Timelines

Learning Objectives Represent financial scenarios on a timeline, identifying present value, future value, and periodic payments. Define and use exponential functions to model the growth of a single investment (lump sum) over time. Apply the formula for the future value of an annuity as a function of time, connecting it to the sum of a geometric series. Apply the formula for the present value of an annuity to solve problems involving loans and mortgages. Manipulate financial functions to solve for different variables, such as the number of time periods (n) using logarithms. Analyze and compare different financial plans by modeling them as functions on a timeline. Ever wondered how a small, regular saving can grow into a huge amount for a car or college? ⏳ We can map this jou...

TermDefinitionExample TimelineA graphical representation of time as a line, where specific points (0, 1, 2, ..., n) represent distinct time periods (e.g., years, months). In the context of functions, the timeline represents the discrete domain of the independent variable, `n`.A 5-year loan with monthly payments would be represented on a timeline with points from 0 (today) to 60 (the 60th month). Present Value (PV)The current worth of a future sum of money or stream of cash flows, given a specified rate of return. It is the value at time `n=0` on the timeline.The PV of a loan is the initial amount of money you borrow, e.g., $20,000 for a car loan. Future Value (FV)The value of a current asset or cash flow at a specified date in the future. It is the value at a future time `n` on the timeli...

Future Value of a Lump Sum (Compound Interest) FV = PV(1 + i)^n Use this function to find the future value (FV) of a single investment (PV) after `n` compounding periods at an interest rate of `i` per period. This is an exponential function where the base is (1+i). Future Value of an Ordinary Annuity FV = R * [((1 + i)^n - 1) / i] Use this function to find the total accumulated value of a series of `n` regular payments (R). This formula sums a geometric series of all payments, each compounded to the end of the timeline. Present Value of an Ordinary Annuity PV = R * [(1 - (1 + i)^-n) / i] Use this function to find the lump sum value today (PV) that is equivalent to a series of `n` future payments (R). This is commonly used to calculate loan amounts or payments.