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ZingPath: Linearity, Slope, and Intercepts

The Concept of Linearity

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Linearity, Slope, and Intercepts

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The Concept of Linearity

Algebra-1

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Given a verbal description, you will determine if a relationship is linear by creating a table or a graph.

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Now You Know

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

  • Create tables that represent linear relationships, given a verbal description.
  • Translate from a table of values to a graphical representation.
  • Determine if a relationship is linear from a table or a graph.

Everything You'll Have Covered

The concept of linearity encompasses a broad variety of mathematical concepts, such as constant ratio, proportionality, rate of change, homogeneity, and slope. The most fundamental of these is special type of relationship between quantities called a linear relationship. This describes a diversity of mathematical, scientific, and practical phenomenon, and helps mathematicians to understand more complicated types of relationships.

Consider a train that, after leaving its station, travels at the fixed speed of 50 mph. During any hour, the train's distance from the station increases by 50 miles. Alternatively, we can also see that the train's distance increases by 100 miles after two hours or by 25 miles after half an hour.

Each of these descriptions has a common characteristic: the ratio of change in distance to change in time is constant, and this is unaffected by the train's position relative to the station. This is the constant rate of change:

The key insight to make is that, if we measure the change in distance on any time interval, then the ratio of change is equal to 50 mph. The constancy of this ratio holds for any time interval, be it very tiny, arbitrarily chosen, or unrealistically huge:

These calculations are expressions of the fact that the change in distance is proportional to the change in time. For this reason, the train's distance from the station is linearly related to the time it travels.

Variables are used to represent unknown quantities in a relationship. These variables can be classified as dependent and independent. In the example above, time is most naturally considered as the independent variable because it is causally unaffected by the train's distance from the station. The train's distance ostensibly depends on how long the train has travelled, whereby the variable representing distance is called the dependent variable.

It is important to note that, although some situations might feel natural, such a categorization of variables is wholly determined by the viewpoint we would like to take. For example, a circle's circumference is linearly related to its diameter. In this situation, one could view the circumference as the independent variable and the diameter as the dependent variable. This would be reasonable, for instance, if one sought to determine a circle's diameter based on its circumference. On the other hand, it could be perfectly reasonable to reverse these designations, so that the diameter is independent and the circumference is dependent.

Linear relationships can be represented by graphs, equations, and tables. As graphs, linear relationships are represented by straight lines in the Cartesian plane.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Students should understand the concept of rate of change.
Course Algebra-1
Type of Tutorial Concept Development
Key Vocabulary dependent variable, graph, independent variable