ACT Math: Average Rate of Change—Secant Line Slope Between Two Points

Average rate of change on an interval [x₁, x₂] is (f(x₂)-f(x₁))/(x₂-x₁). This is the slope of the line connecting two points on a curve (the secant line). Example: For f(x)=x², find average rate of change from x=1 to x=3. f(1)=1, f(3)=9. Rate=(9-1)/(3-1)=8/2=4. The average rate of change tells you how much the function increases (or decreases) per unit of x on that interval. If rate is positive, function is increasing; if negative, decreasing; if zero, no change on average.

Contrast: Instantaneous rate of change is the slope at one point (the derivative, calculus); average rate is slope between two points (algebra).

Calculation 1: f(x)=3x+2. Find average rate from x=1 to x=4. f(1)=5, f(4)=14. Rate=(14-5)/(4-1)=9/3=3. (Linear functions have constant rate=slope.) Calculation 2: f(x)=x². Find average rate from x=2 to x=5. f(2)=4, f(5)=25. Rate=(25-4)/(5-2)=21/3=7. Calculation 3: f(x)=√x. Find average rate from x=1 to x=4. f(1)=1, f(4)=2. Rate=(2-1)/(4-1)=1/3≈0.33. Always compute f at both endpoints, subtract outputs, divide by change in x.

Verify: For linear f(x)=3x+2, rate should equal slope 3. ✓

Problem 1: f(x)=x²+1, from x=0 to x=2. f(0)=1, f(2)=5. Rate=(5-1)/(2-0)=4/2=2. Problem 2: f(x)=-2x+7, from x=1 to x=3. f(1)=5, f(3)=1. Rate=(1-5)/(3-1)=-4/2=-2. (Decreasing.) Problem 3: f(x)=x³, from x=1 to x=2. f(1)=1, f(2)=8. Rate=(8-1)/(2-1)=7. Problem 4: f(x)=1/x, from x=1 to x=2. f(1)=1, f(2)=0.5. Rate=(0.5-1)/(2-1)=-0.5. Complete all four daily until average rate calculations are automatic.

Visualize: Average rate is the slope of the line drawn between the two points on the curve.

Average rate of change questions appear in 1-2 ACT Math sections, testing whether you understand how to measure function behavior over an interval. The formula is simple and always the same; solving is mechanical once you plug in numbers correctly.

Master this concept in one study session. By test day, calculating average rates becomes quick.