Understanding Negative Slope: Definition and Examples

Negative slope is essential for understanding graphs, coordinate geometry, and real-life scenarios like economics or motion. Recognising it helps students quickly interpret whether relationships decline or increase, boosting exam scores and practical problem-solving. This fundamental graph concept appears regularly in maths curriculums.

Formula Used in Negative Slope

The standard formula is: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \), where m is the slope between two points on a line. If m is negative, the line has a negative slope.

Here’s a helpful table to understand negative slope more clearly:

Negative Slope Table

Scenario	X Increases	Y Increases?
Positive Slope	Yes	Yes
Negative Slope	Yes	No
Zero Slope	Yes	No Change

This table shows how the pattern of negative slope appears in relationships where, as x increases, y decreases, unlike positive or zero slopes.

What Is a Negative Slope?

A line has a negative slope if it trends downward from left to right on a graph. It means that while the x-value rises, the y-value falls. Mathematically, the slope (m) is negative. This often models an inverse relationship: as one variable increases, the other decreases. For example, in a linear graph the slope’s sign can quickly tell you about the relationship’s direction.

How to Describe Negative Slope in Words

To describe negative slope in words, simply say: "As x increases, y decreases." Or, "The line falls as it moves from left to right." This is different from a positive slope, where both increase together.

Worked Example – Solving a Problem

Let’s find the slope between two points and check if it is negative:

1. Write the two points: A(6, 4) and B(10, -2)
2. Use the slope formula:

3. Substitute: \( x_1 = 6, y_1 = 4; x_2 = 10, y_2 = -2 \)

4. Calculate \( m = \dfrac{-2 - 4}{10 - 6} = \dfrac{-6}{4} = -1.5 \)

5. Since the result is negative, the line has a negative slope.

For more examples using slope, check out linear equations in two variables or the general straight line equation page.

Practice Problems

• Given points (2, 7) and (5, 1), find their slope and say if it is negative.
• Sketch a line with a negative slope using any two points on grid paper.
• Give a real-world example where negative slope occurs.
• Which of the following slope values indicate a negative slope: 3, -2, 0, -0.5?
• Find the equation of a line with negative slope passing through (0, 2) and (6, -1).

Common Mistakes to Avoid

• Confusing negative slope with zero or positive slope values.
• Mixing up which variable represents change in x (run) and change in y (rise) when using the formula.
• Forgetting that steepness or direction depends on the sign of m in the line’s equation.

Real-World Applications

The concept of negative slope frequently appears in fields like economics, physics, and landscaping. For example, in demand graphs, as price rises, quantity demanded falls, showing a negative slope. You’ll also see it in road gradients and any situation involving a downward trend. With Vedantu, students can explore more about coordinate geometry and how slopes apply in real-world graphs.

Page Summary

We explored the idea of negative slope, its definition, calculation, and uses. You learned how to solve slope problems, avoid common errors, and recognise negative slope patterns in real situations. Keep practicing with Vedantu’s maths resources for better understanding and exam confidence.

Related Topics: Learn more about Slope, Gradient, Equation of a Straight Line, and Graphing of Linear Equations for deeper understanding and easy exam revision.