Mrs. Valentine's Math and Science



Objective: I will be able to identify a relation as a function, evaluate a function for given values, graph a function, and determine the domain and range of a function.Vocabulary:FunctionRelationDomainRangeLinear FunctionIndependent VariableDependent VariableEvaluating a FunctionGraph of a FunctionZeros of a FunctionVertical Line TestY-interceptNotes:RelationsA relation is any set of ______________ _________. -77406528321000The set of all first components is called the domain (______________)The set of all second components is called the range (______________)FunctionsA relation in which each member of the ______________ corresponds to exactly _____ member of the ____________ is a _______________. A function _______________ have one x-value with ________ or more y-valuesA function _______ have two different x-values with the ___________ y-value.Functions as EquationsFunctions are usually given as __________________ rather than sets of ordered pairs. Ex: ___________________________In the example above, x, is the __________________ variable because it can be assigned ________ _____________ from the domain. The ____________________ variable is y because its value ________________ on x. NOTE: Not all equations with the variables x and y define functions. ________ x-value must produce ____________ _______ y-value for the equation to represent a function._______________ _______________When an equation represents a function, the function is often ______________ by a _____________ such as f, g, h, F, G, or H. (NOTE: _____ ___________ can be used.)The special notation _______Read as “___________” or “___________”__ is the __________ and ________ is the _____________ value of the functionRepresents the value of the function at the number xEx: y=0.012x2-0.2x+8.7 becomes ______________________________When a number is input for x, it _______________ x in the special notation (Ex: f(30)). In this case, use 30 as the input and determine the output f(30).Graphs of FunctionsThe graph of a function is the ___________ of its _______________ _________. To graphChoose ____pute fx by ___________________ fx at x.________ the ordered pairs.________ the points.All functions with equations of the form __________________ graph straight lines and are called________________ ______________________________ __________ ________________ ______ _________ in the regular coordinate system _________________ _____________. Can use the vertical line test to determine _____ ___ _________ is a _______________. Values of x paired with _____ or more different values of __ form a vertical line.If any vertical line intersects a graph in _____________ _______ _____ point, the graph ________ _______ _____________ y as a function of x.Obtaining Information from a GraphAt the right or left of a graph you will find closed dots, open dots, or arrows. ______________ dots – the graph does not ____________ beyond this point, and the point ____________ to the graph.__________ dots – the graph does not extend beyond this point, and the point does _________ __________ to the graph._____________ – the graph extends __________________ in the direction in which the arrow points.Identifying Domain and Range from a GraphInterval Notation_______________ _____________ indicate endpoints that are ______________ in an interval._______________ indicate endpoints that are not included in an intervalParentheses are always used with _______ ____ ______Examples: Set-Builder Notation vs. Interval NotationSet-Builder: __________________Interval Notation: ___________Identifying InterceptsThe _______ of a function f are the x-values for which fx=0. (_________________).To find the y-intercept, find the point at which the graph ___________ the y-axis (____).A function can have ___________ than one __-intercept but at __________ _______ __-intercept.Practice:For each of the following relations, determine the domain and range. {(Stern, 95), (Cowell, 95), (Beck, 90), (Winfrey, 82), (McGraw, 82)} {(0,9.1), (10, 6.7), (20,10.7), (30,13.2), (42, 21.7)}For each of the following sets, determine if the relation is a function and explain your answer.3. {(95, Stern), (95, Cowell), (90, Beck), (82, Winfrey), (82, McGraw)}4. {(0,9.1), (10, 6.7), (20,10.7), (30,13.2), (42, 21.7)}Do the following equations define y as a function of x? Explain.5. x2+y=4 6. x2+y2=4 Using the function fx=x2+3x+5, evaluate the function at the indicated values.7. f28. fx+39. f-xGraph each of the following functions on the same graph.196215088900010. fx=2x11. gx=2x+4Use the vertical line test to determine if each of the following graphs represents a function.12. 13. 14. For each of the graphs below, give the domain and range in both set notation and interval notation. Identify any intercepts.15. 16. 