Topic 2.5 Exponential Function Context and Data Modeling: construct an exponential model from a context or data set, interpret the initial value and growth or decay factor, and use the model to make predictions.

What this topic is asking

The College Board (Topic 2.5) wants you to construct an exponential model $f(t) = a \cdot b^{t}$ from a context or a data set, interpret its initial value $a$ and growth or decay factor $b$, and use it to predict. You should build a model from two data points, from a stated rate, or by exponential regression on technology.

The model and its parameters

The initial value sets the starting height; the factor sets how fast it multiplies. Both must be interpreted with units in a modelling answer.

Building a model from two points

When you have two data points $(t_1, y_1)$ and $(t_2, y_2)$, find $b$ from the ratio of outputs over the input gap, then solve for $a$. If one point is the initial value $(0, a)$, the work is shorter: $a$ is read directly and $b$ comes from any second point.

Fitting data with regression

When the data are noisy rather than exact, the calculator section lets you run exponential regression, which finds the $a$ and $b$ that best fit all the points. The output is a model of the same form $a \cdot b^t$, and you interpret its parameters the same way. Regression also reports how well the model fits, which feeds into comparing competing models in Topic 2.6.

A point that earns marks is interpreting the growth factor as a rate. A base of $b = 1.0845$ is "about $8.45\%$ growth per year", and a base of $b = 0.794$ is "about $20.6\%$ decay per day". Translating the bare factor into a percentage with the correct period and the correct direction (growth or decay) shows you understand what the model says about the world, which is exactly what the free-response rubric rewards over a number left unexplained.

A second useful habit is checking a prediction against the structure of the model. Doubling times and half-lives give quick sanity checks: if a sample halves every $3$ days, then after $9$ days only an eighth remains, so a predicted mass should match $\frac{1}{8}$ of the start. Using these structural checkpoints catches arithmetic slips before they cost a final answer.

Try this

Q1. A car worth $30{,}000$ loses $15\%$ of its value each year. Write a model for its value after $t$ years. [1 point]

• Cue. Decay factor $b = 1 - 0.15 = 0.85$, so $V(t) = 30{,}000 \cdot 0.85^{t}$.

Q2. An investment doubles every $7$ years. What is the growth factor $b$ per year? [1 point]

• Cue. $b^7 = 2$, so $b = 2^{1/7} \approx 1.104$, about $10.4\%$ per year.