Change > Difference quotient
12345Difference quotient

## Theory

Here you see part of a graph of a function $y=f\left(x\right)$.

The mean change of this function $f$ on the interval $\left[a,b\right]$ is:
$\frac{\left(\Delta y\right)}{\left(\Delta x\right)}=\frac{\left(f\left(b\right)-f\left(a\right)\right)}{\left(b-a\right)}$

This ratio of the difference between the values of the function at either end of the interval and the difference between the two values of $x$ is called the difference quotient of the function $f$ on the interval $\left[a,b\right]$.

In the graph of $f$ this difference quotient is equal to the slope of the line through $A\left(a,f\left(a\right)\right)$ and $B\left(b,f\left(b\right)\right)$.
This line is called a secant line $\mathrm{AB}$.