Section 2: Slope

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 3.2 Slope 247 Version: Fall 2007 3.2 Slope In the previous section on Linear Models, we saw that if the dependent variable was changing at a constant rate with respect to the independent variable, then the graph was a line. If the rate was positive, then as we swept our eyes from left to right, the line rose upward, the dependent variable increasing with increasing changes in the independent variable. If the rate was negative, then the graph fell downward, the dependent variable decreasing with increasing changes in the independent variable. You may have also learned that higher rates led to steeper lines (lines that rose more quickly) and lower rates led to lines that were less steep. In this section, we will connect the intuitive concept of rate developed in the previous section with a formal definition of the slope of a line. To start, let’s state up front what is meant by the slope of a line. Slope is a number that tells us how quickly a line rises or falls. If slope is a number that is directly connected to the “steepness” of a line, then we should have certain expectations. Expectations. 1. Lines with positive slope should slant uphill (as our eyes sweep from left to right). 2. Lines with negative slope should slant downhill (as our eyes sweep from left to right). 3. Because any horizontal line neither slants uphill nor downhill, we expect that it should have slope equal to zero. 4. Lines with a larger positive slope should rise more quickly than lines with a smaller positive slope. 5. If two lines have negative slope, then the line having the slope with larger absolute value should fall more quickly than the other line. It remains to define how to compute the slope of a particular line. Whatever definition we choose, it should conform with the expectations outlined above. We also would like the definition of slope to conform with the concept of rate developed in the previous section. Thus, we make the following definition. Definition 1. The slope of a line is the rate at which the dependent variable is changing with respect to the independent variable. Note how the word “change” is used Definition 1 . It is important to understand that the change in some quantity can be positive, negative, or zero. For example, if Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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248 Chapter 3 Linear Functions Version: Fall 2007 the temperature outside is 40 F when I leave my home at 6 AM, and at noon the temperature is 65 F, then the change in temperature is a positive 25 F. On the other hand, if the temperature outside is 65 F at noon, and the temperature is 50 F when I return home in the evening, then the change in temperature is a negative 15 degrees Fahrenheit. In calculating the change in a quantity, follow this rule.
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