The Function Showroom: Three Machines, Three Fingerprints
π― Today's mission briefing
We are learning to tell the linear, quadratic and exponential families apart by how they grow, so we can read any table of values like a mechanic reads an engine.
You'll know you've got it when:
- I can describe each family's growth rule: linear adds a constant, quadratic has constant second differences, exponential multiplies by a constant
- I can identify which family produced a table of values by checking differences and ratios
- I can explain why fast-growing is not the same as exponential β and prove it with a table
The three pay offers
Connects to what your guest already knows and makes them curious. Activating prior knowledge is one of the strongest predictors of new learning.
You land a 30-day summer job and the boss, who fancies herself a comedian, offers three pay schemes. Offer A: a flat $100 every day. Offer B: $1 on day one, $4 on day two, $9 on day three β the day number squared. Offer C: one cent on day one, doubling every day after. Most people rank them A, B, C and feel clever for dodging the one-cent trap in reverse. By the end of the month, Offer C pays more per DAY than the other two pay for the entire job combined. Today you learn to see that coming from the first three rows of the table.