SECTION 1.5 Lines 33 is increased and the rise remains the same, the staircase becomes less steep. If the run is kept the same but the rise is increased, the staircase becomes more steep. This important characteristic of a line is best defined using rectangular coordinates. Figure 48 Rise = y2 – y1 Q = (x2, y2) P = (x1, y1) L Run = x2 – x1 Slope of L is m = (a) x1 x2 y2 y1 y2 – y1 _______ x 2 – x1 , x1 ≠ x2 Q = (x1, y2) P = (x1, y1) L Slope is undefined; L is vertical (b) x1 y2 y1 x x y y Figure 49 Triangles ABC and PQR are similar (equal angles), so ratios of corresponding sides are proportional. Then the slope using P and Q is ( ) ( ) − − = y y x x d B C d A C , , 2 1 2 1 which is the slope using A and B. x y R A C B y2 – y1 Q = (x2, y2) P = (x1, y1) x2 – x1 DEFINITION Slope Let ( ) = P x y , 1 1 and ( ) = Q x y , 2 2 be two distinct points. If ≠ x x , 1 2 the slope m of the nonvertical line L containing P and Q is defined by the formula = − − ≠ m y y x x x x 2 1 2 1 1 2 (1) If = x x , 1 2 then L is a vertical line and the slope m of L is undefined (since this results in division by 0). Figure 48(a) illustrates the slope of a nonvertical line; Figure 48(b) illustrates a vertical line. As Figure 48(a) illustrates, the slope m of a nonvertical line may be viewed as = − − = = − − = = Δ Δ m y y x x m y y x x y x y x Rise Run or as Change in Change in 2 1 2 1 2 1 2 1 That is, the slope m of a nonvertical line measures the amount y changes when x changes from x1 to x .2 The expression Δ Δ y x is called the average rate of change of y with respect to x . Two comments about computing the slope of a nonvertical line may prove helpful: • Any two distinct points on the line can be used to compute the slope of the line. (See Figure 49 for justification.) Since any two distinct points can be used to compute the slope of a line, the average rate of change of a line is always the same number. In Words The symbol Δ is the Greek uppercase letter delta. In mathematics, Δ is read “change in,” so Δ Δ y x is read “change in y divided by change in x .” • The slope of a line may be computed from ( ) = P x y , 1 1 to ( ) = Q x y , 2 2 or from Q to P because − − = − − y y x x y y x x 2 1 2 1 1 2 1 2
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