chapter1Notes - Chapter 1 Review of Straight Lines We start...

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Chapter 1 Review of Straight Lines We start with a brief review of properties of straight lines, since these properties are fundamentally important to our understanding of more advanced concepts (tangents, slopes, derivatives). Skills described in this introductory material will be required in many contexts. 1.1 Geometric ideas: lines, slopes, equations Straight lines have some important geometric properties, namely: The slope of a straight line is the same everywhere along its length. Definition: slope of a straight line: y x y x Figure 1.1: The slope of a line (usually given the symbol m ) is the ratio of the change in the y value, ∆ y to the change in the x value, ∆ x . We define the slope of a straight line as follows: Slope = y x v.2005.1 - September 4, 2009 1
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Math 102 Notes Chapter 1 where ∆ y means “change in the y value” and ∆ x means “change in the x value” between two points. See Figure 1.1 for what this notation represents. Equation of a straight line Using this basic geometric property, we can find the equation of a straight line given any of the following information about the line: The y intercept, b , and the slope, m : y = mx + b. A point ( x 0 , y 0 ) on the line, and the slope, m , of the line: y - y 0 x - x 0 = m Two points on the line, say ( x 1 , y 1 ) and ( x 2 , y 2 ): y - y 1 x - x 1 = y 2 - y 1 x 2 - x 1 Remark: any of these can be rearranged or simplified to produce the standard form y = mx + b , as discussed in the problem set. 1.2 Examples and worked problems The following examples will refresh your memory on how to find the equation of the line that satisfies each of the given conditions. (Note: you should also be able to easily sketch the line in each case.) (a) The line has slope 2 and y intercept 4. (b) The line goes through the points (1,1) and (3,-2). (c) The line has y intercept -1 and x intercept 3. (d) The line has slope -1 and goes through the point (-2,-5). Solutions: v.2005.1 - September 4, 2009 2
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Math 102 Notes Chapter 1 (a) We can use the standard form of the equation of a straight line, y = mx + b where m is the slope and b is the y intercept to obtain the equation: y = 2 x + 4 (b) The line goes through the points (1,1) and (3,-2). We use the fact that the slope is the same all along the line. Thus, ( y - y 0 ) ( x - x 0 ) = ( y 1 - y 0 ) ( x 1 - x 0 ) = m. Substituting in the values ( x 0 , y 0 ) = (1 , 1) and ( x 1 , y 1 ) = (3 , - 2), ( y - 1) ( x - 1) = (1 + 2) (1 - 3) = - 3 2 . (Note that this tells us that the slope is m = - 3 / 2.) We find that y - 1 = - 3 2 ( x - 1) = - 3 2 x + 3 2 , y = - 3 2 x + 5 2 . (c) The line has y intercept -1 and x intercept 3, i.e. goes through the points (0,-1) and (3,0). We can use the method in (b) to get y = 1 3 x - 1 Alternately, as a shortcut, we could find the slope, m = y x = 1 3 . (Note that ∆ means “change in the value”, i.e. ∆ y = y 1 - y 0 ). Thus m = 1 / 3 and b = - 1 ( y intercept), leading to the same result. (d) The line has slope -1 and goes through the point (-2,-5). Then, ( y + 5) ( x + 2) = - 1 , so that y + 5 = - 1( x + 2) = - x - 2 , y = - x - 7 .
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