Negative Slope indicates Line slopes Downward from Left to Right

Theorem

Let $\LL$ be a straight line with a slope which is negative.

Then $\LL$ slopes downward from left to right.

Proof

Let $\LL$ have a slope which is negative.

Expressed in slope-intercept form, $\LL$ can be written:

$y = x \tan \psi + c$

where:

$\psi$ is the angle between $\LL$ and the $x$-axis

$c$ is the $y$-intercept.

By construction:

$90 \degrees < \psi < 180 \degrees$

Hence by Shape of Tangent Function:

$\tan \psi < 0$

Hence the result.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): line: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 2.