If slope is a number that measures the steepness of a line then one would

# If slope is a number that measures the steepness of a

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If slope is a number that measures the steepness of a line, then one would expect that a steeper line would have a larger slope. You Try It! EXAMPLE 4. Graph two lines, the first passing through the points P ( 3 , 2) Compute the slope of the line passing through the points P ( 2 , 3) and Q (2 , 5). Then compute the slope of the line passing through the points R ( 2 , 1) and S (5 , 3), and compare the two slopes. Which line is steeper? and Q (3 , 2) and the second through the points R ( 1 , 3) and S (1 , 3). Calcu- late the slope of each line and compare the results. Solution: The graphs of the two lines through the given points are shown, the first in Figure 3.47 and the second in Figure 3.48 . Note that the line in Figure 3.47 is less steep than the line in Figure 3.48 . 5 5 5 5 x y P ( 3 , 2) Q (3 , 2) Figure 3.47: This line is less steep than the line on the right. 5 5 5 5 x y R ( 1 , 3) S (1 , 3) Figure 3.48: This line is steeper than the line on the left. Remember, the slope of the line is the rate at which the dependent variable is changing with respect to the independent variable. In both Figure 3.47 and Figure 3.48 , the dependent variable is y and the independent variable is x .
190 CHAPTER 3. INTRODUCTION TO GRAPHING Subtract the coordinates of point P ( 3 , 2) from the coordinates of point Q (3 , 2). Slope of first line = Δ y Δ x = 2 ( 2) 3 ( 3) = 4 6 = 2 3 Subtract the coordinates of the point R ( 1 , 3) from the point S (1 , 3). Slope of second line = Δ y Δ x = 3 ( 3) 1 ( 1) = 6 2 = 3 Note that both lines go uphill and both have positive slopes. Also, note that the slope of the second line is greater than the slope of the first line. This is consistent with the fact that the second line is steeper than the first. Answer: The first line has slope 2, and the second line has slope 4 / 7. The first line is steeper. In Example 4 , both lines slanted uphill and both had positive slopes, the steeper of the two lines having the larger slope. Let’s now look at two lines that slant downhill. You Try It! EXAMPLE 5. Graph two lines, the first passing through the points P ( 3 , 1) Compute the slope of the line passing through the points P ( 3 , 3) and Q (3 , 5). Then compute the slope of the line passing through the points R ( 4 , 1) and S (4 , 3), and compare the two slopes. Which line is steeper? and Q (3 , 1) and the second through the points R ( 2 , 4) and S (2 , 4). Cal- culate the slope of each line and compare the results. Solution: The graphs of the two lines through the given points are shown, the first in Figure 3.49 and the second in Figure 3.50 . Note that the line in Figure 3.49 goes downhill less quickly than the line in Figure 3.50 . Remember, the slope of the line is the rate at which the dependent variable is changing with respect to the independent variable. In both Figure 3.49 and Figure 3.50 , the dependent variable is y and the independent variable is x .