Slope of Secant

Theorem

Let $f: \R \to \R$ be a real function.

Let the graph of $f$ be depicted on a Cartesian plane.

Let $AB$ be a secant of $f$ where:

$A = \tuple {x, \map f x}$
$A = \tuple {x + h, \map f {x + h} }$

Then the slope of $AB$ is given by:

$\dfrac {\map f {x + h} - \map f x} h$

Proof

The slope of $AB$ is defined as the change in $y$ divided by the change in $x$.

Between $A$ and $B$:

the change in $x$ is $\paren {x + h} - x = h$
the change in $y$ is $\map f {x + h} - \map f x$.

Hence the result.

$\blacksquare$