Integrated Math 3 / Unit 9: Sequences, Series and Functions Review
9.6 Choosing a Function Model
Notes
Look at how the outputs change when inputs go up by equal steps.
- Constant first differences mean a linear function.
- Constant second differences mean a quadratic function.
- A constant ratio between outputs means an exponential function.
- In the long run, an exponential with $b > 1$ outgrows any polynomial.
- Example: a taxi fare of $3 plus $2 per mile is linear, since each mile adds $2.
Practice
The table has outputs $5, 10, 20, 40$ for $x = 0, 1, 2, 3$. What type of function fits?
Each output is 2 times the one before, which is a constant ratio.
The table has outputs $2, 5, 10, 17$ for $x = 0, 1, 2, 3$. What type of function fits?
First differences are $3, 5, 7$ and second differences are all 2.
Which function is largest for very large $x$?
Any exponential with base greater than 1 eventually exceeds every polynomial.
Ready for more? Take the Unit 9 test.