ACT Math: Exponential vs. Linear Growth—Identify and Compare

Linear growth: y=mx+b. Output increases by a constant amount each step. Example: y=2x+1 increases by 2 for each 1-unit increase in x (at x=0, y=1; at x=1, y=3; at x=2, y=5). On a graph, linear is a straight line. Exponential growth: y=ab^x. Output multiplies by a constant factor each step. Example: y=2(3)^x multiplies by 3 each time x increases by 1 (at x=0, y=2; at x=1, y=6; at x=2, y=18). On a graph, exponential is a curve that gets steeper. Linear grows predictably and slowly. Exponential grows slowly at first, then explosively fast. This difference is crucial: a pandemic spreading exponentially overwhelms hospitals; a disease spreading linearly is manageable.

Over time, exponential always surpasses linear (regardless of starting rates).

Scenario 1 (Compare at points): Linear y=5x vs. exponential y=2(1.5)^x. At x=0: linear=0, exponential=2. At x=5: linear=25, exponential=2(7.59)≈15.2. At x=10: linear=50, exponential=2(57.67)≈115.3. Exponential starts smaller but eventually surpasses linear. Scenario 2 (Doubling): Linear doubling (y=2x) vs. exponential doubling (y=2^x). Linear: 2, 4, 6, 8, 10... Exponential: 1, 2, 4, 8, 16, 32... Exponential's rate increases; linear's is constant. Scenario 3 (Real-world): Population growing 1,000 people/year (linear, rate=1000) vs. 5% per year (exponential, rate=1.05). After 10 years, linear adds 10,000 total. Exponential multiplies by 1.05^10≈1.629, so a population of 10,000 becomes 16,290. Exponential means multiplicative growth; linear means additive growth.

Identify by context: "doubles every X" = exponential. "Increases by X each year" = linear.

Problem 1: "A bank account earns 3% interest annually." Type: Exponential (multiplies by 1.03 each year). Formula: A=P(1.03)^t. Problem 2: "A plant grows 2 cm per week." Type: Linear (adds 2 cm each week). Formula: h=2t (plus initial height). Problem 3: A bacteria population: hour 0: 100, hour 1: 200, hour 2: 400. Type: Exponential (doubles). Formula: N=100(2)^t. At hour 5? N=100(2)^5=3200. Problem 4: A car depreciates $3,000 per year starting at $30,000. Type: Linear. Formula: V=30000-3000t. At 5 years? V=30000-15000=15000. Complete all four daily until you quickly identify growth type and create equations.

Verify exponential by checking the ratio between consecutive terms (should be constant).

Exponential vs. linear questions appear in 2-3 ACT Math sections, testing whether you recognize growth patterns and can model them. Identifying the type correctly (exponential or linear) lets you choose the right formula and solve confidently.

Master this distinction in one study session. By test day, recognizing growth types becomes automatic.