Negative Slope
A negative slope refers to a line that is trending downwards as it moves from left to right. The rise to run ratio of a line with a negative slope is negative. It can be calculated using the formula m = (y_{2}  y_{1})/(x_{2}  x_{1}) = Tan θ = f'(x) = dy/dx. The negative slope signifies that, if one quantity is decreasing another quantity is increasing. The line with a negative slope makes an obtuse angle with the positive xaxis. Let us learn more about the negative slope, its graph, calculation, with the help of examples, and FAQs.
1.  What Is A Negative Slope? 
2.  Graph Of Negative Slope 
3.  How To Calculate Negative Slope? 
4.  Examples On Negative Slope 
5.  Practice Questions 
6.  FAQs On Negative Slope 
What Is A Negative Slope?
A negative slope signifies that the coordinates of the two given points are negatively related. A line with a negative slope has m < 0 and the angle θ made by this line with the positive xaxis is an obtuse angle such that 90º < θ < 180º. The rise to run ratio of a line with a negative slope is negative. Here the rise is the change in y value, which is represented as Δy, and the run is the change in x value, which is represented as Δx.
Negative Slope = rise/run = Δy/Δx
A negative slope signifies that the two variables are inversely related. Here, as the x value increases the y value decreases. Alternatively, we can also observe that as the x value decreases the y value increases. Angle made by a line with a negative slope is in the anticlockwise direction with respect to the positive xaxis or a line parallel to the xaxis is an obtuse angle. This obtuse angle is greater than a right angle(90º+θ), and the slope of the line is a negative slope.
m = Tan(90º+θ) = Tanθ
The line with a negative slope is trending downwards from left to right. This can be observed in the graph below.
Graph of Negative Slope
The concept of the negative slope gives an inverse relationship between the two quantities. The two quantities are represented graphically across the xaxis and the yaxis, and the line is plotted to represent the relationship between these two variables. As the value of the quantity represented along with the xaxis increases, the value of the other quantity represented along with the yaxis decreases. The inverse relationship of the increase of the x value, with the decrease of the y value is represented by the negative slope of the line.
Graphically the line with a negative slope is one that falls as it moves from left to right. The line with a negative slope makes an obtuse angle with the positive xaxis, in the anticlockwise direction.
How To Calculate Negative Slope?
The negative slope of a line can be computed using two methods. The negative slope of a line can be computed either from the points on the line or the angle made by the line with the positive xaxis. For the two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope can be calculated using the formula m = \(\dfrac{(y_2  y_1)}{(x_2  x_1)}\).
Also if θ is the angle made by the line with a positive xaxis in the anticlockwise direction, the slope of the line can be computed with the tangent of this angle θ. The angle made by a line with a negative slope is always an obtuse angle and can be taken as θ=90º + α. And we compute the slope using the formula m = Tan(90º + α) = Tanα
For a given equation of a curve f(x), the slope of the curve is the slope of the tangent at the point on the curve and is calculated by taking the differentiation of the function. m = f'(x) = dy/dx.
Let us check a few examples of negative slopes.
 The price of a commodity and the quantity demanded to have a negative relationship. As the price increases, the quantity purchased is decreasing and the graph of such a line has a negative slope.
 The age of a person and the average distance walked by the person per day if represented along the xaxis and yaxis, has a sloping line from left to right.
 The car moving down a sloping road if observed is with a negative slope.
 The children sliding down an inclined plane is another simple example of a negative slope.
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Examples of Negative Slope

Example 1: Find the slope of the line passing through the points (3, 4), and (9, 7), and inform if it is a positive slope or a negative slope.
Solution:
The given two points are \((x_1, y_1)\) = (3, 4), and \((x_2, y_2)\) = (9, 7)
The slope of a line passing through the two points is m = \(\dfrac{(y_2  y_1)}{(x_2  x_1)}\)
m = \(\dfrac{(7  4)}{(9  3)}\)
m = \(\dfrac{3}{6}\)
m = 1/2
Therefore the line connecting these two points has a positive slope.

Example 2: Find the point at which a line passing through a point (4, 4) cuts the xaxis if it is having a negative slope of 0.8.
Solution:
The given point is \((x_1, y_1)\) = (4, 4)
The negative slope of the line is m = 0.8, and the point on the xaxis is \((x_2 , y_2)\) = (a, 0).
The slope of a line passing through the two points is m = \(\dfrac{(y_2  y_1)}{(x_2  x_1)}\).
m = \(\dfrac{0  4}{a  4}\)
0.8 = \(\dfrac{4}{a  4}\)
0.8 = 4/(a  4)
0.8(a  4) = 4
0.8a  3.2 = 4
0.8a = 4 + 3.2
0.8a = 7.2
a = 7.2/0.8
a = 9
Therefore the point on the xaxis is (9, 0).
FAQs on Negative Slope
How Do You Compute The Negative Slope Of A Line?
Negative slope refers to the slope of a line that is sloping downwards as we are moving from left to right. The angle made by a line with a negative slope is an obtuse angle with respect to the positive xaxis. A negative slope gives an inverse relationship between two variables. As the value of the x variable increase, the value of the y variable decreases.
What Can You Say About the Coordinates Of A Points Lying On A Line With Negative Slope?
For a line with a negative slope and passing through the two points \((x_1, y_1)\) and \((x_2, y_2)\), if the set of ordinates increases, the set of abscissa decrease. Also sometimes the ordinate value decreases, and the abscissa increases.Here we can observe that as x_{2 }> x_{1}, we have y_{2}< y_{1}, OR as x_{2 }< x_{1}, we have y_{2 }> y_{1}.
How To Find The Negative Slope Of A Line From The Two Given Point?
The slope of a line connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula m = \(\dfrac{(y_2  y_1)}{(x_2  x_1)}\). The slope is the ratio of the difference between the y coordinate values, and the difference between the x coordinate values.
What Can We Understand About A Line Having Negative Slope?
The negative slope signifies that the line is sloping downwards from left to right. Here the relationship between the two variables represented in the graph along the xaxis and the yaxis is inverse. The increase in one variable results in a decrease in the other variable. Similarly we can have lines with a zero slope, or a positive slope.
How Can You Differentiate Between Two Lines  One With Negative Slope And Another With Positive Slope?
The negative slope and the positive slope are based on the sign of the slope value. The ve value of the slope gives the negative slope and the +ve value of the slope gives the positive slope. The line with a negative slope is sloping downwards as it moves from left to right and the line with a positive slope is inclined as we are moving from left to right.
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