GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A financial analyst is comparing two investment options. Investment A grows linearly, starting at $1000 and increasing by $50 each year. Investment B grows exponentially, starting at $800 and increasing by 5% each year. Which function correctly models Investment B's value, V(t), after t years?
- AV(t) = 1000 * (1.05)^t
- BV(t) = 800 + 0.05t
- CV(t) = 800 * (1.05)^t
- DV(t) = 1000 + 50t
Show answer & explanationAnswer & explanation
Correct answer: C. V(t) = 800 * (1.05)^t
Exponential growth functions are of the form V(t) = P₀(1 + r)^t, where P₀ is the initial amount and r is the growth rate as a decimal. Investment B starts at $800 (P₀) and grows at 5% (r = 0.05).
Why the other options are wrong
- A. This uses the initial value of Investment A with the growth rate of Investment B.
- B. This attempts to model Investment B linearly, using simple interest, not exponential growth.
- D. This models Investment A, which is linear, not Investment B.
Modeling Exponential Growth
An exponential growth function models situations where a quantity increases by a constant percentage over equal time intervals. It is defined by an initial amount and a growth factor.
- General form: y = a * b^x or y = P₀(1 + r)^t
- P₀ (or 'a') is the initial amount.
- r is the growth rate (as a decimal).
- b (or 1+r) is the growth factor.
- t (or 'x') is the number of time periods.
Memory trick: Linear adds, Exponential multiplies – know the difference!