Notes
Exponential functions
- $y = a \cdot b^x$: $a$ is the initial value and $b$ is the factor. Growth has $b > 1$; decay has $0 < b < 1$.
- With rate $r$: growth $b = 1 + r$, decay $b = 1 - r$. Compound interest: $A = P(1 + r)^t$.
- Half-life problems halve the amount each period.
Geometric sequences
- Multiply by a common ratio $r$: $a_n = a_1 \cdot r^{n-1}$.
Linear vs exponential
- Linear: constant difference (add). Exponential: constant ratio (multiply).
- Exponential growth eventually beats any linear growth.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Decay needs a base between 0 and 1.
$a_8 = 6 \cdot 2^7 = 6 \cdot 128 = 768$.
The factor $1.08 = 1 + 0.08$ means 8% growth.
One hour is three doubling periods: $100 \cdot 2^3 = 800$.
Only the last table has a constant ratio of 2 between outputs.
$900(0.8)^2 = 900(0.64) = 576$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Plan A: $2,000 is invested at 5% interest compounded annually. Plan B: $2,000 is invested and $150 is added at the end of each year (no interest).
(a) Write a function for the value of Plan A after $t$ years.
Show the answer and scoring guide
$A(t) = 2000(1.05)^t$.
Check off the points you earned:
(b) Find the value of Plan A after 3 years.
Show the answer and scoring guide
$2000(1.05)^3 = 2000(1.157625) = 2315.25$, so $2,315.25.
Check off the points you earned:
(c) Plan B is worth $2000 + 150t$. After 3 years, which plan is worth more and by how much? Which plan will eventually be worth more over a long time, and why?
Show the answer and scoring guide
Plan B after 3 years is $2000 + 450 = 2450$. Plan B is worth more by $2450 - 2315.25 = 134.75$, so $134.75. Over a long time Plan A is worth more because exponential growth multiplies by a constant factor and eventually exceeds any linear growth.
Check off the points you earned: