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2. Functions and Graphs 2.1 Functions Cartesian Coordinate System y 9 x2 II I III IV A function is a rule (process or method) that produces a correspondence between two sets of elements such that to each element in the first set, there is corresponds one and only one element in the second set. The first set is called the domain and the second set is called the range. 5 1 a w A x 10 2 b x B y 15 3 y C y x x 1, x 1 x f f(x) x exactly one y function (rule for finding y) For every input, there is exactly one output. Rule: Compute y by dividing x by 5 f x x 5 4 y 3x 8 y2 x2 9 Vertical line test An equation defines a function if each vertical line in the coordinate system passes through at most one point in the graph of the equation. If any vertical line passes through two or more points on the graph of the equation, then the equation does not define a function. If a function is specified by an equation and the domain is not indicated, then we assume the domain is the set of all real number replacements of the input variable that produce real values for the output variable. In many applied problems, the domain is determined by practical considerations within the problem. meaningful input – domain corresponding output – range Finding the Domain 1 1 y y 4 x y x 4 x The symbol f x For any element x in the domain of the function f, the symbol f x represents an element in the range of f , which corresponds to x in the domain of f. If x is an input value, then f x is the corresponding output value. If x is not in the domain of f, then f is not defined at x, and f x does not exist. 12 f x g x 1 x2 h x x 1 x2 f 6, f 2, g 2, f 2, f 0 g 1 h10 Domains? f x x2 2x 7 f a f a h f a h f a f a h f a h Applications Cost function Cost=(fixed costs)+(variable costs) C a bx Price-Demand function p m nx Revenue function Revenue=(number sold) * (price per item) Profit function P R C A camera manufacturer wholesales to retail outlets across the US. The company produced price-demand data per the following table: x(millions) p($) 2 87 5 68 8 53 12 37 The company then modeled the data to get the price-demand function p x 94.8 5 x , 1 x 15 2.2 Elementary Functions: Graphs and Transformations f x x2 g x x2 4 h x x 4 2 k x 4x 2 Library of Elementary Functions (pictures on p.60) f x x h x x 2 m x x3 n x x p x 3 x g x x x 64 x 12.75 Vertical and horizontal shifts f x x f x x 4 f x x 5 f x x 4 f x x 5 Reflections, Stretches, and Shrinks f x x f x 2 x f x 0.5 x f x x f x 2 x Graph Transformations Vertical translation k 0 y f x k k 0 Horizontal translation h 0 y f x h h 0 Reflection y f x Vertical expansion/contraction A 1 y Af x 0 A 1 y x 3 1 Piecewise defined functions x , x 0 x x, x 0 Utilities Easton Utilities uses the rates from the table below to compute each customer’s monthly natural gas bill. $0.7866 per CCF for the first 5 CCF $0.4601 per CCF for the next 35 CCF $0.2508 per CCF for all over 40 CCF 1.2 Linear Function and Straight Lines Intercepts x-intercept y-intercept Linear and Constant functions A function f is a linear function if f x mx b , m 0 , where m and b are real numbers. The domain and range are both the set of all reals. If m 0 , then f is the constant function f x b , in which the domain is the set of all reals, and the range is the constant b. 3 Graph f x x 4 2 Find the x- and y- intercepts 3 Solve for x: x 4 0 2 Graphs of Ax By C If A 0 If B 0 In the Cartesian plane, the graph of any equation of the form Ax By C , where A, B and C are real constants (A and B not both 0), is a straight line. Every straight line in the Cartesian plane is a graph of this type. Vertical and horizontal lines are a special case of this equation. x 4 y6 2 x 3 y 12 Slope of a Line If a line passes through two distinct points P x1 , y1 and P2 x2 , y2 , then its slope is given by 1 y rise y2 y1 m x1 x2 x run x2 x1 Find the slope of a line from 1,2 to 4,3. Find the slope of a line from 1,3 to 2,3. Find the slope of a line from 2,3 to 3,3. Find the slope of a line from 2,4 to 2,2. Slope-intercept form The equation y mx b, m is the slope, 0, b is the y-intercept, is called the slope-intercept form of an equation of a line. 2 Find the slope and y-intercept of y x 3 3 Write the equation of a line with a slope 2 3 and y-intercept 0,2 . Point-Slope form An equation of a line with slope m that passes through the point x1 , y1 is y y1 m x