10.2 Exponential Functions: Graphing, Growth, and Decay

Quick Answer: An exponential function has the form y = a·b^x (b > 0, b ≠ 1). Its graph passes through (0, a), has horizontal asymptote y = 0, domain all reals, and range (0, ∞) when a > 0. Growth uses b > 1; decay uses 0 < b < 1. From context, write y = a(1 + r)^t for growth or y = a(1 - r)^t for decay.

Florida's B.E.S.T. benchmarks MA.912.AR.5.3, MA.912.AR.5.4, and MA.912.AR.5.6 cover graphing exponential functions, writing growth and decay models from tables, graphs, and contexts, and applying them in real-world problems. These live in the Non-Linear Relationships reporting category (31-38% of the test).

General Form and Key Features

The general exponential function is y = a·b^x, where a is the initial value (the y-intercept), b is the base or growth/decay factor, and x is the exponent variable. Conditions: a ≠ 0, b > 0, b ≠ 1. The parent function y = b^x passes through (0, 1) and (1, b).

Key features for graphing:

Transformations

A vertical shift changes the asymptote. For y = a·b^x + k, the asymptote becomes y = k and the range becomes (k, ∞) when a > 0. For y = a·b^(x - h) + k, the horizontal shift is h units; the graph still has asymptote y = k. Florida items often test the asymptote shift directly.

Worked graphing example: graph y = 2·(1.5)^x. The y-intercept is (0, 2). Because b = 1.5 > 1, the graph rises as x increases and approaches y = 0 as x decreases. Plot a few points: x = -1 gives y = 2/1.5 ≈ 1.33; x = 1 gives y = 3; x = 2 gives y = 4.5. Connect with a smooth increasing curve. Domain: all reals. Range: (0, ∞). Asymptote: y = 0.

Writing Models from a Table

If consecutive y-values form a constant ratio, the model is exponential. The common ratio is b; the initial value (the y-value when x = 0) is a.

Worked example: a table gives x = 0, 1, 2, 3 with y = 5, 10, 20, 40. Check ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2. The constant ratio is 2, so b = 2. The initial value is 5, so a = 5. The model is y = 5·2^x.

If the initial value is not at x = 0, solve for a using one known point. For example, if y = 80 when x = 3 and the ratio is 2, then a·2^3 = 80, so a = 10.

Growth and Decay from Context

Translating a verbal description uses these forms:

• Growth: y = a(1 + r)^t, where r is the growth rate as a decimal and t is the time in periods.
• Decay: y = a(1 - r)^t, where r is the decay rate as a decimal.

The base becomes (1 + r) for growth or (1 - r) for decay. This is how Florida items word population, depreciation, and radioactive decay problems.

Worked example: a town of 12,000 grows at 4% per year. Model: P(t) = 12,000(1.04)^t. After 8 years: P(8) = 12,000(1.04)^8 ≈ 12,000 × 1.3686 ≈ 16,423. Note this is a multiplicative model, not a linear one.

Worked decay example: a car worth $28,000 depreciates 15% per year. V(t) = 28,000(0.85)^t. After 5 years: V(5) = 28,000(0.85)^5 ≈ 28,000 × 0.4437 ≈ $12,424.

Linear vs Exponential

The diagnostic is the pattern of change:

• Linear: constant difference. Each step adds the same amount. The first differences of y are constant.
• Exponential: constant ratio. Each step multiplies by the same factor. The ratio of consecutive y-values is constant.

Florida items may give a table and ask which model fits best. Compute first differences; if not constant, compute ratios. If ratios are constant, the model is exponential.

Half-Life and Repeated Doubling

A half-life of h units gives y = a(1/2)^(t/h); a doubling time of d units gives y = a·2^(t/d). Example: a 60-gram sample with 12-year half-life after 36 years gives 60(1/2)^(36/12) = 60/8 = 7.5 g.

EOC Trap Summary

1. Using y = a·x^b (power function) instead of y = a·b^x (exponential). The variable must be in the exponent.
2. Writing (1 - r) for growth or (1 + r) for decay — the sign of r must match the direction.
3. Forgetting to convert a percentage rate to a decimal (5% becomes 0.05).
4. Claiming the range is all reals when a > 0 — exponential outputs are always positive (for a > 0).
5. Mixing up the asymptote when a vertical shift is present; the asymptote moves to y = k, not y = 0.

Exponential items reward pattern recognition: check for a constant ratio, write the model, and then substitute the requested time value. Keep units straight when the rate is annual but the question asks about months.