Distinguish linear from exponential growth (constant difference versus constant ratio), construct linear and exponential functions from descriptions, tables, or two points, and interpret their parameters (initial value, rate of change, growth factor) in context.

What this topic is asking

The Regents Algebra I exam (the Linear, Quadratic, and Exponential Models, F-LE, and Building Functions, F-BF, clusters) wants you to tell linear growth from exponential growth, to build each kind of function from a description, a table, or two points, and to interpret the parameters: the initial value, the rate of change or slope, and the growth or decay factor. Modeling questions are among the most frequent constructed-response items.

Linear versus exponential

The defining test is how a quantity changes from one step to the next.

In a table where $x$ increases by 1 each row, look at the outputs. If they go $3, 7, 11, 15$ (adding 4 each time), the model is linear with rate 4. If they go $3, 6, 12, 24$ (multiplying by 2), it is exponential with growth factor 2. This single check answers most Part I model-identification questions.

Building a linear model

A linear function needs a rate of change (slope) and an initial value (the $y$-intercept). From two points, the slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$; from a description, it is the per-unit amount ("\25$per month"). The initial value is the amount at$x = 0$("a$\$50$ sign-up fee").

For a phone plan costing a \50$fee plus$\$25$ per month, $C(m) = 50 + 25m$. The $50$ is the starting cost and the $25$ is the slope.

Building an exponential model

An exponential function needs an initial value $a$ and a growth factor $b = 1 + r$ (growth) or $b = 1 - r$ (decay), where $r$ is the percent change written as a decimal.

Interpreting parameters and comparing models

Every parameter has a meaning to state in context. In $f(x) = a(b)^x$, $a$ is the value at the start, and $b$ tells the per-step multiplier (and thus the percent change). A clarifying idea the Regents tests is that exponential growth eventually outpaces any linear growth: even if a linear model is larger at first, a positive exponential rate means the exponential function overtakes it and stays ahead. When a question gives a table or two models and asks which is greater "in the long run", the exponential one wins for large inputs. Always justify a linear-versus-exponential claim by naming the constant difference or constant ratio, because that is where the credit sits.

Try this

Q1. A table shows outputs $5, 10, 20, 40$ as $x$ goes $0, 1, 2, 3$. Linear or exponential? [1 credit]

• Cue. Constant ratio of 2, so exponential: $f(x) = 5(2)^x$.

Q2. Write a linear model for a tank starting at 80 liters and draining 6 liters per minute. [2 credits]

• Cue. $L(t) = 80 - 6t$: initial value 80, rate $-6$.