x1 , which is called the point-slope form of an equation of a line. Find the equation of a line that has a slope ½ and passes through the point 4,3. Find the equation of a line that passes through the two points 3,2 and 4,5. Equations of a line (summary) Standard form Ax By C Slope-intercept form y mx b Point-slope form y y1 m x x1 Horizontal line y b Vertical line xa The management of a company that manufactures skateboards has fixed costs of $300 per day and total costs of $4300 per day at an output of 100 pairs of skateboards per day. Assume cost C is linearly related to the output x. Find the slope of a line joining the points associated with outputs of 0 and 100. Find an equation of the line relating output to cost. Graph the cost equation from 0 x 200 Supply and Demand At a price of $9/box of oranges, the supply is 320k boxes while the demand is 200k. At $8.50/box, supply is 270k, while demand is 300k. (x,p) Find price-supply equation p mx b . Find price-demand equation p mx b . Find equilibrium price and quantity. Price-Demand (skip) At the beginning of the 21st century, the world demand for crude oil was about 75 million barrels per day and the price of a barrel fluctuated between $20 and $40. Suppose the daily demand for crude oil is 76.1 million barrels when the price is $25.52 per barrel and the demand drops to 74.9 million barrels when the price rises to $33.68. Assuming a linear relationship between the demand x and the price p, find a linear function that models the price- demand relationship for crude oil. Use this to predict the demand if the price rises to $39.12. The daily supply of crude oil also varies with the price. Suppose that the daily supply is 73.4 million barrels when the price is $23.84, and this supply rises to 77.4 million barrels when the price rises to $34.24. Assuming a linear relationship between supply x and price p, find a linear function that models the price-supply relationship for crude oil. Use this model to predict the supply if the price drops to $20.98 per barrel. The price tends to stabilize at the point of intersection of the demand and supply functions. This point is called the equilibrium point. 2.6 x 167 6.8 x 543 2.3 Quadratic Functions If a, b, and c are real numbers with a 0 , then the function f x ax 2 bx c is a quadratic function and its graph is a parabola. Solution methods: Square root property Factoring Quadratic Formula Completing the Square Sketch a graph of f x x 2 5 x 3 in the rectangular coordinate system, and find its intercepts. Solve the quadratic inequality x 2 5x 3 0 graphically and symbolically. How many intercepts are possible? Vertex form: f x a x h k f x 2 x 2 16 x 24 Given the quadratic function f x 0.5 x 2 6 x 21 Find the vertex form for f. Find the max/min of the function. State the range. Discuss the relationship between f x and g x x2 . A camera manufacturer wholesales to retail outlets across the US. The company modeled supply and demand data to get the price-demand function p x 94.8 5 x , 1 x 15. The revenue function is therefore R x xp x x94.8 5x . What price will maximize revenue? Given production costs at C x 156 19.7 x , what amount of cameras will maximize profit? What is the wholesale price that will maximize profit? Where are our break-even points? 2.4 Polynomial and Rational Functions Constant fcn: f x b Linear function: f x mx b Quadratic function: f x ax 2 bx c Cubic function: f x ax 3 bx 2 cx d A polynomial function is a function that can be written in the form f x an x n a2 x 2 a1 x a0 for n, a non-negative integer, called the degree of the polynomial. The coefficients, an , an 1 , a1 , a0 are real numbers with an 0 . The domain of a polynomial function is the set of all real numbers. Turning points and x intercepts of Polynomials (skip) The graph of a polynomial function of positive degree can have at most n 1 turning points and can cross the x axis at most n times. Polynomial Root Approximation (skip) If r is a zero of the polynomial P x x n an1 x n1 an2 x n2 a1 x a0 then r 1 maxan 1 , an 2 a1 , a0 . Approximate (to four decimal places) the real zeros of P x 2 x 4 5 x3 4 x 2 3x 6 Using the length of a fish to estimate its weight is of interest to both scientists and sport anglers. The data in the table gives the average weights of lake trout for certain lengths. Find a polynomial model that can be used to find the weights of lake trout for certain lengths. x 