Negative Slope - Meaning, Graph, and Examples

Negative Slope - Meaning, Graph, and Examples

TL;DR

A negative slope means a line falls from left to right - as x increases, y decreases. It is found with m=Δy/Δxm = \dfrac{\Delta y}{\Delta x}m=ΔxΔy​, and it comes out negative whenever the two variables move in opposite directions. This article covers what negative slope means, how to calculate it, its graph, real examples, and the common mistakes.

What Is A Negative Slope?

A negative slope describes a line that falls as it moves from left to right: as the x-values increase, the y-values decrease. The slope of a line, written mmm, measures how steeply it rises or falls, and it is the ratio of vertical change to horizontal change:

m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Here rise (Δy\Delta yΔy) is the change in the vertical direction and run (Δx\Delta xΔx) is the change in the horizontal direction, computed with the rise-over-run method. The slope is negative when moving to the right (positive run) makes the line go down (negative rise). This is the opposite of a positive slope, which rises left to right, and it sits on the same slope scale used across coordinate geometry.

How Do You Calculate A Negative Slope?

You calculate a negative slope exactly like any slope - the result simply comes out below zero. Take two points on the line, (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​), and apply the slope formula. The sign takes care of itself.

  1. Step 1 - label the points. Pick any two points on the line and call them (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​).

  2. Step 2 - subtract in the same order. Compute Δy=y2−y1\Delta y = y_2 - y_1Δy=y2​−y1​ and Δx=x2−x1\Delta x = x_2 - x_1Δx=x2​−x1​. Keep the same point "first" in both subtractions.

  3. Step 3 - divide. The slope is m=Δy/Δxm = \Delta y / \Delta xm=Δy/Δx. A negative result confirms a falling line.

In the slope-intercept form y=mx+cy = mx + cy=mx+c, the coefficient mmm is the slope directly - so a line like y=−2x+5y = -2x + 5y=−2x+5 has slope −2-2−2 without any calculation. A negative slope also makes an obtuse angle (between 90° and 180°) with the positive x-axis, whereas a positive slope makes an acute angle.

Examples of Negative Slope

Example 1

Find the slope of the line through the points (1, 5) and (4, 2).

Apply the slope formula, keeping (1, 5) as the first point:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 5}{4 - 1} = \frac{-3}{3} = -1

Final answer: m=-1.

Example 2

A student finds the slope through (2, 7) and (6, 1) by computing 2−67−1=−46=−23 and reports a negative slope. Find the error and the correct slope.

The correct method always puts the change in y over the change in x, in the same point order:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{6 - 2} = \frac{-6}{4} = -\frac{3}{2}

Final answer: m=−3/2.

Example 3

State the slope of the line y=−4x+9.

In slope-intercept form y=mx+cy = mx + cy=mx+c, the coefficient of xxx is the slope:

m=−4

Final answer: the slope is −4.

Example 4

A line passes through (−2, 6) and (3, −4). Find its slope.

Apply the formula:

m = \frac{-4 - 6}{3 - (-2)} = \frac{-10}{5} = -2

Final answer: m=−2.

Example 5

Which is steeper: a line of slope −2 or a line of slope −5?

The line with slope −5 is steeper.

Example 6

A pool loses water steadily: it holds 800 litres at time 0 hours and 500 litres at time 3 hours. Find the slope of volume against time and say what it means.

Treat time as xxx and volume as yyy:

m = \frac{500 - 800}{3 - 0} = \frac{-300}{3} = -100

Final answer: m = -100 litres/hour.

Where Negative Slope Earns Its Keep

A negative slope is how any downward trend gets measured, and the applications are everywhere a quantity falls over time or against another.

Common Mistakes With Negative Slope

Mistake 1: Flipping rise and run

Don't do this: Put the change in x on top and the change in y on the bottom.

The correct way: Always put the change in y (Δy) over the change in x (Δx).

Mistake 2: Losing the negative sign

Don't do this: Mishandle a minus sign.

The correct way: Handle each subtraction carefully.

Mistake 3: Confusing steepness with sign

Don't do this: Decide which negative slope is "bigger" based on value.

The correct way: Compare absolute values for steepness.

Conclusion

Frequently Asked Questions