Lesson 2: Properties of Linear Functions
Module 5: Linear Functions
Lesson 2 Summary
In this lesson you investigated these questions:
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How are linear functions uniquely defined by their properties?
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How can the properties of linear functions be used to solve problems?
A line is uniquely defined by two points on the line or by a slope and a point. In other words, there is only one possible line that can pass through a particular pair of points. Likewise, while there may be many lines that share a common slope, there is only one line that possesses that slope and also passes through a given point.
In this lesson you studied the properties of linear functions.
You learned about the intercepts of a line and observed that all x-intercepts have the form (x, 0) and that all y-intercepts have the form (0, y). You also learned that a line can have one, two, or infinitely many intercepts.
In this lesson you also examined the domain and range of a linear relation. You learned that both the domain and the range of a diagonal line are always elements of the real numbers. Only in the case of horizontal lines is the range restricted to a single value while the domain remains unrestricted. A vertical line is not a linear function since it fails the vertical line test. However, the range of a vertical line is an element of the real numbers and its domain is limited to a single value.
This lesson reinforces the concept that the slope of a linear function is constant. Whenever you see the graph of a linear function, you will know that the change in the dependent variable is constant relative to the change in the dependent variable. Whether the context refers to an athlete who is running at a steady pace or a car that is consuming gasoline at a constant rate, each situation represents a constant change.
In upcoming lessons you will study the equations of linear relations. You will see that just as different athletes might have different approaches to training, there are different ways of expressing linear relations.
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