Chapter 1 | Functions and Graphs
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Figure 1.16 For any linear function, the slope ( y 2 − y 1 )/( x 2 − x 1 ) is independent of the choice of points ( x 1 , y 1 ) and ( x 2 , y 2 ) on the line.
Definition Consider line L passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ). Let Δ y = y 2 − y 1 and Δ x = x 2 − x 1 denote the changes in y and x , respectively. The slope of the line is (1.3) m = = Δ We now examine the relationship between slope and the formula for a linear function. Consider the linear function given by the formula f ( x ) = ax + b . As discussed earlier, we know the graph of a linear function is given by a line. We can use our definition of slope to calculate the slope of this line. As shown, we can determine the slope by calculating ( y 2 − y 1 )/( x 2 − x 1 ) for any points ( x 1 , y 1 ) and ( x 2 , y 2 ) on the line. Evaluating the function f at x =0, we see that (0, b ) is a point on this line. Evaluating this function at x =1, we see that (1, a + b ) is also a point on this line. Therefore, the slope of this line is ( a + b )− b 1−0 = a . y 2 − y 1 x 2 − x 1 y Δ x . We have shown that the coefficient a is the slope of the line. We can conclude that the formula f ( x ) = ax + b describes a line with slope a . Furthermore, because this line intersects the y -axis at the point (0, b ), we see that the y -intercept for this linear function is (0, b ). We conclude that the formula f ( x ) = ax + b tells us the slope, a , and the y -intercept, (0, b ), for this line. Since we often use the symbol m to denote the slope of a line, we can write f ( x ) = mx + b to denote the slope-intercept form of a linear function. Sometimes it is convenient to express a linear function in different ways. For example, suppose the graph of a linear function passes through the point ( x 1 , y 1 ) and the slope of the line is m . Since any other point ( x , f ( x )) on the graph of f must satisfy the equation m = f ( x )− y 1 x − x 1 ,
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