17. Obj.: I will be able to evaluate and graph exponential functions. I will be able to recognize the natural base e and use it in an exponential function.VocabularyExponential FunctionBaseNatural BaseNatural Exponential FunctionAsymptoteNotesEvaluate Exponential FunctionsFunctions whose equations contain a _______________ in the ______________ are called exponential functions.The exponential function f with base b is defined by_____________ or _____________ where x is any ________ number__ is a ___________ constant other than one (b>0 and b≠1) called the baseUse a _____________________ to evaluate exponential functions. Graphing Exponential FunctionsRecall that ______________ and ______________-82867512030600In these cases, when the function is in the form bx, x must be ______________, causing ___________ at irrational domain values. The function fx=bx can be graphed at all values of x, meaning the graph will be _______________________.Graphing _______________ values of x and evaluate for fx.Plot the points and connect in a _______________ __________. Note that graphs of exponential functions have a horizontal ________________ (a line they approach but ____________ ____________ or cross).Characteristics of fx=bxDomain:____________; Range: ___________y-intercept: __________; no ___-intercepts_________________ function that has an ______________Horizontal asymptote: _____________(__-axis)Transformations: fx=abc(x-h)+kTransformationEquationDescriptionVertical translationk>0 shift upward k unitsk<0 shift downward k unitsHorizontal translationh>0 (x-h) shift right h unitsh<0 (x+h) shift left h unitsReflectionnegative a reflects over x-axisnegative c reflects over y-axisVertical stretch/shrinka>1 vertically stretches graph0<a<1 vertically shrinks graphHorizontal stretch/shrinkc>1 horizontally shrinks graph0<c<1 horizontally stretches graph The Natural Base eDefined as the value that ______________ approaches as ___ approaches __________. e≈2.718281827 or ________________ when rounded to the nearest hundredthThe function fx=ex is called the _______________ exponential function.PracticeApproximate each number using a calculator. Round to three decimal places1.4-1.5 2.e2.3 Identify the base in each of the following exponential functions.3.f(x)=ex 4.g(x)=10x Explain how each of the following functions was transformed from its parent function (f(x)=2x ).5.hx=2*2x 6.gx=2x-1+27.mx=-2x 8.kx=12*2-3x Graph and determine the domain and range of the following functions. Include the graph and equation of any asymptotes9.hx=3x 10.g(x)=12x 11.f(x)=2x+3 12.h(x)=2x-1 Obj.: I will be able to use compound interest formulas. VocabularyCompound InterestPrincipalCompounded SemiannuallyCompounded QuarterlyContinuous CompoundingNotesCompound InterestInterest computed on your ______________ investment as well as on any _________________ _________________.The initial amount invested is called the _______________, P.The annual percentage rate of interest, r, is compounded once per year.For one year, the total amount in such an account can be represented byThe formula for compound interest over time isInterest can be compounded multiple times per year. When interest is compounded n times a year, we say that there are n compounding periods per year. Name# Compounding periods/yrLength of Each PeriodSemiannual CompoundingQuarterly CompoundingMonthly CompoundingDaily CompoundingThe above formula can be adjusted for the number of compounding periods in a year.This formula gives the total amount with a principal investment, __, compounded __ times per year at an annual rate of __ over __ years._______________________ Compounding Compound interest where the ________________ of _____________________ periods _________________ ______________.The formula for compound interest is PracticeIf you invest $2500.00 in an account, what is the balance in the account and the amount if interest after 4 years if you earn 1.7% interest compounded annually? A credit card company charges 12.9% annual interest. If they compound interest monthly, how much will you owe for every dollar you do not pay off for a year?An initial deposit amount of $5,000.00 is made into a savings account that compounds 7.1% interest annually. How much is in the account at the end of five years?After 80 years of 5.8% interest compounded monthly, an account has $102,393.44. What was the original deposit amount?The value of a $25,000.00 car depreciate at a rate of 12% per year. What will the car be worth in 5 years?He charges 35% annual interest compounded continuously. How much does the gambler owe the loan shark at the end of one year? You take out a 15-year, $50,000.00 loan for college at 5% annual interest, compounded continuously. How much will you have paid back by the end of the 15 years?What is the principal for a continuously compounded account earning 3.9% for 15 years that now has a balance of $2,500,000.00?Obj.: I will be able to identify logarithmic functions and their properties. I will be able to convert between logarithmic and exponential forms. VocabularyLogarithmic FunctionLogarithmExponential FormInverse Properties of LogarithmsCommon Logarithmic FunctionNatural Logarithmic FunctionNotesInverse FunctionsOnly one-to-one functions have inverses that are functions. Pass the horizontal line testInverse found by switching x and y, then solve for yIf fa=b then f-1b=a (switch the domain and range)f(f-1x)=x and f-1(fx)=xGraph of f-1 is the reflection of f about the line y=xLogarithmic FunctionsThe inverse of y=bx is x=byNew notation needed to solve for y, called logarithmic notationThe inverse function of the exponential function with base b is called the logarithmic function with base b.y=logbx is equivalent to by=x for x>0 and b>0, b≠1y=logbx is called logarithmic formby=x is called exponential formKnowing where the base and exponent are in each form will allow you to convert between the forms.Evaluating LogarithmsRemembering that logarithms are exponents makes it possible to evaluate some logarithms by inspection. Ex: Evaluate log232Ask: 2 to what power gives 32? 