The Four Cases of Slope | Moosa Academy

Every straight line falls into one of four slope cases. Learn to recognise them from the picture and you can predict the sign — or the absence — of the slope before doing any calculation, which makes a wrong answer easy to spot.

Concept The four cases

Rising — slope positive.

Falling — slope negative.

Horizontal — slope zero.

Vertical — slope undefined.

Case 1 Rising — positive slope

As increases, increases too. Reading left to right, the line climbs.

Both differences share the same sign, so the quotient is positive.

The larger the value, the steeper the climb.

Case 2 Falling — negative slope

As increases, decreases. Reading left to right, the line descends.

The two differences have opposite signs, so the quotient comes out negative. If your calculation gives a positive slope for a visibly falling line, an order has been reversed somewhere.

Case 3 Horizontal — zero slope

A horizontal line has the same -value everywhere, so the rise is zero:

Zero divided by any non-zero number is 0, so the slope is exactly zero — a real value, not a missing one.

Case 4 Vertical — undefined slope

A vertical line has the same -value everywhere, so the run is zero:

Division by zero has no meaning, so the slope is undefined. This is genuinely different from a zero slope: horizontal lines have a slope of 0, vertical lines have none at all.

Example Predicting before calculating

Classify the lines through these pairs of points.

and

— rising.

and

— falling.

and

— horizontal.

and

: the run is

— vertical, undefined.

⟹ 2, −2, 0, and undefined

Note Mistakes to avoid

Swapping the horizontal and vertical cases — horizontal is 0, vertical is undefined.

Writing "no slope" for a horizontal line; it has a slope, and that slope is 0.

Reporting a positive slope for a line that clearly falls.

Attempting to divide by zero instead of declaring the slope undefined.

Skipping the visual check that would have caught a sign error.

Summary

A rising line has a positive slope; a falling line has a negative one.
A horizontal line has zero rise, so its slope is exactly 0.
A vertical line has zero run, so its slope is undefined.
Zero and undefined are different: one is a value, the other is no value at all.
Predict the case from the picture first — it catches sign errors immediately.