Integrated Math 1 / Unit 4: Linear Functions and Arithmetic Sequences
4.6 Linear Modeling
Notes
Many situations change at a constant rate and can be modeled with $y = mx + b$.
- Find the rate of change (slope) and the starting value ($y$-intercept).
- Interpret in words: "starts at $b$ and changes by $m$ per unit."
- Example: a candle 20 cm tall burns 2 cm per hour: $h = 20 - 2t$. It burns out when $20 - 2t = 0$, so $t = 10$ hours.
- A table is linear if the outputs change by a constant amount for equal steps in input.
Practice
A candle is 20 cm tall and burns 2 cm each hour. After how many hours does it burn out?
Solve $20 - 2t = 0$, so $t = 10$.
A table shows $x$ = 0, 2, 4, 6 with $y$ = 10, 16, 22, 28. Which equation fits?
The $y$-value rises 6 for every 2 in $x$, so the slope is 3. At $x = 0$, $y = 10$.
A consultant charges $C = 15h + 40$ dollars for $h$ hours. What does the 40 mean?
The constant term is the value when $h = 0$.
Ready for more? Take the Unit 4 test.