10 14 18 22 26 30 34 38 44 y 5 12 26 56 96 152 226 326 536 Rational Functions A rational function is any function that can be n x written in the form f x , where n x and d x d x are polynomials and d x 0 . The domain is the set of all real numbers such that d x 0 . Find the domain and the intercepts for the rational function 3x f x 2 x 4 3x Graph f x 2 x 4 (skip) Finding vertical and horizontal asymptotes n x Let f x be a rational function in lowest terms. d x To find a vertical asymptote, solve d x 0 for x. If a is a real number such that d x 0 , then x a is a vertical asymptote of the graph of y f x . (if a is a zero of both n x and d x , then f x is not in lowest terms. Factor out x a from both. Horizontal asymptote (divide by highest pwr of x) 1) If the degree of n x is less than the degree of d x , y 0 is a horizontal asymptote. 2) If the degree of n x is equal to the degree of d x , then y a is a horizontal asymptote, b where a and b are the leading coefficients of n x and d x . 3) If the degree of n x is greater than the degree of d x , there is no horizontal asymptote. Graphing Rational Functions 3x Given the rational function f x x2 4 Find intercepts and equations for any vertical or horizontal asymptotes Using this information and additional points as necessary, sketch a graph of f for 7 x 7 and 7 y 7. x 1 Find asymptotes 2 x 1 A company that manufactures computers has established that, on the average, a new employee can assemble N t components per day after t days of on-the-job training, as given 50t by N t , t 0 . Sketch a graph of N, t4 0 t 100 , including any vertical or horizontal asymptotes. 2.2 Exponential Functions Power function: f x x 2 Exponential fcn: g x 2 x A function f represented by f x b x where b 0, b 1, is an exponential function with base b. The domain of f is the set of all real numbers and the range of f is the set of all positive real numbers. 2 x is a reflection of 2 x Basic Properties of f x b x , b 0, b 1 1. All graphs will pass through the point 0,1. 2. All graphs are continuous curves with no holes or jumps. 3. The x-axis is a horizontal asymptote. 4. If b 1, then b x increases as x increases. 5. If 0 b 1, then b x decreases as x increases. Sketch a graph of f x 1 4 x , 2 x 2 2 Properties of Exponential Functions a x a y a x y ax a x y ay a a x y x y ab a xb x x ax ay x y ax bx a b Natural Exponentiation For calculation purposes, assume one puts $1 in a savings account for 1 year at 100% interest. Compounding m A1 Annual 1 2 Monthly 12 2.613035 Daily 365 2.714567 Hourly 8760 2.718127 Minutely 525,600 2.718279 Secondly 31,536,000 2.718282 Continuous Euler’s number – irrational, like e 2.718281828459... f x e x - natural exponentiation Calculator exercise e1 e 0.5 e 2.56 Exponential functions with base e are defined by y ex y e x Cholera bacteria multiplies exponentially by cell division as given approximately by N N 0e1.386t , where N is the number of bacteria present after t hours and N 0 is the number initially present. If we start with 25 bacteria, how many bacteria will be present In 0.6 hour? In 3.5 hours? Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react woth nitrogen to produce radioactive Carbon-14. Carbon-14 enters all living tissues through CO2, which is first absorbed by plants. Carbon-14 is maintained at a constant level until the organism dies, at which point, it decays according to A A0e 0.000124t . If 500 mg is present in a sample from a skull at the time of death, how much is present after 15,000 years? After 45,000 years? Compound Interest If P (present value) dollars is invested an annual rate of interest r, compounded m times per year, then after t years, the account will contain A (future value) dollars, where mt r A P 1 m If $1000 is invested at 10%, compounded monthly, how much will be in the account after 10 years? Continuous compounded interest mt r A P 1 m Starting with P=100, r=0.08, and t=2 years, examine m as m increases without bound. Compounding m A Annual 1 116.64 Semiannually 2 116.9859 Quarterly 4 117.1659 Weekly 52 117.3367 Daily 365 117.349 Hourly 8760 117.351 If a principal P is invested at an annual rate r compounded continuously, then the amount A at the end of t years is given by A Pe rt . If $5000 is invested for two years at an annual rate of 8%, what is the value of the account if the interest is Compounded daily? Compounded continuously? Simple Interest: