Algebra II - Exponential and Logarithmic Functions - Quick Guides | Mathematics

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Study GuideAlgebra II–Exponential and Logarithmic Functions1.Exponential FunctionsWhat Is an Exponential Function?Anexponential functionis a function where thevariable is in the exponent.It has the general form:Where:•bis thebase•b > 0•b ≠ 1•xis any real numberKey IdeaExponential functions growvery quickly.For example:•(22= 4)•(23= 8)•(27= 128)After only a few steps, the numbers become very large. That is why exponential growth is calledrapidgrowth.Example 1—Graphing (y = 2x)

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Study GuideFigure 1.Exponential growth of 2.To graph an exponential function, we first create atable of values.xy = (2x)Ordered Pair-3(1/8)(-3, 1/8)-2(1/4)(-2, 1/4)-1(1/2)(-1, 1/2)01(0,1)12(1,2)24(2,4)38(3,8)Steps to Graph1.Plot these points on thecoordinate plane.2.Connect them smoothly.Important Observations•The graphpasses through (0,1).

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Study Guide•The graphnever touches the x-axis.•As (x) becomes more negative, (y) getscloser to 0.Thex-axis becomes a horizontal asymptote(the curve approaches it but never touches it).Example 2—Graphing (f(x) = (1/2)x)Figure 2.Exponential growth of 1/2.Now consider the function:Table of Valuesx(f(x))Ordered Pair-38(-3,8)-24(-2,4)-12(-1,2)01(0,1)11/2(1,1/2)

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Study Guide21/4(2,1/4)31/8(3,1/8)What Happens Here?Unlike (2x), this graphdecreases as x increases.This is calledexponential decay.Key Observations•The graph still passes through(0,1).•The curve approaches thex-axis but nevertouches it.Important Property of Exponential FunctionsAll exponential functions:contain the point:becauseGrowth vs DecayExponential GrowthOccurs when:Example:

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Study GuideThe graphincreases rapidly.Exponential DecayOccurs when:Example:The graphdecreasesas (x) increases.Example 3—Graphing (y = 3x) and (x = 3y)Figure 3.In this graph,y= 3xandx= 3y.Now we compare two functions:•(y = 3x)•(x = 3y)Points for (y = 3x)

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Study Guidexy-31/27-21/9-11/3011329327Ordered pairs:(-3, 1/27)(-2, 1/9)(-1, 1/3)(0, 1)(1, 3)(2, 9)(3, 27)Points for (x = 3y)Here weswitch x and y values.yx-31/27

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Study Guide-21/9-11/3011329327Ordered pairs:(1/27, −3)(1/9, −2)(1/3, −1)(1, 0)(3, 1)(9, 2)(27, 3)Relationship Between the GraphsThe graphs of•(y = 3x)•(x = 3y)areinverse functions.Inverse graphs aremirror images across the line (y = x).Key Characteristics of Exponential Functions

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Study Guide1.Domain:All real numbers2.Range:Positivereal numbers3.The graphalways passes through (0,1)4.Thex-axis is a horizontal asymptote5.Ifb > 1 → growth6.If0 < b < 1 → decaySummaryAnexponential functionhas the form:where (b > 0) and (b≠1).Key Ideas•Exponential functions grow or shrinkvery quickly.•Every exponential graphpasses through (0,1).•Thex-axis acts as an asymptote.•Ifb > 1, the graph showsgrowth.•If0 < b < 1, the graph showsdecay.•Inverse exponential functions arereflections across (y=x).2.Quiz: Exponential Functions1. QuestionFor the functionwhat is the value ofy when (x = 4)?

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Study GuideAnswer Choices•12•27•81Correct Answer:81Why This Is CorrectSubstitute (x = 4)into the function.So the value ofy is 81.2. QuestionFor the functionwhat is the value ofy when (x =-5)?Answer Choices•−243•1/81•1/243Correct Answer

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Study GuideWhy This Is CorrectSubstitute (x =-5).Use the negative exponent rule:So the value ofy is 1/243.3. QuestionFor the functionwhich is itsinverse function?Answer Choices•(x = 3y)•(x = y3)•(3 = xy)Correct Answer

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