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The Concept of Slope

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The Concept of Slope

Algebra Foundations

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The concept of slope will be introduced as a rate of change between dependent and independent variables, and as a geometric concept.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Interpret slope as a rate of change between dependent and independent variables.
  • Interpret the geometric meaning of slope and compare the slopes of different lines.
  • Justify that vertical lines have undefined slope and horizontal lines have zero slope.
  • Calculate the slope of a linear model and interpret its meaning in the context of the model.

Everything You'll Have Covered

The slope of a straight line in the plane is a measure of the line's steepness. The steepness of a line is directly related to the magnitude of its slope.

Arithmetically, a line's slope is defined as the ratio of vertical change to horizontal change. That is, given two points and on the line, the ratio is equal to the line's slope. This formula is known as the slope formula. The slope formula can be used to understand slope as it relates to horizontal and vertical lines.

A horizontal line undergoes no vertical change, and so the y-coordinates of any two points on the line are equal. For example, if is a horizontal line through the point , then any other point on has the form , where is some real number. The slope of l is given by the slope formula:

The slope is zero because the line has no vertical change. On the other hand, a vertical line undergoes no horizontal change, meaning that any two points have equal x-coordinates. For example, any point on the vertical line passing through has the form . Substituting these points into the slope formula results in division by zero:

It is straightforward to see that no vertical line has a well-defined slope. Unfortunately, there is a common misconception that vertical lines should have infinite slope. Although the slope of a line becomes arbitrarily large as the line becomes steeper, the slope of a vertical line is undefined.

The concept of slope provides a means to geometrically interpret rates of change. A rate of change is a measure of how quickly some quantity changes with respect to other related quantities. The vague nature of this description is necessitated by the wide variety of circumstances under which rates of change can be discussed. Therefore, it is helpful to limit the discussion to linear relationships.

Consider two varying quantities, represented by x and y, such that each value of x is uniquely associated to some value of y. For example, the quantities could be distance and time, circumference and diameter, or income and wage. Two such quantities are linearly related if, for any two pairs and of corresponding values in the relationship, the ratio

is constant. The above ratio equals the rate of change in y with respect to x. The apparent equivalence of this formula to the slope formula is not an accident. Consider that each linear relationship is represented by a line and, conversely, any nonvertical straight line represents a two-variable linear relationship. Therefore, by viewing all nonvertical lines as representations of linear relationships, slope and rate of change can be viewed as coincident measures; slope measures steepness, rate of change measures change in a relationship, and the two are constant and equal for every linear relationship.

Tutorial Details

Approximate Time 35 Minutes
Pre-requisite Concepts Students should compare and order numbers, plot points on a graph and read points from a graph; and understand the meanings of quotients, rate of change, dependent/independent variables, lines, and negative numbers.
Course Algebra Foundations
Type of Tutorial Concept Development
Key Vocabulary horizontal line, linearity, linear model