Quick answer

Both express rise over run between two points.

Formula

  • (y_2 − y_1)/(x_2 − x_1)
  • (f(b) − f(a))/(b − a)

Introduction

Clarify vocabulary before exams and lab reports so you do not lose points on notation alone.

Start from the definition in what is average rate of change if the interval language is new to you.

Then align symbols with the formula guide so (x_1, y_1) maps cleanly to (a, f(a)).

Use the Average Rate of Change Calculator to plot points and compare the dashed data path with the solid secant whose slope is your rate.

Similarities and differences

Similarity: both compare vertical change to horizontal change between two specified locations.

Slope language often appears in line equations and coordinate geometry.

Average rate language often appears with functions, tables, and applied contexts like velocity or revenue growth.

Difference in emphasis: rate language reminds you to interpret units and interval meaning, not only compute a number.

Side-by-side fractions

  • Coordinate form: (y_2 − y_1)/(x_2 − x_1)
  • Function form: (f(b) − f(a))/(b − a)
  • Same arithmetic when points represent the same relationship

Translate (x_1, y_1) to (a, f(a)) by naming the function value at the input.

Keep subtraction order paired: if you use y_2 − y_1 on top, use x_2 − x_1 below.

  1. Translate labels. Map coordinate pairs to function notation when the problem uses f(x).
  2. Pick one vocabulary set per answer. Mixing terms is fine in explanation, but show one clear fraction.
  3. Compute and compare. Both forms should give the same value for the same geometric points.
  4. Interpret in context. Write a sentence with units when the problem is applied.

Line through two points

Points (2, 5) and (8, 17): slope = (17 − 5)/(8 − 2) = 12/6 = 2.

Define f(x) as the line through those points. Then (f(8)−f(2))/(8−2) also equals 2.

The numeric agreement is why many textbooks treat the terms as interchangeable on function graphs.