UNIT 11 STUDY GUIDE Key Features of the graph of Exponential functions in the form The graphs all cross the y-axis at (0, 1) The x-axis is an asymptote. Equation of the asymptote is y=0 Domain: Range: Key Features of the graph of The domain is the positive real numbers, and the range is all real numbers. The graphs all cross the -axis at. A point on the graph is always (b, 1). {where b = the base} The y-axis is an asymptote. Equation of the asymptote is x=0 End behavior: as and as Inverse of the exponential function
Key Features of the graph of The graph crosses the -axis at. The y-axis is an asymptote. Equation of the asymptote is x=0 The graph passes through the point End behavior: as Inverse of the exponential function and as. {Where is the reciprocal of the base.} From what we have seen of these sets of graphs of functions, can we state the relationship between the graphs of and, for? If, then the graphs of and are reflections of each other across the -axis. To graph log functions on the calculator: Select the Y= button. Use log base button on calculator.
Inverse Functions: If two functions whose domain and range are a subset of the real numbers are inverses, then their graphs are reflections of each other across the diagonal line given by in the Cartesian plane. In general, to find the formula for an inverse function of a given function : Write using the formula for. Interchange the symbols and to get. Solve the equation for to write as an expression in. Then, the formula for is the expression in found in step (iii). The functions and are inverses of each other.
How do you find the inverse of a function? Swap ordered pairs: If your function is defined as a list of ordered pairs, simply swap the x and y values. Remember, the inverse relation will be a function only if the original function is one-to-one. Solve algebraically: Solving for an inverse relation algebraically is a three step process: 1. Set the function = y 2. Swap the x and y variables 3. Solve for y Review of Transformations of Functions:
Solving a Log Equation Graphically Finding the intersection points using the calculator 1) Graph each equation 2) Hit 2 nd Trace #5 3) Move cursor close to the point of intersection and hit enter 3 times. SOLUTIONS:
Exponential Growth and Decay Functions: Exponential Growth Functions Exponential Decay Functions Exponential growth occurs when a quantity increases by the same factor over equal intervals of time. Exponential decay occurs when a quantity decreases by the same factor over equal intervals of time. A function in the form, where, is an exponential growth function. time A function in the form, where, is an exponential decay function. time ending amount Initial Amount Rate of growth ending amount Initial Amount Rate of decay For exponential growth, the value inside the parentheses is greater than 1 because is added to 1. For exponential decay, the value inside the parentheses is less than 1 because is subtracted from 1. Example: A house purchased 5 years ago for $100,000 was just sold for $135,000. Assuming exponential growth, approximate the annual growth rate, to the nearest percent. annual growth rate
Rewriting Exponential Functions in Terms of Monthly Rate of Growth or Decay where time in years Writing More Difficult Exponential Growth and Decay Functions What if the word problem does not give you the rate of change as a percent? Exponential growth look for keywords such as doubles, triples, quadruples,... Exponential decay look for keywords such as halves (half-life), a third,... a. The half-life of a medication is the time is takes for the medication to reduce to half of its original amount in a patient s bloodstream. A certain antibiotic has a half-life of one hour. A patient takes 500 milligrams of the medication. b. An initial population of 30 mice doubles each year. Write an equation that models this population after t years.
What if the word problem mentions the growth or decay rate over a specific period of time where t = time elapsed and h = half-life time a. The half-life of a medication is the time is takes for the medication to reduce to half of its original amount in a patient s bloodstream. A certain antibiotic has a half-life of one hour. A patient takes 500 milligrams of the medication. b. The half-life of a medication is the time is takes for the medication to reduce to half of its original amount in a patient s bloodstream. A certain antibiotic has a half-life of about 3 hours. A patient takes 500 milligrams of the medication. Example: One of the medical uses of Iodine 131 (I 131), a radioactive isotope of iodine, is to enhance x-ray images. The half-life of I 131 is approximately 8.02 days. A patient is injected with 20 milligrams of I 131. Determine, to the nearest day, the amount of time needed before the amount of I 131 in the patient s body is approximately 7 milligrams. A( t) 20(0.5) 8.02 t
LOCATION OF REGRESSIONS IN THE CALCULATOR These non-linear regressions are also found using the graphing calculator. All types of regressions on the calculator are prepared in a similar manner. Your regression options can be found under STAT CALC (scroll for more choices) EXPONENTIAL y = ab x LOGARITHMIC (Natural Log) y = a + blnx To enter data on a calculator: To enter a list: STAT button 1: Edit Enter list under L1, L2, etc. Hit 2 nd button then QUIT (in yellow above MODE) when list is entered. To clear a list: STAT button 4: ClrList Enter list name to be cleared (L1, L2, etc) ENTER Another option: Scroll up to select the list to be cleared. Then hit CLEAR and enter. The list will now be cleared. To get L1 or L2 back if you accidently delete it: STAT button 5: setup editor Hit ENTER Will see words DONE on home screen To view edited list: STAT button 1: EDIT Now you can view sorted list
Monthly Loan Formulas Tips when working with monthly loan formulas: Read the question twice. Be sure to pay attention to what each variable stands for. The second time, write down the numbers that correspond to each variable. Determine the variable that they are asking you to find. Substitute the values in the given formula. Try to simplify any expressions within the formula before solving for the remaining variable. If you are solving for an exponent, you will have to use logs. A down payment is the amount you would pay prior to taking out a loan. This amount gets subtracted from the total purchase price. Monthly Payment Example #1:
Monthly Payment Example #2: Jim is looking to buy a vacation home for $172,600 near his favorite southern beach. The formula to compute a mortgage payment, M, is where P is the principal amount of the loan, r is the monthly interest rate, and N is the number of monthly payments. Jim's bank offers a monthly interest rate of 0.305% for a 15-year mortgage. With no down payment, determine Jim's mortgage payment, rounded to the nearest dollar. Algebraically determine and state the down payment, rounded to the nearest dollar, that Jim needs to make in order for his mortgage payment to be $1100.
Compound Interest Formula must memorize! A = Final amount P = initial amount r = rate n = number of compounding periods t = time Compounding periods: Annually once per year (n = 1) Semi-annually twice per year (n = 2) Quarterly four times per year (n = 4) Monthly twelve times per year (n = 12) Compound Interest Example #1:
Compound Interest Example #2: Seth s parents gave him $5000 to invest for his 16th birthday. He is considering two investment options. Option A will pay him 4.5% interest compounded annually. Option B will pay him 4.6% compounded quarterly. Write a function of option A and option B that calculates the value of each account after n years. Seth plans to use the money after he graduates from college in 6 years. Determine how much more money option B will earn than option A to the nearest cent. Algebraically determine, to the nearest tenth of a year, how long it would take for option B to double Seth s initial investment.