Choosing Between Linear and Exponential Models on the Digital SAT

Quick Answer: Choosing between linear and exponential models requires identifying whether a quantity changes by a constant amount (linear) or a constant percentage or multiplier (exponential) over equal intervals. When in doubt, plot the given data points in the Desmos graphing calculator to visually confirm if the trend is a straight line or a curve.

mindmap
  root((Model Choice))
    Linear
      Constant Amount
      Adds or Subtracts
      Straight Line
      y = mx + b
    Exponential
      Constant Percent or Factor
      Multiplies or Divides
      Curved Line
      y = a(b)^x

What Is Choosing Between Linear and Exponential Models?

On the Digital SAT, you will frequently be asked to read a scenario or look at a data table and determine the best mathematical model to represent the situation. The core difference lies in how the dependent variable changes over time or across equal intervals. Understanding this distinction is a foundational skill in the Problem-Solving & Data Analysis domain as outlined by the College Board.

A linear model is used when a quantity changes by a constant amount over equal intervals. For example, if a savings account grows by exactly $50 every month, the growth is linear. This is closely related to finding unit rates, where the rate of change (the slope) remains identical no matter where you check it on the graph.

An exponential model, on the other hand, is used when a quantity changes by a constant percentage or a constant multiplier over equal intervals. If a savings account grows by 5% every month, or if a bacteria population doubles every hour, the growth is exponential. Because the amount added increases as the total size increases, the graph forms a curve rather than a straight line. Recognizing these patterns is just as important as mastering direct and inverse variation when analyzing SAT data.

Step-by-Step Method

1. Step 1: Read for Keywords — Scan the problem for words that indicate the type of change. "Increases by 10," "adds $5," or "drops by 2" point to linear models. "Increases by 10%," "doubles," "halves," or "retains 90%" point to exponential models.
2. Step 2: Check the Intervals — Ensure the time steps or $x$-values are increasing by equal, constant intervals (e.g., every 1 year, every 5 minutes).
3. Step 3: Analyze the Differences or Ratios — If given a table of values, subtract consecutive $y$-values. If the differences are constant, it's linear. If the differences change but dividing consecutive $y$-values gives a constant ratio, it's exponential.
4. Step 4: Formulate the Equation — Use $y = mx + b$ for linear scenarios, placing the constant difference as $m$. Use $y = a(b)^x$ for exponential scenarios, placing the constant ratio as $b$.

Desmos Shortcut

The built-in Desmos Calculator is an incredibly powerful tool for these questions, especially if you are given a table of values and aren't sure which model fits best.

Click the "+" button in Desmos and add a Table. Type in your $x$ and $y$ values. Click the magnifying glass icon ("Zoom Fit") to automatically adjust the window. If the dots form a perfect straight line, the model is linear. If they curve upward or downward, it's exponential. You can even ask Desmos to find the exact equation by typing y1 ~ mx1 + b (for linear regression) or y1 ~ ab^x1 (for exponential regression) on a new line. Whichever model gives an $r^2$ value of $1$ (or closest to $1$) is the correct choice.

Worked Example

Question: A scientist is observing two different populations of cells in a lab. Population A starts with 1,000 cells and increases by 200 cells every hour. Population B starts with 1,000 cells and increases by 20% every hour. Which of the following statements correctly describes the models that best represent the two populations over time?

A) Both populations are best represented by linear models. B) Both populations are best represented by exponential models. C) Population A is best represented by a linear model, and Population B is best represented by an exponential model. D) Population A is best represented by an exponential model, and Population B is best represented by a linear model.

Solution:

Let's analyze Population A first. The problem states it "increases by 200 cells every hour." Because it is increasing by a constant amount (200) over equal intervals (every hour), the rate of change is constant. This is the definition of a linear model. The equation would be $y = 200x + 1000$.

Now let's analyze Population B. It "increases by 20% every hour." Because it is increasing by a constant percentage, the actual number of cells added will grow larger each hour as the base population grows. This requires a constant multiplier (1.20). This is the definition of an exponential model. The equation would be $y = 1000(1.20)^x$.

Therefore, Population A is linear and Population B is exponential.

The correct answer is C.

Common Traps

1. Confusing growth factors with constant addition — Based on Lumist student data, 60% of students initially confuse the exponential growth factor $(1+r)$ with the decay factor $(1-r)$, but an equally common trap is seeing a "5% increase" and treating it as adding $5$ (linear) instead of multiplying by $1.05$ (exponential). Always distinguish between percentages and raw amounts.

2. Misreading graph scales — Our data shows that 35% of errors in Problem Solving & Data Analysis involve misreading graph axes or scales. The SAT will sometimes present an exponential curve on a graph with a logarithmic axis, making it look like a straight line, or trick you by changing the step size on the $x$-axis in a data table. Always verify the intervals before calculating your differences or ratios.

FAQ

How do I know if a word problem is linear or exponential?

Look for keywords describing the rate of change. "Increases by 5 each year" means the model is linear (a constant amount), while "increases by 5% each year" or "doubles" means it is exponential (a constant multiplier).

Can I use Desmos to figure out the model type?

Yes! You can add a table in Desmos, enter your data points, and look at the shape of the graph. A straight line indicates a linear model, while a curve that gets progressively steeper or flatter indicates an exponential model.

What is the standard formula for each model type?

Linear models use the format $y = mx + b$, where $m$ is the constant amount added per interval. Exponential models use $y = a(b)^x$, where $b$ is the constant multiplier or growth/decay factor.

How many Choosing Between Linear and Exponential Models questions are on the SAT?

Problem-Solving & Data Analysis makes up approximately 15% of the Digital SAT Math section. On Lumist.ai, we have 20 practice questions specifically focused on identifying and choosing between linear and exponential models.