Average
Rate of Change
ARC
The change in the value of a quantity divided
by the elapsed time. For a function, this is the change
in the y-value divided by the change in the x-value
for two distinct points on
the graph.
Any of the following formulas can be used.
ARC = average rate of change = \(\frac{{\Delta y}}{{\Delta x}} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{f\left( {{x_2}} \right) - f\left( {{x_1}} \right)}}{{{x_2} - {x_1}}} = \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\)
Note: This is the same thing as the slope of the secant
line of a curve.
See
also
Instantaneous
rate of change, mean value theorem
Worked Example
Problem: Find the average rate of change of f(x) = x² from x = 1 to x = 4.
Step 1: Identify the two x-values: x₁ = 1 and x₂ = 4.
x1=1,x2=4 Step 2: Evaluate the function at each x-value.
f(1)=12=1,f(4)=42=16 Step 3: Compute the change in y (the numerator).
Δy=f(4)−f(1)=16−1=15 Step 4: Compute the change in x (the denominator).
Δx=4−1=3 Step 5: Divide to find the average rate of change.
ΔxΔy=315=5 Answer: The average rate of change of f(x) = x² from x = 1 to x = 4 is 5. This means that on average, the function's output increases by 5 units for every 1-unit increase in x over this interval.
Another Example
This example uses a real-world context (projectile motion) and demonstrates the important edge case where the average rate of change is zero even though the function is not constant — the function increases and then decreases within the interval.
Problem: A ball is thrown upward. Its height in feet after t seconds is given by h(t) = −16t² + 64t + 5. Find the average rate of change of the ball's height from t = 1 to t = 3 seconds.
Step 1: Identify the interval: t₁ = 1 and t₂ = 3.
t1=1,t2=3 Step 2: Evaluate h(t) at each endpoint.
h(1)=−16(1)2+64(1)+5=−16+64+5=53 Step 3: Evaluate h at the second endpoint.
h(3)=−16(3)2+64(3)+5=−144+192+5=53 Step 4: Apply the average rate of change formula.
3−1h(3)−h(1)=253−53=20=0 Answer: The average rate of change is 0 feet per second. Even though the ball rose and then fell during this interval, the height at t = 3 equals the height at t = 1, so the average rate of change over the whole interval is zero.
Frequently Asked Questions
What is the difference between average rate of change and instantaneous rate of change?
Average rate of change measures how a function changes over an entire interval [x₁, x₂] and equals the slope of the secant line through those two points. Instantaneous rate of change measures how the function is changing at a single point and equals the slope of the tangent line at that point. Instantaneous rate of change is found by taking the derivative, which is the limit of the average rate of change as the interval shrinks to zero.
Is average rate of change the same as slope?
For a linear function, yes — the average rate of change between any two points equals the constant slope of the line. For a nonlinear function, the average rate of change equals the slope of the secant line connecting two specific points, which generally changes depending on which two points you choose. So the average rate of change gives you a slope, but it is the slope of a secant line rather than the curve itself.
Can the average rate of change be negative or zero?
Yes. The average rate of change is negative when the function's output decreases over the interval (f(x₂) < f(x₁)). It equals zero when the function has the same output at both endpoints, even if the function varies in between. A positive average rate of change means the output increased overall.
Average Rate of Change vs. Instantaneous Rate of Change
| Average Rate of Change | Instantaneous Rate of Change |
|---|
| What it measures | Change over an interval [x₁, x₂] | Change at a single point x = a |
| Formula | (f(x₂) − f(x₁)) / (x₂ − x₁) | lim as h→0 of (f(a+h) − f(a)) / h |
| Geometric meaning | Slope of the secant line | Slope of the tangent line |
| Requires calculus? | No — only algebra is needed | Yes — requires limits or derivatives |
| Number of points needed | Two distinct points | One point (with a limiting process) |
Why It Matters
Average rate of change appears throughout algebra, precalculus, and calculus. In algebra, it connects directly to the slope formula you already know. In calculus, it is the starting point for understanding derivatives: the instantaneous rate of change is defined as the limit of the average rate of change as the interval width approaches zero. Real-world applications include calculating average speed, average cost per unit, and average growth rates in science and economics.
Common Mistakes
Mistake: Subtracting the x-values or y-values in the wrong order, such as computing (f(x₂) − f(x₁)) / (x₁ − x₂).
Correction: Always match the order in the numerator and denominator. If you subtract x₁ from x₂ in the denominator, you must subtract f(x₁) from f(x₂) in the numerator. Reversing one but not the other flips the sign of your answer.
Mistake: Assuming the function behaves the same way throughout the interval just because the average rate of change is a certain value.
Correction: The average rate of change only tells you the net change over the interval divided by its width. The function could increase and decrease within the interval. For example, an average rate of change of zero does not mean the function is constant — it means the function ends at the same value it started.
