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Unformatted text preview: Gregory Johnson Math 21-126 Homework 1 1 Lines Deﬁnition 1.1. • A vertical line is a set of all points satisfying the equation x = c for some ﬁxed constant c. • Let f (x) = mx + b be a linear function. We call the graph of f a (nonvertical) line. m is called the slope and b is called the y -intercept. Ex. 1.2. Given 2 points P1 = (x1 , y1 ) and P2 = (x2 , y2 ), ﬁnd an equation of the line that passes through both. • Case 1 x1 = x2 : The vertical line x = x1 passes through both points. • Case 2 x1 = x2 : We need to ﬁnd f (x) = mx + b such that f (x1 ) = y1 and f (x2 ) = y2 . That is, we need to ﬁnd m and b such that mx1 + b = y1 and mx2 + b = y2 . Solving the ﬁrst equation for b we get b = y1 − mx1 . Substituting this into the second equation we get mx2 + y1 − mx1 = y2 . Solving for m we end up with: mx2 + y1 − mx1 = y2 mx2 − mx1 = y2 − y1 m(x2 − x1 ) = y2 − y1 y2 − y1 m= x2 − x1 Therefore b = y1 − f (x) =
y2 −y1 x2 −x1 x1 and x + y1 − y2 − y1 x2 − x1 x1 y2 − y1 x2 − x1 Ex. 1.3. The line y = 3 x − 2 is the unique line passing through the points 2 (2, 1) and (−2, −5). This is illustrated in the ﬁgure below. Figure 1: y = 3 x − 2 2 Thank You. Drive Thru. ...
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