To find the average rate of change , you compare how much a quantity changes to how much the input changes over a given interval.

Quick Scoop: Core Idea

Think “slope between two points”:

Average rate of change=change in outputchange in input=f(b)−f(a)b−a\text{Average rate of change}=\frac{\text{change in output}}{\text{change in input}}=\frac{f(b)-f(a)}{b-a}Average rate of change=change in inputchange in output​=b−af(b)−f(a)​

  • f(a)f(a)f(a) = value of the function at the starting input aaa.
  • f(b)f(b)f(b) = value of the function at the ending input bbb.
  • It’s the same idea as slope: “rise over run.”

Step‑by‑step: How to do it

  1. Identify the interval
    • You’ll be given two x‑values, like “from x=ax=ax=a to x=bx=bx=b.”
  1. Find the function values
    • Compute f(a)f(a)f(a) and f(b)f(b)f(b) by plugging into the function (or reading from a table/graph).
  1. Use the formula

Average rate of change=f(b)−f(a)b−a\text{Average rate of change}=\frac{f(b)-f(a)}{b-a}Average rate of change=b−af(b)−f(a)​

 * Subtract outputs on top, subtract inputs on bottom (keep the same order in both).
  1. Include units if you have them
    • Example: meters per second, dollars per year, degrees per day.

Mini Example (Concrete Numbers)

Suppose f(x)=x2f(x)=x^2f(x)=x2 and you want the average rate of change from x=2x=2x=2 to x=3x=3x=3.

  • f(2)=4f(2)=4f(2)=4, f(3)=9f(3)=9f(3)=9.
  • Plug into the formula:

Average rate of change=9−43−2=51=5\text{Average rate of change}=\frac{9-4}{3-2}=\frac{5}{1}=5Average rate of change=3−29−4​=15​=5

So the average rate of change is 5 (units per 1 x‑unit).

Another viewpoint: From data or graphs

You can do the same thing even without a neat formula:

  • From a table: pick the two x‑values in the interval, read off their y‑values, and apply the same fraction (y2−y1)/(x2−x1)(y_2-y_1)/(x_2-x_1)(y2​−y1​)/(x2​−x1​).
  • From a graph: find the coordinates of the two points, then use the slope formula between them.

Real‑world intuition

Average rate of change works like:

  • Average speed: total distance/total time\text{total distance}/\text{total time}total distance/total time.
  • Average cost change: change in cost/change in items\text{change in cost}/\text{change in items}change in cost/change in items.

Even if the function wiggles up and down in between, this number tells you the overall change per unit over that whole interval, not every little fluctuation.

TL;DR:
Use f(b)−f(a)b−a\displaystyle \frac{f(b)-f(a)}{b-a}b−af(b)−f(a)​: plug in the start and end x‑values, subtract outputs, subtract inputs, then divide. It’s just the slope between two points on the graph of the function.

Information gathered from public forums or data available on the internet and portrayed here.