Integrated Math 2 / Unit 5: Modeling with Functions
5.3 Comparing Growth Rates
Notes
Different function types grow at different speeds.
- A linear function adds a constant amount. A quadratic grows faster. An exponential with $b > 1$ eventually beats any linear or quadratic function.
- A quadratic can be ahead for small $x$, but the exponential overtakes it later.
- Example: $2^x$ versus $x^2$: at $x = 3$, $8 < 9$, at $x = 4$ they tie at 16, and at $x = 5$, $32 > 25$.
- To compare, make a table of values or check a few key points.
Practice
For very large $x$, which function has the greatest values?
An exponential with base greater than 1 eventually outgrows linear and quadratic functions.
Compare $5x^2$ and $2^x$. At $x = 10$, which is greater?
$2^{10} = 1024$ and $5(10)^2 = 500$. At $x = 5$ the quadratic was ahead, but the exponential overtakes it.
Plan A saves $100 plus $10 per week. Plan B starts with $1 and doubles each week, giving $2^w$ dollars after week $w$. Which statement is true at the end of week 10?
Plan A has $100 + 10(10) = 200$. Plan B has $2^{10} = 1024$.
Ready for more? Take the Unit 5 test.