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Slope

Definition of Slope

Words The slope is the value of the quotient .

Model

Interpretation of the Slope

Draw three lines whose equations are y = x , y = 2 x , and . These lines look like this in the coordinate plane.

Check the slopes of each of the lines. Note that the line with largest slope of 2 is the steepest, and that the line with least slope of is the least steep.

The figure above shows lines of various slopes.

Slopes of Horizontal and Vertical Lines

"What slope should a horizontal line have?"

Since a horizontal line always has rise equal to zero, the slope will always be zero divided by a positive number, and so the slope is zero. Note that the x-axis has zero slope.

"What about vertical lines?"

Since the run is always zero and division by zero is undefined, the slope is undefined. It is sometimes useful to think of them as having "infinite slope," but since infinity is not a number, this is not a precise statement.

Negative Slope

Notice that so far all slopes have been positive numbers, and all lines have sloped upward from left to right. For lines drawn in the coordinate plane, the standard direction to move along them is from left to right and bottom to top.

Key Idea

Lines with positive slope rise to the right and lines with negative slope rise to the left.

For positive slopes, the larger the number, the more steeply the line slopes upward.

For negative slopes, the larger the absolute value of the negative number, the more steeply the line slopes downward.

The next figure shows lines of both positive and negative slopes.

Algebraic Formula for Slope

Let's summarize the whole discussion by introducing the algebraic formula for slope. To do this, we draw two points in the coordinate plane that correspond to the ordered pairs (x₀ , y₀) and (x₁ , y₁), as in the figure below.

The rise is y₁ - y₀ and the run is x₁ - x₀ . Slope is rise divided by run, which gives the formula

Values can be substituted into this formula once the coordinates are given. It does not matter which points are designated as (x₀ , y₀) and (x₁ , y₁). However, the first x in the denominator must come from the same coordinate pair as the first y in the numerator.