Domain:x*(Range:y+0Dcx±0Ranee:y>0(a)(b)FIGURE 1.14 Graphs of the power functions f(x) =x"for part (a)a = 1andforpart (b)a =2UF11(a) ) =sin(b) (x) = cos.xFIGURE 1.19'Graphs of the sine and cosine functions(b)eNFIGURE-1.18Graphs of thre
2016/11/15 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y=log2x=1og±3105y= logiot3.y-2-J210.500.50.500.5(a) y = 2', y = 3',y= 10(b) y-2.y 3-f.y 10~FIGURE1.21Graphs of four logarithmicFIGURE 1.20 Graphs of exponential fiunctions.functions.FunctionWhere inereasingWhere decreasingJ=x05x<810050JNowhere00X:00y = 1/xNowhere00≤x<0and0<x<00J = 1/x200<x<00<x<003.Increasing versus Decreasingy=Vi0≤X<00NowhereJ=00<X50Functions0x<00if the graph ofafunction climbs orrises as you movefrom left to right, the function is increasingIfthegraphofa functiondescends orfalls asyoumovefrom left to right, the function is decreas ing.DEFINITIONSAfunction yf(x) is aneven funetion ofx if j() - J(t)4.Even Functions and Oddodd funetion ofx if f(x)= f(x),Functions:Symmetryfor every x in the function's domain
2016/11/15 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. Increasing versus Decreasing Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If the graph of a function climbs or rises as you move from left to right, the function is increasing. If the graph of a function descends or falls as you move from left to right, the function is decreasing. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4. Even Functions and Odd Functions: Symmetry Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The graph ofan evenfunction is symmetric aboutthe y-axisThe graph of anodd fiunctionis symmetric abouthe origin.(a)(a)ihFIGURE1.24(a) When we add the constant term 1 to the functionFIGURE 1.23 (a) The graph of y = x2y x2, the resulting function y x? + 1 is still even and its graph is(an even function) is symmetric about thestll symmetric about the y-axis. (b) When we add the constant term 1 toy-axis (b) Thegraphof y =+ (an oddthe function y = x, the resulting function y = x + 1 is no longer odd.function) is symmetric about the origin.bThe symmetry about the origin is lost (Example 7)f(x)5.FunctionsDefinedinPiecesFIGURE 1.8To graph thefunction y = f(x) shown here,we apply different formulas todifferent parts of its domain(Example 4).Absolute Value Properties1.|-aHa2. [a·b/-[al-(bl3210-/al3./g[b]FIGURE1.7The absolute valuefunction has domain (oo, oo)4.la+b|≤al+[b]and range [0, oo]
2016/11/15 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5. Functions Defined in Pieces Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Value Properties 1.| | | | a a 2. |a b|=|a||b| | | 3. | |= | | a a b b 4. |a+b| |a|+|b|
=x2 +2-26.Tansformations of GraphsofFunctions=x22FIGURE1.29To shift thegraphof f(x) = x up (or down), we addpositive (ornegative)constants to2unitsthe formula for f (Example 3aand b).Add a positiveAdd a negativeconstantto.x.constant to.x.Shift Formulas= (x 2)2Vertical Shitsy=fx) +kShifsthe graph ofupkunits ifk> 0Shifts it down|| units ifk < 0Horizontal Shiftsy= f(x + h)Shifts the graph of lefr h units if A > 0Shifts it righr|a/units ifh <0FIGURE1.30Toshiftthegraphofy=xtotheleft, we add a positive constant to x. To shift thegraph to theright, we addanegativeconstant to(Example 3c),Vertical and Horizontal Scaling and Reflecting FormulasFore>1,y"cfo)Stretches the graph of f vertically by a factor of ey(x)Compresses the graph of f vertically bya factor ofc.y = f(ex)Compressesthegraphofhorizontallybyafactorof.y =f(x/c)Stretches the graph of f horizontally by a factor of e.Fore = 1,y = f(x)Refletsthe graphoffacross the-axisFIGURE 1.31 Shifing the graph ofy =J(x)Reflects the graph of J across the y-axisy = x|2 units to the right and 1 unitdown (Example 3d)
2016/11/15 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6. Tansformations of Graphs of Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley