfu mu b an expunentta1 tunetmn c a primer tunetmn SOLUT ONS 1t 1s a h earfu ctm W a e k uvvt at ts ru1e nastunn fx 7 mx b We can use tne fur m Wu gwe pumtstu s we Sn uvvvve k uvvt at x 7 324K 5 We can use etnerune uttne gwen pmntstu nd A Let suse 181 1 81 2 1 7 817 3241 5 2 b 7 817 324 2 b 7 7 243 Thus w 1 nean ts ru1e 1 x 7 324K 7 243 V f 1 an expunent1a1 funmun We knew 1ts ru1e nas tunn fx 7 abx We can use tne Wu gwen pmntstu nd Wu Equatmns 1nvu1v1ng nd 5 1 81 2 1 7 817 115 3 729 2 3 7 729 7 ab 1mHa systems uf edua uns by furmmg ratms Let s b 1n Seetmn 1H Mudu1e 2 we sawed s r try a d erent rnetnud nere tne subs tu on rnetnu Let s start by SEIMng tne rst Equatmn fur a 817 115 0 Now we can substitute the expression for a in the second equation 729ab3 2 729b3 2 72981b2 2 3 81 b 3 912 3 b J5 3 we don39t need ixg since the base ofan exponential function is always positive Now that we know what b is we can use the fact that a to find a Thus if f is exponential its rule is fx 273 Since f is a power function we know that its rule has form fx kxp We can use the two given points to find two equations involving k and p 1 81 2 f1 81 k1P 3 729 2 f3 729 k3P We can use the first equation to immediately find k 81k1P 3 k 81 Now we can find p by substituting k 81 into the second equation 729813P E P 3 81 3 note that this could be solved with logarithms P 3 9 3 ifthe solution weren39t so obvious 3 p2 Thus if f is a power function its rule is fx 81x2 Graphs of Power Functions For a power function y kxp the greater the power of p the faster the outputs grow Below are the graphs of six power functions Notice that as the power increases the outputs increase more and more quickly As x increases without bound written Jr gt00quot higher powers of x get a lot larger than ie dominate lower powers of x Note that we are discussing the longterm behavior of the function 1 2 3 4 32 yzxzayzxi yX4 yzxsi The graphs of y x y x As x approaches zero written quotx gt0quot the story is completely different If x is between 0 and 1 x3 is largerthan x4 which is largerthan x5 Try x 01 to confirm this For values of x near zero smaller powers dominate On the graph below notice how on the interval 0 1 the linear power function y x dominates power functions of larger power 1 Thegraphsofyx yx32 yx2yx3 yx yx EXAMPLE Use your graohhg eateutator to graph fx 1000 and w x tor x gt 0 Cumpare the Dngrterm behavmr or these two ruhotohs cum HERE BE sure to turn up the yo1urhe on your cumputer Cou1o the graph or x1000gt and g x e x gt10007 Tu oete h e x hterseot agam tor sorhe ya1ue rmmE vmere tese graphs mters or BEL 1et s so1ye the Equatmn x 0 x g0 L1000x3 0 31000 shoe the oh1y x 1000 hterseot when x gt 1000 so1ut1ohs to m Equatmn are F0 a o F1000 the graph or and g ohty hterseot at F0 and F1000 so they do not EXAMPLE Use your graohhg eateutator to graph the oowertuhouoh fx x3 and the exoohehttat fun m gltxgt 2 tor x gt 0 Compare the ungrterm behavmr these WEI funmuns DUCK HERE amp Kev Poim39 Ahy pusmve mErEaSmg exooheht1a1 tuhmoh eyehtuauy gruvvs taster thah anypEIWErfunmun

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