A P1 rt mt r Compound Interest: A P1 m Continuously compounded interest: A Pe rt 2.6 Logarithmic Functions Inverse Functions Reversible actions f x 8 x Gal to pints g x f 1 x x 8 Pts to gal ABC…XYZ CDE…ZAB HELP JGNR Open door, get in, close door, start engine Shut off eng, open door, get out, close door x7 2 2x 7 Notation x 1 f 1 x % time skies cloudy in Augusta GA 1 2 3 4 5 6 7 8 9 10 11 12 43 40 39 29 28 26 27 25 30 26 31 39 f 3 f 1 A function is a one-to-one function if, for elements c and d in the domain of f, c d implies f c f d (different inputs result in different outputs) If f is a one-to-one function, then the inverse of f is the function formed by interchanging the independent and dependent variables for f. If a, b is on the graph of f, then b, a is on the graph of f 1 x . f :a b f 1 : b a Let f x x 3 2 . Find and verify inverse func Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Common Logarithms g x log x x 10 k log x log 10 k k The common logarithm of a positive number x, denoted log x , is defined by log x k if and only if x 10k where k is a real number. The function given by f x log x is called the common logarithm function. log 1 1 log 1000 log 10 Y1 10 X Y2 log X Logarithms with other bases a 1 The logarithm with base a of a positive number x, denoted log a x , is defined by log a x k if and only if x a k where a 0, a 1, and k is a real number. The function given by f x log a x is called the logarithmic function with base a. (a logarithm is an exponent) If x 2k for some k, then log 2 x k . Natural logarithms If x e k for some k, then ln x k . John Napier (1550-1617) Calculator exercise log 3,184 ln 0.000349 log 3.24 log 5 25 2 1 log 9 3 2 1 log 2 2 4 43 64 6 36 3 1 2 8 y log 4 16 log 2 x 3 y log 8 4 log b 100 2 Properties of Logarithmic functions For positive numbers m, n, and a 1, r 1. log b 1 0 2. log b b 1 3. log b b x x 4. blogb x x 5. log b M log b N log b MN M 6. log b M log b N log b N 7. log b M p p log b M 8. log b M log b N M N wx log b yz log b wx 3 5 e x ln b ln x ln b 3 2 log b 4 log b 8 log b 2 log b x 2 3 log x log x 1 log 6 log x 2.315 ln x 2.386 10 x 2 ex 3 3x 4 Change of Base formula Let x, a 1, and b 1 be positive real numbers. Then, log b x log a x log b a e x ln b b x log 5 38.25 How long (to the next whole year) will it take money to double if it is invested at 20% compounded annually? mt r Compound Interest: A P1 m 3. Mathematics of Finance 3.1 Simple Interest I P r t Interest on a loan of $100 at 12% for 9 months Amount: Simple Interest A P P r t P1 rt Amount due on a loan of $800 at 9% for 4 months Present value of an investment How much should you pay to get $5000 in 9 months at 10%? At P P r t If you buy a 180-day treasury bill with maturity value of $10,000 for 9,893.78, what rate of interest is earned? You finance the sale of your car at 10% over 270 days, for $3500. 60 days later, you sell the note for $3550. What rate of interest will the buyer get? Transaction size Commission $0-2499 $29+1.6% $2500-9999 $49+0.8% 10,000+ $99+0.3% An investor purchases 50 shares of stock at $47.52/share. After 200 days, the investor sells the stock for $52.19/share. Using the commission table, find the annual rate of interest earned. 3.2 Compound Interest A P1 rt $1000 deposited at 8%, compounded quarterly A1 A2 A3 An A0 1 r n mn r An A0 1 m A P1 i n $1000 deposited at 8% over 5 years Compound interest for various periods Annually A P1 i n Semiannually r i A P1 i n m Quarterly A P1 i n Monthly A P1 i n nt r A P 1 n Present Value How much should you invest now at 10% compounded quarterly to have $8000 toward the purchase of a car in 5 years? Growth Rate A recent growth oriented mutual fund has grown from $10,000 to $126,000 in the last 10 years. What interest rate, compounded annually, would produce the same growth? Growth time How long will it take $10,000 to grow to $12,000 if it is invested at 9% compounded monthly? Y1 100001.0075 n Y2 12000 A P1 I ^ N =0 Annual Percentage Yield Bank Rate Compounded Advanta 4.93 monthly DeepGreen 4.95 daily CharterOne 4.97 quarterly Liberty 4.94 continuously Which CD has the best return? If a principal is invested at the annual (nominal) rate r compounded m times a year, then the annual percentage yield is m r APY 1 1 m The annual percentage yield is also referred to as the effective rate or true interest rate. Find the APYs for each of the banks above. Computing the