SAT Average Rate of Change: Understanding Secant Lines and Calculating Change Over Intervals

Average rate of change over an interval [a,b] is the slope of the secant line connecting two points on the curve: [f(b)−f(a)]/(b−a). This is different from instantaneous rate (derivative), which the SAT does not require. Average rate of change tells you the overall trend: is the function increasing or decreasing, and how steeply, on average, over the interval?

A car traveling 120 miles in 2 hours has an average rate of 60 mph, even if it accelerates and slows during those hours. The secant-line formula captures this average behavior.

Step 1: Identify the interval [a, b]. Step 2: Calculate f(a) and f(b). Step 3: Compute the numerator: f(b)−f(a). Step 4: Compute the denominator: b−a. Step 5: Divide to get the rate. This straightforward process prevents errors. Remember: the numerator is change in output (Δy); the denominator is change in input (Δx).

For tables or graphs, read the values directly; for equations, substitute to find f(a) and f(b). The method is the same.

Example 1: f(x)=x². Average rate of change from x=1 to x=3. f(1)=1, f(3)=9. Rate=[9−1]/(3−1)=8/2=4. Example 2: A table shows temperature: 8 AM is 60°, 2 PM is 80°. Average rate of change=20 degrees/6 hours≈3.3°/hour.

Common error: using f(b)−f(a) without dividing by b−a, forgetting that rate requires a denominator showing the time or distance interval.

For three days, compute average rate of change from equations, tables, and graphs. Include positive rates (increasing), negative rates (decreasing), and zero rates (flat). By day three, you will recognize which representation format (equation, table, graph) is fastest for each problem and calculate confidently.

On test day, average rate problems are quick points: use your five-step routine and move on.