Construct and graph exponential functions, distinguish exponential from linear growth, and interpret growth and decay models (TN A1.F.IF.D.7e, A1.F.LE.A.1, A1.F.LE.A.2).

What this topic is asking

The Linear, Quadratic, and Exponential Models standards cover exponential functions. A1.F.LE.A.1 distinguishes exponential from linear situations; A1.F.LE.A.2 builds exponential models; A1.F.IF.D.7e graphs them, showing intercepts and end behavior. The model forms are not on the reference sheet, so memorize growth $y = a(1 + r)^t$ and decay $y = a(1 - r)^t$.

The growth and decay models

A percent change becomes a base near $1$. Memorize these, since they are not on the reference sheet.

A $5\%$ increase gives base $1.05$; a $5\%$ decrease gives base $0.95$. The base is not the rate $r$ itself.

Graphing exponentials

The graph of $f(x) = ab^x$ has two reliable features:

• $y$-intercept at $(0, a)$: set $x = 0$, since $b^0 = 1$.
• Horizontal asymptote at $y = 0$: the curve approaches but never touches the $x$-axis (for the basic model).

Growth curves rise to the right and flatten toward the asymptote on the left; decay curves fall to the right toward the asymptote.

Linear versus exponential

The defining test (A1.F.LE.A.1): a quantity is linear if it changes by a constant amount per step (add the same number), and exponential if it changes by a constant factor or percent per step (multiply by the same number). "$50$ per month" is linear; "$5\%$ per month" is exponential. From a table, check whether successive differences are constant (linear) or successive ratios are constant (exponential).

How TNReady examines this topic

• Numeric response. Write a model and evaluate it at a given time.
• Multiple choice. Choose the growth or decay model, with base-versus-rate and linear-versus-exponential distractors.
• Graphing / inline choice. Identify the $y$-intercept, the asymptote, or whether a table is linear or exponential.

A clarifying idea is that exponential functions are the continuous cousins of geometric sequences: both multiply by a constant factor, so the base $b$ plays the role of the common ratio $r$.

Why exponential eventually beats linear

A key TNReady idea (A1.F.LE.A.3) is that a quantity growing exponentially eventually exceeds one growing linearly, no matter how large the linear rate or how small the exponential rate. The reason is that linear growth adds a fixed amount each step, while exponential growth multiplies, so the increments themselves keep getting bigger. Early on, a line with a steep slope can be far ahead of a slow-growing exponential, which is why a table or graph might show the linear function winning at first. But because the exponential's step size compounds, it accelerates and overtakes the line, then pulls away for good. This is the mathematics behind compound interest outpacing simple savings, and behind why "$5\%$ growth" beats "$50 per year" once enough time passes. Recognizing the constant-percent signature, and remembering that it always wins in the long run, is exactly what the standard tests.

Try this

Q1. Write a model for \800$growing at$3%$per year, and find the amount after$1$ year. [2 points]

• Cue. $A = 800(1.03)^t$; at $t = 1$, $A = 824$.

Q2. Is "a tank loses $10$ liters per hour" linear or exponential? [1 point]

• Cue. Linear: a constant amount lost per hour (constant difference).