M151Q Precalculus · Unit 2: Powers, exponentials, logarithms · on Exam 2
MFG 4.1–4.2 Exponential growth, decay, and exponential functions
Study notes for Exponential growth, decay, and exponential functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Exponential change multiplies by the same factor each period: $A(t) = A_0 b^t$ with growth factor $b = 1 + r$ (growth) or $b = 1 - r$ (decay). Percent change per period, not a fixed amount.
- Linear vs exponential from a table: constant differences mean linear; constant ratios mean exponential.
- $f(x) = ab^x$ with $b>0$, $b\ne1$: domain all reals, range $(0,\infty)$, horizontal asymptote $y = 0$, $y$-intercept $a$. $b>1$ increases; $0
- Find $a$ and $b$ from two points: divide the equations to eliminate $a$, solve for $b$, then back-substitute.
- Solve $b^x = b^y$ by matching bases: $8^x = 32 \Rightarrow 2^{3x} = 2^5 \Rightarrow x = 5/3$.
Key formulas
Exponential model
$$A(t) = A_0 b^t,\qquad b = 1 + r\ \text{(growth)},\quad b = 1 - r\ \text{(decay)}$$
Two points
$$\frac{ab^{x_2}}{ab^{x_1}} = \frac{y_2}{y_1} \Rightarrow b = \left(\frac{y_2}{y_1}\right)^{1/(x_2-x_1)}$$
Worked example
A population is 5000 and grows 6% per year. Write $P(t)$, find $P(10)$, and the growth factor over a decade.
Show the solution
$P(t) = 5000(1.06)^t$. $P(10) = 5000(1.06)^{10} \approx 8954$. Decade factor $1.06^{10} \approx 1.79$: about 79% growth per decade, not 60%.
Common mistakes
- Adding 6% ten times instead of multiplying by 1.06 ten times.
- Using b = 0.06 for "6% growth" (it is 1.06).
- Assuming exponential graphs reach zero: they approach the asymptote but never touch it.
Exam tipGrowth factor = 1 + rate. Decay of 15% is a factor of 0.85. Say the factor before writing the formula.
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