25=32log232=5Basic Logarithmic PropertiesProperties Involving 1logbb=1 because 1 is the exponent to which b must be raised to obtain b (b1=b)logb1=0 because 0 is the exponent to which b must be raised to obtain 1 (b0=1)Inverse Properties of Logarithms (for b>0 and b≠1)logbbx=xblogbx=xCommon LogarithmsLogarithmic function with base 10 is called the common logarithmic functionUsually expressed as fx=logx.Uses LOG key on calculatorProperties:Natural LogarithmsLogarithmic function with base e is called the natural logarithmic function.Usually expressed as fx=lnxUses LN key on calculatorProperties:PracticeWrite the equations in their equivalent exponential forms.1.log232=5 2.log6216=3 3.log864=2 4.log1000=4 Write the equations in their equivalent logarithmic forms. 5.162=256 6.14-4=256 7.7-3=1343 8.3612=6 Evaluate the logarithms to find exact answers without a calculator.9.log416 10.log218 11.log1100 12.lne Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group f(t), after t months, was modeled by the function ft=85-16lnt+1, 0≤t≤1213. What was the average score on the original exam? 14. What was the average score after 5 months?Obj.: I will be able to graph a logarithmic function including any transformations, and I will be able to determine its domain.VocabularyDomain of Logarithmic FunctionNotes________________ of Logarithmic FunctionsUse the fact that ______________________ functions are the _________________ of _______________________ functionsRecall that the graph of a logarithmic function is the __________________ of its _____________ over the line ____________.The ____________ will determine the shape of the graph: Characteristics of Logarithmic Functions in the form fx=log bxDomain: __________ ; Range: ___________Graphs of all logarithmic parent functions pass through _______There is _____ y-interceptAsymptote: _______Transformations of fx=a log bcx-h+kTransformationEquationDescriptionVertical translationk>0 shift upward k unitsk<0 shift downward k unitsHorizontal translationh>0 (x-h) shift right h units (asymptote: x = c)h<0 (x+h) shift left h units (asymptote: x = – c)Reflectionnegative a reflects over x-axisnegative c reflects over y-axisVertical stretch/shrinka>1 vertically stretches graph0<a<1 vertically shrinks graphHorizontal stretch/shrinkc>1 horizontally shrinks graph0<c<1 horizontally stretches graph PracticeGraph the following functions.1.fx=2x and gx=log2x 2.fx=14x and gx=log14x Describe the transformations of each function from its parent function.3.fx=12log(x+1) 4.fx=logx-1 5.fx=ln?(5-x) 6.fx=-2log6x Find the domain of each logarithmic function. Give the equation of the vertical asymptote7.fx=ln(x-2) 8.fx=logx+3 9.fx=log5x+62 10.fx=-log2(7-x) Obj.: I will be able to expand and/or condense logarithmic expressions using the product, quotient, and power rules. I will be able to convert from one base to another using the change-of-base property.VocabularyProduct RuleQuotient RulePower RuleExpanding a Logarithmic ExpressionCondense a Logarithmic ExpressionNotesThe _______________ RuleRecall that when multiplying the same base, exponents are added: _________________Since ______________ are _____________, the product rule can be applied:b, M & N are _____________ ________ numbers; b≠1When the product rule is applied, it is called _______________ a logarithmic expression.The _______________ RuleRecall that when dividing the same base, exponents are subtracted: ________________Since logarithms are exponents, the _____________ ________ can be applied:b, M & N are positive real numbers; ___________The ____________ RuleRecall that when raising an exponential expression to a power, multiply exponents: ______________________Since logarithms are exponents, the ____________ _________ can be applied:b & M are positive real numbers; b≠1; p is ________ __________ numberNote that the graph of the expanded form may have a ________________ ___________ than the original.Expanding & Condensing Logarithmic ExpressionsSometimes, ____________ ________ ______ rule may need to be used when ________________ or ___________________ a logarithmic expression.Condensing a logarithmic function uses the ____________ of the above properties.________________ of logarithms must be ____ before you can condense them using the product and quotient rules.Change-of-Base PropertyFor any logarithmic bases a and b, and any positive number M, Base __ is the ______________ base while base __ is the ______ base being introduced.This property can be used with _____________ logarithms and ______________ logarithms, as well.PracticeUse properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.1.log5(7*3) 2.log9(9x) 3.logx100 4.log5125x 5.logbx2yz 6.log8x64 Use properties of logarithms to condense each logarithmic expression.7.2 logbx+3 logby 8.log296-log23 9.13[5lnx+6-lnx-lnx2-25] 10.logx+log15+log(x2-4)-log(x+2)Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.11.log513 12.log0.319 Obj.: I will be able to solve exponential and logarithmic equations. I will be able to apply solving these equations to practical applications.VocabularyExponential EquationLogarithmic EquationNotesExponential EquationsAn exponential equation is an ____________ containing a _____________ in an _________________. Some exponential equations can be solved by expressing each side as a power of the _________ _________. If b > 0 and b≠1, and _____________, then ________.Most _______________ be written so that each side has the same base. Use ________________!If ________, then __________________If the exponential equations involves base _____, use ____________ logarithms. If it involves any other base, use ________________ logarithms.Using the Definition of a Logarithm to Solve Logarithmic EquationsA logarithmic equation is an _______________ containing a __________________ in a __________________ ________________.Some logarithmic equations can be expressed in the form ____________Re-write these in _____________________ form (____________) and solve for the variable. Check _______________ _______________. Include only values for which ________.Note: ___ can still be a ______________ number so long as _______.Some logarithmic equations can be expressed in the form ____________________Use the _____________________ property to write without logarithms (__________) and solve for the variable._____________ proposed solutions. Include only values for which _______ and _______.