Exponential vs Linear Growth
Practice identifying and writing models for constant-rate versus constant-percent change.
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Questions Covered in This Set
11 cards to master
Core distinction between linear and exponential growth
Linear ADDS the same amount each period; exponential MULTIPLIES by the same factor each period.
Linear model formula (and meaning of parts)
y = mx + b, where b is the starting value and m is the amount added each period.
Exponential model formula (and meaning of parts)
y = a · b^x, where a is the starting value and b is the growth factor per period.
How do you build b from a percent increase of r%?
b = 1 + r/100. Example: 8% growth → b = 1.08; 150% growth → b = 2.5.
How do you build b from a percent decrease of r%?
b = 1 − r/100. Example: 8% decay → b = 0.92; halving → b = 0.5.
Which words signal exponential rather than linear?
A percent attached to a repeating time period, 'doubles every ___', 'decays by half', 'multiplied by ___ each ___'.
How do you tell from a table whether data is linear or exponential?
Check differences (constant → linear) and ratios (constant → exponential).
A quantity doubles every 5 hours, t in hours. Write the model.
P = P₀ · 2^(t/5); general template y = a · b^(t/length of one period).
Grows 3% per month but t is in years — what's the model?
A = a(1.03)^(12t), since there are 12 monthly periods per year.
How do linear and exponential graphs differ?
Linear is a straight line forever; exponential growth curves upward ever more steeply, and exponential decay flattens toward a horizontal asymptote (never touching it).
A car worth $24,000 loses 15% per year. Model and value after 4 years?
V = 24000(0.85)^t; at t = 4, V ≈ $12,528.