Annual Nominal Rate Given the Effective Rate A savings and loan wants to offer a CD with a monthly compounding rate that has an effective rate of 7.5%. What annual nominal rate compounded monthly should they use? 3.3 Future Value of an Annuity; Sinking Funds Ordinary Annuity - payments made at the end of each time interval Future Value - sum of all the payments made plus all interest earned. $100 deposit every 6mo, 6% semiannually, 3 yrs 1 i n 1 S ni i Future Value of an Ordinary Annuity 1 i n 1 FV PMT PMTs ni i What is the value of an annuity at the end of 20 years if $2,000 is deposited each year into an account earning 8.5% compounded annually? How much of this value is interest? Sinking Funds Suppose the parents of a newborn child decide that on each of the child's birthdays up to the 17th year, they will deposit $PMT in an account that pays 6% compounded annually for a college fund. What should the annual deposit be for the amount to be $80,000 after the 17th deposit? Computing the Payment for a Sinking Fund A company estimates that it will have to replace a piece of equipment at a cost of $800,000 in 5 years. To have this money available in 5 years, a sinking fund is established by making equal monthly payments into an account paying 6.6% compounded monthly. How much should the payment be? How much interest is earned during the last year? Growth in an IRA Jane deposits $2000 annually into a Roth IRA that earns 6.85% compounded annually. Due to a change in employment, these deposits stop after 10 years, but the account continues to earn interest until Jane retires 25 years after the last deposit. How much is in the account at Jane's retirement? Approximating Interest Rates A person makes monthly deposits of $100 into an ordinary annuity. After 30 years, the annuity is worth $160,000. What annual rate compounded monthly has this annuity earned during this 30-year period? 3.4 Present Value of an Annuity; Amortization How much should you deposit in an account paying 6% compounded semiannually in order to be able to withdraw $1000 every 6 months for the next 3 years? Present Value of an Ordinary Annuity 1 1 i n PV PMT i What is the present value of an annuity that pays $200 per month for 5 years if money is worth 6% compounded monthly? Retirement Planning Recently, Lincoln Benefit Life offered an ordinary annuity that earned 6.5% compounded annually. A person plans to make equal annual deposits into this account for 25 years in order to make 20 equal annual withdrawals of $25,000. How much must be deposited annually to accumulate sufficient funds to provide for these payments? How much total interest is earned during this entire 45 year process? Amortization Suppose you borrow $5,000 from a bank to buy a car, and agree to repay the loan in 36 equal monthly payments. If a bank charges 1% monthly on the unpaid balance (12% annually compounded monthly), how much should each payment be to retire the total debt? Monthly Payment and Total Interest on an Amortized Debt You buy a TV set for $800 over 18 mo at 1½ % monthly. What is the payment? How much interest will you pay? Amortization Schedules If you borrow $500 over 6mo at 1% monthly, how much is interest and how much is principal? Equity in a home A family purchased a home 10 years ago for $80,000. The home was financed by paying 20% down and signing a 30 year mortgage at 9%. The market value of the house is now $120,000. How much equity does the family have? Automobile financing You have negotiated a price of $25,200 for a new pickup. Options are 0% financing for 48 months, or a $3000 rebate (loan at 4.5% compounded monthly). Problem Solving Strategy for Finance Problems 1. Determine whether the problem involves a single payment or a sequence of periodic payments. a. Simple/Compounded interest problems have a single present and single future value. b.Annuities may be concerned with a present/future value, but involve a sequence of payments. 2. Single payment - simple/compound interest? a. Simple interest is usually used for durations of a year or less b. Compound interest for longer periods. 3. Sequence of payments a. increasing in value? - future value problem b. decreasing value? - present value problem (amortization) Interest: I P r t Simple Interest: A P1 rt A P1 i n Compound Interest: Future value of an 1 i 1 n ordinary annuity: FV PMT i 1 1 i Present value of an n ordinary annuity: PV PMT i