121616949-math.17

# 121616949-math.17 - 1.1 Lines 3 EXAMPLE 1.1 According to...

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1.1 Lines 3 EXAMPLE 1.1 According to the 1990 U.S. federal income tax schedules, a head of household paid 15% on taxable income up to \$26050. If taxable income was between \$26050 and \$134930, then, in addition, 28% was to be paid on the amount between \$26050 and \$67200, and 33% paid on the amount over \$67200 (if any). Interpret the tax bracket information (15%, 28%, or 33%) using mathematical terminology, and graph the tax on the y -axis against the taxable income on the x -axis. The percentages, when converted to decimal values 0.15, 0.28, and 0.33, are the slopes of the straight lines which form the graph of the tax for the corresponding tax brackets. The tax graph is what’s called a polygonal line , i.e., it’s made up of several straight line segments of different slopes. The first line starts at the point (0,0) and heads upward with slope 0.15 (i.e., it goes upward 15 for every increase of 100 in the x -direction), until it reaches the point above x = 26050. Then the graph “bends upward,” i.e., the slope changes to 0.28. As the horizontal coordinate goes from
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Unformatted text preview: and continues with slope 0.33. See ﬁgure 1.2 . 10000 20000 30000 50000 100000 ..................................................................................................................................................................................................................... • • Figure 1.2 Tax vs. income. The most familiar form of the equation of a straight line is: y = mx + b . Here m is the slope of the line: if you increase x by 1, the equation tells you that you have to increase y by m . If you increase x by Δ x , then y increases by Δ y = m Δ x . The number b is called the y-intercept , because it is where the line crosses the y-axis. If you know two points on a line, the formula m = ( y 2-y 1 ) / ( x 2-x 1 ) gives you the slope. Once you know a point and the slope, then the y-intercept can be found by substituting the coordinates of either point in the equation: y 1 = mx 1 + b , i.e., b = y 1-mx 1 . Alternatively, one can use the “point-slope” form of the equation of a straight line, which is: ( y-y 1 ) / ( x-x 1 ) = m . This...
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