______________________ Both SidesWrite both sides of the equation as _________________ of the __________________ _________. Remember that ________________PracticeSolve each exponential equation. Express the solution in terms of natural logarithms and then use the calculator to approximate an answer to two decimal places. 1.3x=81 2.6x=1296 3.40e0.6x+2=242 4.54x-7-3=10 Solve each logarithmic equation. Reject any value of x that produces a logarithm of a negative number or zero. 5.5ln(2x)=20 6.7+3lnx=6 7.log2x+log2x-7=3 8.log2x-4=3 The formula A=18.9e0.0055t models the population of New York State, A, in millions, t years after 2000. What is the population of New York in 2000? When will the population of New York reach 19.6 million? The pH of a solution ranges from 0 to 14. An acid solution has a pH less than 7. Pure water is neutral and has a pH of 7. Normal, unpolluted rain has a pH of about 5.6. The pH of a solution is given by pH=-logx where x represents the concentration of the hydrogen ions in the solution in moles per liter. The most acidic acid rainfall ever had a pH of 2.4. What was the hydrogen ion concentration? Express the answer as a power of 10, then round to the nearest thousandth. The formula A=Pert describes the accumulated value, A, of a sum of money, P, the principal, after t years at annual percentage rate r (in decimal form) compounded continuously. Complete the table for a savings account subject to continuous compounding.Amount InvestedAnnual Interest RateAccumulated AmountTime t in years$50005.8%Triple the amount investedObj.: I will be able to model exponential growth and decay. I will be able to use logistic growth models for limited growth applications and to re-write an exponential equation in the natural base.VocabularyExponential GrowthExponential DecayHalf-LifeCorrelation EffectNotesModel Exponential Growth and DecayWith exponential ___________ or ___________, quantities grow or decay at a rate ____________ __________________ to their __________. Given by the model:If _______, the function models the amount, or size, of a _____________ entity. ____ is the _______________ amount of the ____________ entity at time ____________ is the __________ at time _______ is a _________________ representing _____________ _________Ex: population growthIf _______, the function models the amount, or size, of a _____________ entity. ____ is a ______________ representing ______________ _________Ex: half-life______________ Growth ModelsLogistic growth model: function used to ____________ situations in which ______________ is ____________. Mathematical Model:___, ___, and ___ are __________________ with ______ and ________________ is a _________________ ____________ for the graph of the function___ represents the _________ size that A can attain (A cannot be bigger than c)Modeling DataScatter plots can show data that is exponential, logarithmic, and/or linear. A _______________ ____________, r, is a measure of how well the model fits the data. _________; _____ indicates a _______ relation. ______ indicates an ___________ relation. The closer to 1 or -1 r is, the ___________ a model ______ the data. Expressing an Exponential Model in Base eRecall that ____________ is ____________________ to ______________________________ the equation in this ____________ the ________ of growth or decay. Practice:Low interest rates, easy credit, and strong demand from new immigrants have driven up the average sales price of new one-family houses in the US. In 1995, the average sales price was $158,700 and by 2000, it had increased to $207,200. Use the exponential growth model A=A0ekt, in which t is the number of years after 1995, to find the exponential growth function that models the data. According to the model, by which year will the average sales price of a new one-family house reach $300,000?Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14. How much of an original sample of 394kg of c-14 will be remaining after 6942 years?The exponential model A=1081.2e0.022t describes the population, A, of a country in millions, t years after 2010. Use the model to determine when the population of the country will be 3048 million.The exponential model A=838.4e0.026t describes the population, A, of a country in millions, t years after 2005. Use the model to determine when the population of the country will be 1947 million. The function ft=300001+20e-1.5t describes the number of people, f(t), who have become ill with influenza weeks after its initial outbreak in a town with 30,000 inhabitants. How many people were ill by the end of the fourth week? The function ft=5001+83.3e-0.162t describes the population, f(t), of an endangered species of birds t years after they are introduced to a nonthreatening habitat. How many birds are expected in the habitat after 10 years?Write each of the following in terms of base e.y=1004.6xy=2.50.7xFor the scatter plots given below, determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data.10. ................
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