Construct and interpret exponential functions for growth and decay, and interpret their parameters in context (NC.M1.F-LE.1, F-LE.2, F-LE.5).

What this topic is asking

Three standards combine. NC.M1.F-LE.1 asks you to distinguish linear from exponential situations (constant additive change versus constant multiplicative change). NC.M1.F-LE.2 asks you to construct an exponential function from a description, a graph, or two points. NC.M1.F-LE.5 asks you to interpret the parameters (initial value and growth or decay factor) in context.

Linear versus exponential

The first decision is which model fits.

The parts of an exponential model

For $f(x) = ab^x$:

• $a$ is the initial value, the output when $x = 0$ (since $b^0 = 1$).
• $b$ is the growth/decay factor. Growth if $b > 1$; decay if $0 < b < 1$.
• Writing $b = 1 + r$ (growth) or $b = 1 - r$ (decay) shows the rate $r$ as a percent.

A balance growing $4\%$ a year from \500$is$y = 500(1.04)^t$; a population falling$10%$a year from$800$is$y = 800(0.90)^t$.

Building an exponential function

Reading growth and decay rates

The growth factor and the rate are related but different. A factor of $1.04$ is a $4\%$ growth rate ($r = 0.04$); a factor of $0.88$ is a $12\%$ decay rate ($r = 0.12$). To find the rate, compare the factor to $1$: subtract for growth, or subtract from $1$ for decay.

How the NC Math 1 EOC examines this topic

• Gridded response. Evaluate an exponential model at a given input.
• Multiple choice. Choose the exponential function for a context, or identify growth versus decay.
• Calculator-active. Exponential evaluations usually sit in the calculator-active section.

Exponential functions are the continuous cousins of geometric sequences (the common ratio is the growth factor) and are central to comparing function families. Evaluating $ab^x$ uses the exponent properties.

Why a constant percent means exponential

The deep distinction in F-LE.1 is between adding and multiplying. Linear growth adds the same amount each step, so it climbs at a steady pace. Exponential growth multiplies by the same factor each step, so the increase itself grows: $4\%$ of a larger balance is a larger dollar amount than $4\%$ of a smaller one. That is why exponential graphs curve upward ever more steeply (or, for decay, flatten toward zero), while linear graphs stay straight. Recognizing "constant percent" or "doubling" or "halving" as the fingerprint of exponential behavior lets you pick the right model immediately, which is the most common decision the EOC tests in this strand.

Try this

Q1. Write a function for a \1000$investment growing$5%$ per year. [1 point]

• Cue. $y = 1000(1.05)^t$ (factor $1 + 0.05 = 1.05$).

Q2. A substance halves every day, starting at $80$ g. Find the amount after $3$ days. [2 points]

• Cue. $y = 80(0.5)^t$; $y = 80(0.5)^3 = 80(0.125) = 10$ g.