Worked Examples
Four money problems, graphed — the same lines, slopes, and piecewise functions students plot in this module.
Figures current · August 2026
y = 150x + 500slope = the habit
$500 in the account already, $150 added every month. The y-intercept is where you started and the slope is the habit — and only one of the two is still in your control.
$4,100after 24 monthsstarted with $500
→Raising the deposit to $175 tilts the whole line and ends $600 higher — changing the slope beats any one-time deposit.
tax = Σ (rate × slice)slice by slice
A single filer with $60,000 of taxable income in 2026. Each slice of that income is taxed at its own rate: 10% on the first $12,400, 12% up to $50,400, and 22% only on what is left above it.
$7,912total federal income tax13.2% effective · 22% marginal
→A raise into the next bracket never re-taxes the income below it — only the new slice is taxed at the new rate.
0.02s − 95 = 0.01ssolve for s
One card charges a $95 annual fee and pays 2% back. The other is free and pays 1%. Set the two rewards equal and solve for the spending where they tie.
$9,500a year in spendingbelow it, the free card wins
→An annual fee is a fixed cost, so which card is better depends entirely on where your spending lands on the line.
FV = PMT × [((1 + r)ⁿ − 1) ÷ r]n = months
$50 a month for ten years, at an assumed 7% annual return compounded monthly. You contribute $6,000 in total; the exponent supplies the rest.
$8,654after 10 yearscontributed: $6,000
→Double the time and the growth more than quintuples — by year 20 it has passed everything you put in.
Source: IRS Revenue Procedure 2025-32 (2026 tax brackets, single filer). All other figures derived from the inputs shown. Reviewed August 2026.
